DCCCD Phys2425 Lab Manual Fall

By Xiang-Ning Song

PHYSICS 2425University Physics ILaboratory Manual

M2

M1

Acknowledgments

I would like to express my sincere thanks to all of the people whose support, help, andassistance have been important in the completion of this lab manual.

I am grateful for the support from Ray Canham and Rita Maher. My gratitude goes toAfaf Abughazaleh, Claudiu Rusu, Fred Wittel, and Justin Song for reviewing the manualand providing so many helpful suggestions. I am especially thankful to Claudiu Rusu andJustin Song for assistance in creating some of the graphics.

Xiang-Ning Song

Richland College

Physics 2425

Table of Contents

Lab Guide 5

1. Measurements 11

2. Linear Motion: Measuring g Value 21

3. Equilibrium and Vectors: The Force Table 31

4. Projectile Motion 41

5. Dynamics of a Rolling Cart 49

6. Centripetal Acceleration and the Determination of g 57

7. Kinetic Energy and Potential Energy 63

8. Impulse and Momentum 69

9. Dynamic Carts Collisions: Conservation of Momentum 75

10. The Ballistic Pendulum 83

11. Rotational Inertia of A Wheel 91

12. Simple Harmonic Motion 99

13. Latent Heat and Specific Heat 109

14. Mechanical Equivalent of Heat 119

5

LAB REPORT GUIDE

I. Format of Lab Report

PRE-LAB FORMS

You are expected to have completed a Pre-Lab form for each of the experiments that wedo this semester, unless instructed otherwise by your teacher. These Pre-Lab formsconsist of basic questions designed to familiarize you with the concepts involved in thelab experiment. Lab data are recorded on separate paper during the course of the labperiod. These forms are to be turned in (deposited) on the front desk in the laboratorybefore the class begins.

COMPLETED LAB REPORTS

The lab report containing data, analysis, answers to questions, etc., is to be turned in onthe date specified by your teacher, usually at the beginning of the next lab period. Stackthese next to the Pre-Lab forms that you turn in for that day’s experiment. A stapler isavailable to help you make these reports a neat package.

The complete graded lab report will consist of a Pre-Lab form and final report thatare put together by the lab teacher and given a unit grade. The instructor will decide therelative weighing of each part of the lab report.

NEATNESS

All papers must be reasonably neat and organized. The margins should not have straynotes. If there is not sufficient space available on the lab forms to answer the questions,then you should write the answers on a separate sheet of paper and staple this to the labreport. The neatness, organization, and explanation of your measurements in the labreport represent the quality of your work.

REPORT FORMAT. FORMAL REPORTS

Lab report has the format of the outline below. Formal reports are more detailed, and youare to type these using a word processor. Your lab teacher will specify which labs reportsare to be formal reports.EXPERIMENT TITLEAUTHOR’S NAMELAB PARTNERS’ NAMESOBJECTIVES OF THE EXPERIMENTBASIC THEORY. Develop the calculation relations from first principles.DATA AND SAMPLE CALCULATIONS with units and appropriate significant digits.Please show only one sample of each non-trivial type.ERROR ANALYSIS. This is described in this LAB GUIDECONCLUSIONS. What have we learned? You may critique the lab experiment.

6

QUESTIONS. Answer these on separate sheet of paper unless adequate space is availablein the informal reports

ORIGINALITY

Work in the laboratory is usually performed with one or more lab partners who shareyour data. You are expected to discuss the lab work and the data with partners, but pleasewrite your own sentences. You are to understand the mathematics you use in your report.You are to construct your own graphs and make independent calculations (for example, aslope). Do not copy work you do not understand.

II. Experimental Error and Data Analysis

RECORDING DATA. Uncertainty in measurements.LEAST COUNT: The smallest division on a measurement scale is called the least count.Most metric rulers have a least count of 1mm (0.1 cm). The triple beam balance that weuse to measure mass in this lab has a least count of 0.1g.

RECORDING DATA: When you record a measurement you must record all the digitsthat the measuring tool is capable of producing. As a concrete example, considermeasurements of the width, W, of this page with a standard metric (cm) scale. Aftermaking a few measurements, the smallest value that seems reasonable is

Least W = 21.49 cm

The largest value that I made (or seems reasonable) is

Largest W = 21.52 cm

I would record the width as W = W0 ± ∆W = 21.5 ± 0.02 cm. The 0.02 cm is myestimate of the limits of precision for this instrument. I feel that I can measure to 1/5th ofthe smallest division on the scale.

Similarly I have measured the length, L, and I record L= L0 ± ∆L = 28.00 ± 0.02cm. Recording L = 28 cm won’t do since this implies that I don’t know the digit thatfollows the “8”. Another reasonable person might have measured and recorded 28.01cmor perhaps 27.98 cm.

The last digit recorded is an uncertain or estimated digit. You can use the leastcount of the measuring device, repeated measurements of a value, or other reasonableapproach to decide the uncertainty of a measurement and the corresponding precision thatyou should use to record the data. It is responsibility of the student to decide what theuncertainty of a measurement is.

Some numbers are exact numbers. How many ears do you have? We record thisas 2 and not 2.00

7

SIGNIFICANT DIGITS AND SCIENTIFIC NOTATION

As an example let us consider the size of a proton (which is a unit in the nucleusof an atom). A proton’s radius is about 0.000,000,000,000,0075 m. How many significantfigures (or digits) does this number have? It has 2 significant digits. The leading zeros arenot significant in the sense that they relate to the precision of the value. It is rather silly tomake the reader count all those zeros to understand the size of the number. You areexpected to write very large or very small numbers in scientific notation form. For thecase of the proton, r = 7.5 x 10-15 m. We could use an appropriate prefix to indicate themultiplier, femto f = 10-15, r = 7.5 fm.

When you record data you must write all the digits that are significant for themeasuring device and the proper units. In the above example the length of the paper, L, isrecorded to 4 significant digits. The last digit recorded is an uncertain digit.

UNCERTAINTIES FOR ARITHMETIC COMBINATIONS

When you combine numbers through arithmetic, i.e., multiplication, addition, etc., thenumber of significant digits in the result is to agree with the least precise number that wasused in the calculation. As an example, let us compute the area, A, using the data fromprevious example. The area is computed as length times width:

A0 = L0•W0 = (28.00 cm)•(21.50 cm) = 602.0 cm2

The area is recorded using 4 significant digits since the numbers that went into thecalculation was good to only 4 digits.

Do not simply write down all the numbers that might be on your calculator. Exactnumbers, like ‘602’, is what the calculator yields. You have to record it as ‘602.0’ for 4significant digits. When numbers like π or √(2) appear, use the appropriate keys on yourcalculator to make the calculations, then use the data to determine the appropriate numberof significant digits for the final result. Do not round off before you have finishedcalculating.

How does one estimate the relative uncertainty in this area? (Relative uncertaintyis the uncertainty in a number divided by the number.) The relative uncertainty is givenby ∆A/ A0 = (L0∆W + W0∆L)/ A0,

where ∆W and ∆L were both estimated to be 0.02 cm. After arithmetic, the uncertainty of area is ∆A = 0.99 cm2, and then you can express area as A = A0 ± ∆A = 602.0 ± 0.99cm2 .

NOTE ON COMBINATION OF ERRORS: The calculation of the area’s relativeuncertainty as ∆A/ A0 = (∆W/ W0) + (∆L/ L0), using positive values, is a simple approachthat gives a larger relative error that one would expect from an analysis statistics. In thecase of a random source of measurement variations, the correct expression would be ∆A/ A0 = ((∆W/ W0)

2 + (∆L/ L0)2)) ½.

8

PERCENT ERROR. PERCENT DIFFERENCE

The standard values can be found in textbooks or references. You will compute thepercent error of an experimental result using the following basic relation:

% Error = (100%)•(│Your Result – Truth│/ Truth)

Here ‘Truth’ means the standard value of the measured quantity.

The % difference between two different measured results for a quantity iscomputed in much the same way. The % difference is taken as positive.

% Difference = (100%)x (│Difference in Values│/Average Value).

GRAPHS

Choose graph scales so that a graph is as large as practical. Label the axes of the graphwith units and numbers. Simple whole numbers are preferred. You can factor out powersof ten in the scales so that the numbers along the axes are simple whole numbers. Makethe plotted points easy to see, say by small circles or crosses. Make it easy to check theaccuracy of your plots.

As an example consider experimental data that relates the length of a simplependulum to the squared period of the motion which obeys the relation L= g/(4 π2)• T2.The pendulum length is plotted against the squared period, and the slope is computed.

Linear Graph For Pendulum

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6

Period Square (s.s)

Len

gth

of

Pen

du

lum

(m)

The length values are plotted on the y-axis, and the squared time on the x-axis. Theplotted points are marked so that they can be readily seen. A straight line fits these data.

Length(m)

T(s)

T2

(s2)

0.25 1.02 1.04

0.44 1.37 1.88

0.70 1.64 2.69

0.95 1.98 3.92

1.25 2.26 5.11

9

Note that the first and last points are NOT special points. The slope is computed fromtwo points on the line that are far apart. These are not data points. Why? The graph is atool used to average all your data and two arbitrary points on the line weight all the data.You should mark the slope calculation points and list their coordinates. The slope isdefined as the ratio of (the change in the y-values) divided by (the change in the x-values).For the case illustrated, two points on the line are (0.6, 0.15) and (4.0, 0.99).

Slope = {(0.99 – 0.15) cm}/{(4.0 – 0.6)s2} = 0.247 m/s2.

(Note: the free fall acceleration is g = 4π2•(slope) = 9.8 m/s2.)Let’s summarize the requirement for graph. It has an appropriate title. The axes

are labeled with whole numbers, and the units are indicated. The data points are marked.The slope calculation points are labeled. A larger graph would be preferred in a lab report.

10

III. Basic Formulas in Data Analysis

1. Percent error: % error is used to compare your measured value with the standardvalue.

% error = {Measured Value – Accepted Value/(Accepted Value)}•100%

2. Percent difference: % difference is used to compare two measured values.

% difference = {Difference in Values / [Average Value]}•100%={ Value1 – Value2 / [ (Value1 + Value2)/2]} •100%

3. Mean value for a set of N measurements:

xav = [x1 + x2 + .....+xN] / N

4. Deviation from the mean:

Di = xi - xav

5. Mean deviation (uncertainty):

Dav = [D1 + D2 + …+ DN] / N

6. Record the accuracy of the mean value in terms of mean deviation:

Measured Value = xav ± Dav

7. Standard deviation(uncertainty):

σ = NDDD N /)...(22

2

2

1

8. Record the accuracy of the mean value in terms of standard deviation:

Measured Value = xav ± σ

Note: Both the mean deviation and standard deviation represent the dispersion ofexperimental measurements about the mean. They are used as ± terms in yourmeasured value to indicate the precision of your measurement.

Experiment 1

MEASUREMENTSEQUIPMENTTriple Beam BalanceMicrometerVernier CaliperMeter StickMetric ScaleAluminum BlockAluminum and Steel Cylinders

OBJECTIVESUpon completion of this experiment you will be able to:

1. Use basic instruments for measuring length: the metric scale, Vernier caliper and micrometer.

2. Compute the volume of an aluminum block and cylinder.3. Measure the mass of an aluminum block and determine its density.4. Record information to an appropriate number of significant figures.

Figure 1: Instruments of Measurement

CONCEPTSThe following are some standard conversion factors.

1.00 lb = 4.448 N 1.00 inch = 2.54 cm 1.00 ft. = 30.48 cm = 0.3048 m 1.00 mile = 1.609 x 10³ m

By Xiang-Ning Song 11

Most metric scales are calibrated in centimeters (cm) and tenths of cm or millimeters (mm). When measuring length using a metric scale, it is often best not to measure from the end of the scale but to start the measurement at 1.0 cm or another convenient point. The reason for this is that the end of the scale may be worn due to extensive use which might cause an error in the measurement. The following is an example of a measurement using a metric scale.

Left Reading = 1.00 cmRight Reading = 3.45 cmLength L = 2.45 cm

Figure 2: Metric Scale

When using this type of metric scale, a measurement can be made accurately to 0.1 cm and can be estimated to one additional decimal place. Thus, your data, using this centimeter scale, should include two places after the decimal point.A measure of the uncertainty of a measurement with this type of instrument is defined as one-half of the least count where the least count is the smallest measuring increment on the instrument.

Uncertainty = 0.50 x (Least Count)

Thus, for a metric scale where the smallest increment is 0.1 cm, the uncertainty is 0.50 X 0.1 = 0.05 cm. Now length L can be recorded as L = 2.45 ± 0.05 cm.

m2.92cm =>

Figure 3: Vernier Caliper

12

A more accurate instrument for measuring relatively short lengths is the Vernier Caliper. The caliper has a second scale called the vernier which increases its measurement accuracy by a factor of 10. A sketch of a vernier caliper is shown in Figure 3. (Note: There is another more accurate Vernier Caliper in this lab. Your instructor will show you how to use it.)

The vernier caliper can measure accurately to 0.01 cm. In the above measurement, the top (main) scale indicates a reading between 2.9 and 3.0 cm. The bottom (vernier) scale is used to find the next digit by determining which mark on that scale best lines up with a mark on the top scale. In this example, the 0.2 mm (or 0.02 cm) mark lines up. This means that the accurate measurement is 2.92 cm. Note, the least count of the vernier caliper, 0.01 cm, is also regarded as uncertainty. The length can be recorded as L = 2.92 ± 0.01 cm.

The micrometer is the most accurate of our three measuring instruments. It has three scales. A scale on the barrel serves as the main scale, one on the rotating sleeve serves as the second scale, and the vernier scale is on the left side of sleeve. There are millimeter and half-millimeter markings on the barrel. One rotation of the sleeve is divided into fifty divisions and the sleeve progresses one-half millimeter during one rotation. Thus, each division on the sleeve is 0.01 mm or 0.001 cm. The vernier scale is read to 0.001 mm or 0.0001 cm. When using the micrometer, make certain the jaws are closed gently using the clutch mechanism. This assures that the jaws are not over tightened, but are closed with consistent firmness. A sketch of a micrometer follows.

Figure 4: Micrometer

The reading on the this sketch is: Shaft: 6.5 mmSleeve: 0.21 mm

+ Vernier: 0.003 mm Total: 6.713 mm

Note, the micrometer readings are in millimeters and the least count is 0.001 mm.

UNITS CONVERSION: How many cubic inches are in a cubic foot? 1ft3= (1ft3) x (12in/ft) 3 = 1728 in3


CALCULATIONS: The diameter and length dimensions of an aluminum rod were measured to be D = 1.7275 ± 0.0002 cm and 5.08 ± 0.01 cm. Its mass was measured to be M = 32.15g. Calculate the volume, the uncertainty in the volume, and the rod’s density.

Volume: V= (area) x (length) = π (D/2)2 x L= π (1.7275 cm/2)2 x (5.08 cm) = 11.9 cm3.

The relative uncertainty in the volume is ∆V/V = 2x∆D/D + ∆L/L = (2x0.0002/1.7275) + (0.01/5.08) = 0.0022∆V = 0.0262 cm3

Density: ρ= M/V = (32.15 g)/11.9 cm3= 2.70g/cm3

The standard value for aluminum is 2.75 g/cm3

The % error is (100%) x (2.70 – 2.75)/2.75= -1.8% (Note minus sign!)

Note: A less simple but better statistical method of computing the relative uncertainty in a product P=AxBxC…is ∆P/P = ((∆A/A)2 + (∆B/B) 2 + (∆C/C) 2 + …)1/2

PROCEDURE

Part A: HOW TALL ARE YOU?

Have a lab partner help0 you to mark your TWICE on one of the paper sheets on the lab doors. Make one mark with a straight edge flat on your head, turn around 3600, and make a second mark that is independent of the first.

a. Measure the distance from the floor to both marks independently using the centimeter scale and record these data using 4 significant digits in the HEIGHT TABLE.

What is the % difference? (This is defined in the lab guide.)

b. Measure the distance, H3, from the floor to the second mark in inches and record this data using four significant digits. (This last number must be measured and not computed from H2)

c. Compute the ratio of cm/in as the ration H1/H3.

d. The standard value of this ratio is 2.54 cm/in. Compute the % error and record this to 2 significant digits. (The % error will be negative if your ratio is less than 2.54.

14

Part B: LENGTH OF AN ALUMINUM BLOCK: METRIC SCALE MEASUREMENT

You will use the standard metric scale to measure the longest dimension of an aluminum block. Keep in mind that you must record all significant digits. Adjust the scale such that the left end is on some convenient mark. (If the metal of the scale extends to the left of the zero mark, then you may use the scale’s zero mark.) Record the length in both mm and in cm, and include expected variations.

Note: the typical variations in a measurement can be estimated as the difference between a least reasonable value and a largest reasonable value that one might record for a measurement.

Part C: Width of an aluminum block: Vernier Caliper Measurement

Make a measurement of the width of the aluminum block. Record value with its uncertainty in the Width Data Table.

Part D: Aluminum Block Thickness: Measurement with a Micrometer

Remember to always close the micrometer gently using the friction drive thimble. Close the jaws of the micrometer together to determine if the scale reads zero when it is empty. If it is not zero, then you will subtract that reading from the reading of thickness. A typical zero reading might be -0.002 mm. Record the block thickness, T, in BLOCK THICKNESS DATA TABLE.

Part E: Block Mass: Measurement with the triple beam Balance.

Verify that the least count on the triple beam balance is 0.1 g. When you weigh the block, see if you can determine the mass with more precision than to the nearest 0.1 g. Lets take the mass uncertainty to be 0.05 g. Record these data at the bottom of the BLOCK THICKNESS DATA TABLE.

DENSITY CALCULATIONS. Density is defined as mass divided by volume. The volume of the block is the product of (length L) x (width W) x (thickness T). Calculate the density of your aluminum block by dividing its mass by its volume. Finally compute the % error between your value of the density and the textbook value of 2.75 g/cm3.

Part F: Measurement of the Density ρ of a Aluminum/or Steel Rod

Select a rod, and use appropriate tools to measure its length, L, and diameter, D, in cm, and its mass in grams as precisely as you can. Then compute its density and the estimated uncertainty of the density. Record these data in the Aluminum/or Steel Rod Data Table. You are to list the tools used to determine the volume of the rod, show the calculation of the density, ρ, on the Data Sheet. Compute the % error of your value from the standard value for the densities of 2.75 g/cm3 (Al) and 7.86 g/cm3(Steel).


The volume is V= (π D2/4) xL.

The relative uncertainty in the volume is ∆V/V = 2x∆D/D) + ∆L/L.

The uncertainty in the density is ∆ρ/ρ = ∆M/M - ∆V/V.

16

PRE-LAB FORM MEASUREMENTS1. Write the answers for the following in scientific notation with three significant digits.

a. The earth’s circumference, measured through the poles, is exactly 40, 000, 000 m (according to an early standard of the meter). What is the radius of the earth, R?

R = _________________m. b. The surface area of the earth is given by A = 4π × R2. What is the area A?

A = _________________ m2.c. Using the units conversion 1 inch = 2.54 cm, convert the volume V = 302 in3 into a volume in cubic cm (cm3) and in liters. (1 L = 1000 cm3). Show your steps in below!

V = _________________cm3 = ____________ L.

2. Consider the illustration of a standard metric scale across the two lines.

a. Write a short definition of least count, and state what the least count is for this ruler.

b. Show how to determine x, the distance between the two lines. Record data in mm.(See the illustration inside the manual.)

Result: x = ________ mm


3. Measure the length and width of this paper in cm, including uncertainty estimates, then compute the area of this sheet in cm2 with an estimate of its uncertainty.

W = _______±________cm. L = _______±________cm.

A = _______±________cm2.

18

LAB REPORT FORM MEASUREMENTS HEIGHT DATA TABLE

H1(cm) H2(cm) Diff. (cm) % Diff. H3(in) H1/H3(cm/in) % Error

BLOCK LENGTH DATA TABLE. SCALE READINGS (Include the ± ∆L uncertainties.)Left Scale

Reading (mm)Right Scale

Reading (mm)Uncertainty

Estimated (mm)Length L (mm)

Length L (cm)

± ±

BLOCK WIDTH TABLE. VERNIER CALIPER (Include the ± ∆W in recorded values.)W (mm) Recorded W (mm) Recorded W (cm)

± ±

BLOCK THICKNESS TABLE. MICROMETER AND BALANCE MEASUREMENTS.‘ZERO’ (mm) BLOCK (mm) T (mm) [Difference] T (cm) ± ∆ T (cm)

Mass M ± ∆M(g) Density ρ (g/cm3) % Error

ALUMINUM OR STEEL ROD DATA TABLE L ± ∆L (cm) D ± ∆D (cm) Mass M ± ∆M(g) ρ (g/cm3) % Error

Sample Calculation for Aluminum or Steel Rod: 1. Measurement uncertainties:a. Length L: What device did you use to measure L? ___________________________

What is the uncertainty in the length measurement? ∆L = ____________cm

b. Diameter D. What device did you use to measure D? _________________________

What is the uncertainty in the diameter measurement? ∆D = ____________cm

C. Mass. What is a reasonable value for the uncertainty in the mass, ∆M? ∆M = ________

2. Show the density calculation for the aluminum/or steel rod.


QUESTIONS.

1. Convert the following values from British to SI units using an appropriate number of significant digits.a. Sue’s height h is 5 ft and 2.0 inches. What is Sue’s height in meters?

h = ______________b. Sue’s weight is 125.0 lb. What is her mass, M, in kg.

M = ______________

2. Show that there are 106 cm 3 in a 1 m3 volume.

3. A standard scale (ruler) is used to measure the width of this page as follows: The one cm mark with the left edge of the top of the page, and the right edge appears to line up exactly with the 0.5 cm mark between 22 and 23 cm. Which one of the following recordings of the width is wrong, and why is it wrong?

a. W =215.0 mm b. W = 21.5 cm c. W = 21.50 cm

4. The volume of a steel rod is measured as follows: The diameter of the rod is measured with a micrometer to be D = 5.080 ± 0.002 mm. The length is measured to be L = 88.90 ± 0.02 mm. What is the uncertainty in the volume?

5. The density of steel is ρ = 7.86 g/cm 3. Show the conversion to a value in kg/m3units.

20

Experiment 2

Linear Motion: Measuring g Value

EQUIPMENTFree Fall AdapterTwo steel balls (1.27cm and 1.91cm in diameters)One box for catching the ballPicket FenceOne PhotogateRing StandOne soft pad for cushioning the fall of the Picket Fence

Figure 1: Free Fall Adapter

OBJECTIVESUpon completion of this experiment, you will be able to:

1. Properly set up and operate Free Fall Equipments.2. Understand the concept of linear constant acceleration motion.3. Measure the acceleration of the free falling steel ball and the Picket

Fence.

CONCEPTSA freely falling object is an object moving freely under the influence of the gravitational force only. Regardless of its initial motion, the freely falling object experiences a downward acceleration. This acceleration is constant. The motion of a freely falling object is the motion in one dimension under constant acceleration, only this motion is


Two ways to hold and release the ball:

A. Lock and release mechanism, as shown from the top.

B. Hand held and release the ball, as shown from the bottom.

along the vertical. If we choose upward as the positive direction of y ( ay = -g and g=9.8m/s2), the following equations will describe the motion:

Velocity as a function of time: vy = vy0 - g·t (1)

Velocity as a function of displacement: vy2 = vy0

2 - 2·g·(y-y0) (2)

Displacement as a function of time: y-y0 = vy0· t – 1/2·g·t 2 (3)

Displacement as a function of velocity and time: y-y0 = 1/2· (vy0 + vy)· t (4)

PROCEDURE

Part A: Measuring Acceleration with the Free Fall Adapter

1. Turn on the Signal Interface first! Then turn on the computer.

2. Connect the free fall adapter’s stereo phone plug into the Digital Channel 1.3. Run “DataStudio”. Click “Create Experiment”.

4. Click on Channel 1 and choose “Free Fall Adapter” as your digital sensor. Then click on OK.

5. Under the “Display” window, click on the “Table” icon. Drag and drop the Table on “Time of Fall” under the Data window (It is on the top and left corner of the computer screen.).

6. Place the adapter’s timing pad inside box, as shown in the Figure 1. Place them on the floor directly below the release mechanism. Place the smaller diameter steel ball in the release mechanism.

7. Hold the ball at h=0.5m position against a meterstick, as shown in the Figure 1. Click on “Start” button on the computer (upper left corner), release the ball, and then click on “Stop” button. (Note: The computer will not start recording until you release the ball, so you do not have to synchronize releasing the ball with pressing “Rec.”)

8. Record the “Time of Fall”. (Note: If the ball bumps, there are more than one data. Record the reasonable one.)

9. Repeat for 0.75m, 1m, 1.25m, 1.5m and 1.65m.

10. Repeat steps 6-9 for the larger ball (19mm).

11. Turn off the DataStudio.

12. Calculate t2, g, and average g value.22

13. Using graph paper to graph h versus t2 for Ball 2, find the slope and g.

Part B: Freely Falling Picket Fence

1. Connect the photogate’s stereo phone plug to Digital Channel 1 on the interface. Run the DataStudio. Click “Create Experiment”.

2. Click on Channel 1 and choose “Photogate & Picket Fence” as your digital sensor. Then click on OK.

3. Click and drag the Table under “Display” window. Drop the Table on the Position icon under “Data” window.

4. Click on the Velocity under “Data” window. Drag and drop it on Table 1 icon under “Display” window. You should see the Time, Position, and Velocity Table on the screen.

5. Place a soft cushion in a box. Place the box directly under the falling picket fence as shown in Figure 2. Make sure the Fence lands on the cushion.

6. Click the “Start” button. Drop the Picket Fence vertically from just above the photogate and have it move through the photogate’s opening without hitting the gate.

7. Record the time, position, and velocity (Note: Time is in the first column only, not the time for velocity again) data on Data Table 2.


Figure 2: Picket Fence

8. After you write down the data, you can graph velocity versus time on the computer. Click on the “Graph” icon under the “Displays” window. Drag and drop the graph on the “Velocity” icon under the “Data” window.

9. Click the “Fit” button on the bar graph menu (the eighth button). Choose “linear fit”.

10. Record the slope.

11. Finish the calculation for Δtn = tn – tn-1 , vn = (xn - xn-1)/Δtn, and gn= (vn - vn-1)/Δtn

on Data Table 3. Note: Time t and x are from Data Table 2, but not velocity data.

24

PRE-LAB FORMLINEAR MOTION: MEASURING g VALUE

1. Using the given data, draw velocity versus time and find the slope.

T(s) 0.045 0.079 0.107 0.131 0.153 0.173V(m/s) 1.11 1.49 1.79 2.07 2.27 2.48

V(m/s)

T(s)


2. If we say that one object is under free fall motion, what do we mean in physics?

3. A dropping ball initially at rest is dropped 1.5m, calculate (a) the traveling time of the ball and (b) its velocity just before reaching the ground.

26

LAB REPORT FORMLINEAR MOTION: MEASURING g VALUE

Data Table 1

Ball 1 Ball 2h(m) t(s) t2 g=2h/t2

0.500.751.001.251.501.75

h(m) t(s) t2 g=2h/t2

0.500.751.001.251.501.75

Average: gav= Average: gav=

CALCULATIONS1. Calculate t2, g and average g value.

2. Using graph paper to graph h versus t2 for Ball 2, find the slope and g.Based on slope: g=2Slope=_____________________

Data Table 2.

n tn(s) xn(m) vn(m/s)123456

g value from the slope: m/s2.

Data Table 3n Δtn= tn – tn-1 vn = (xn - xn-1)/Δtn gn=(vn-vn-1)/Δtn

1 XXXXXXXX XXXXXXXX XXXXXXXXXXX2 XXXXXXXXXXX3456


t2

h

28

QUESTIONS

1. Compare vn of Data Table 2 with Data Table 3. Is your calculated value same as recorded value from computer? Explain.

2. In this lab you dropped the Picket Fence vertically from just above the photogate.

(a) Calculate the time for the first 5cm length of the Picket Fence passing through the photogate.

(b) Calculate the time for the second 5cm length of the Picket Fence passing through the photogate.


3. If you throw a ball upward with an initial speed of 4m/s at 1.5m above the ground, find

a) the total time of flight for this free fall motion,

b) the final velocity just before touching the ground, and

c) the maximum height with respect to the ground.

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Experiment 3

EQUILIBRIUM AND VECTORS:THE FORCE TABLE

EQUIPMENTForce TableObject Ring with 4 Pulley Strings4 Super PulleysWeight Set4 Weight Hangers

Figure 1: Force Table

OBJECTIVESUpon completion of this lab you will be able to add vectors by graphical, analytical, andexperimental methods. You will understand the concept of physical conditions forequilibrium.

CONCEPTSAn object is in equilibrium if it has neither translational acceleration nor rotationalacceleration. The first condition is met when the net force, the sum of all external forces,is zero. Forces are vectors and this requires their vector sum to be zero. This Force Table

32

experiment illustrates vectors addition, as shown in the figure 1. The sum of tensionforces, applied to a ring at the center by weights suspended from pulleys, is zero.

AN EXAMPLE: Forces A, C, and D act on a particle leaving it in equilibrium. Given thefirst two force descriptions, A = 250 g @ 370 and C = 150 g @ 1430, determine the thirdforce, the equilibrium vector D, by (a) a graphical construction method, and (b) usingvector algebraic method. Assume angles are measured relative to the x-axis unless statedotherwise,

(a) The rule for graphical addition is: the tail of the second vector is placed at thehead of the first vector, then the resultant (sum) vector, R = A + B, is drawn fromthe tail of the first to the head of the second. The equilibrium vector, D = -R, isdrawn from the head of the second to the tail of the first. The result for thisexample, as shown in Figure 2, is D = 254 g @ 2520. The graph paper serves tohelp align the protractor. You should convince yourself that the order of additioncan be reversed.

Figure 2: Graphical Method to Determine the Equilibrium Vector

(b). Vector algebraic method: Equilibrium requires that A + C + D = 0. This is avector sum. It follows that their components sums are also zero:

Ax + Cx + Dx = 0 and Ay + Cy + Dy = 0.

So we must compute terms like Ax = A x cos(θA). Here θA is the angle betweenvector A and the x-axis. The magnitude of A is A, i.e., not printed in bold type orhand written with an arrow on top. Evaluating,

Ax = (250 g) x cos(37o) = 200 g. Ay = (250 g)x sin(37o) = 150 g.Cx = (150 g) x cos(143o) = -120 g. Cy = (150 g) x sin(143o) = 90 g.Dx = -(Ax + Cx) = -80 g. Dy = -(Ay + Cy) = -240 g.


We find the magnitude and direction of D as D =22

yx DD = 253 g. and

θD = tan-1(Dy/Dx) = 252o. So D = 253g @ 252o is answer.

The resultant of A + C is R = -D = 253 g @ 72o. R differs from D by 180o.

Can you see a natural way to extend the vector addition to include three or morevectors? This extension is left as an exercise. The negative of a vector is defined tohave the same magnitude as the vector, but point in the opposite direction.Therefore, -A = 250 g @ 217o.

PROCEDURE

I. Determination of Equilibrant E for A + B + E = 0

1. Apply the following forces to the Force Table.

A = 5 g(holder) + 125 g = 0.130 kg @ 37°

B = 5 g (holder) + 100 g = 0.105 kg @ 143°

2. Determine the magnitude and angle of a third force on the force table that will putthe ring in equilibrium. Call this force E and record your results in Data Table 1.

3. Draw a diagram of the sum of vectors A and B and determine the equilibrantforce from this diagram. The diagram must be drawn to scale (1.0 cm = 10 g or1.0 cm = 20 g) using a metric scale and a protractor. Record the magnitude andthe angle of the equilibrant in the Data Table 3.

NOTE: All angles should be measured counter-clockwise from the zero-degree line.

II. Determination of Equilibrant F for A + (-B) + F = 0


A = 0.130 kg @ 37°

- B = 0.105 kg @ 180° + 143° , where B = 0.105 kg @ 143°

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2. Determine the magnitude and angle of a third force on the Force Table that will putthe ring in equilibrium. Call this force F and record your result in Data Table 1.

III. Determination of Equilibrant G for A + B + C + G = 0


A = 5 g(holder) + 125 g = 0.130 kg @ 37°

B = 5 g (holder) + 100 g = 0.105 kg @ 143°

C = 5 g (holder) + 50 g = 0.055 kg @ 90°

2. Determine the magnitude and angle of a fourth force on the Force Table that willput the ring in equilibrium. Call this force G and record your result in DataTable 1.

IV. Determination of Equilibrant S for P + Q + R + S = 0


P = 5 g (holder) + 100 g = 0.105 kg @ 0°

Q = 5 g (holder) + 63 g = 0.068 kg @ 50°

R = 5 g (holder) + 125 g = 0.130 kg @ 240°

2. Determine the magnitude and angle of a fourth force on the Force Table that willput the ring in equilibrium. Call this force S and record your result in DataTable 1.

3. Draw a diagram of the sum of vectors P, Q, and R and determine theequilibrant force from this diagram. The diagram must be drawn to scale (1.0cm = 10 g or 1.0 cm = 20 g) using a metric scale and a protractor. Record themagnitude and the angle of the equilibrant in the Data Table 4.

Note: For convenience we use the mass to represent the weight in this lab. Themagnitude of force vector should be mass∙g (g=9.8m/s2) and unit in Newton, asshown in Data Table 2.


PRE-LAB FORM

EQUILIBRIUM AND VECTORS: THE FORCE TABLE

1. Four forces, A, B, C, and G, act on a particle leaving it in equilibrium.Given:

A = 130 g @ 37°

B = 105 g @ 143°

C = 55 g @ 90°

Determine G by the vector algebraic method for A + B + C + G = 0

2. Find G by the graphical method using a scale and a protractor. Use ascale factor of 1cm = 10 g. Draw arrow heads on vector ends.

Result: Length in cm: , G = g @ 0.

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Graphical Method to Find G


LAB REPORT FORM

EQUILIBRIUM AND VECTORS: THE FORCE TABLE

Data Table 1: Determination of Equilibrant Forces: E, F, G, and S

EquilibrantM (kg)(Mass)

M · 9.8 (N)(Force)

(Degree)(Direction)

E

F

G

S

Data Table 2: Applied Forces

Applied ForcesM (kg)(Mass)

M · 9.8 (N)(Force)

(Degree)(Direction)

A 0.130 1.27 37O

B 0.105 1.03 143O

-B 0.105 1.03 323O

C 0.055 0.54 90O

P 0.105 1.03 0O

Q 0.068 0.67 50O

R 0.130 1.27 240O

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Data Table 3: Graphical Method to Determine the Equilibrant E for A + B + E = 0Graphical Result: E = _______ kg = _____ deg


Data Table 4: Graphical Method to Determine the Equilibrant S for P + Q + R + S = 0Graphical Result: S = _______ kg = _____ deg

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QUESTIONS

1. (a) Calculate the Equilibrant E by the vector algebraic method for A + B + E = 0

(b) Compare the force table and calculated values of E, what is the % errorbetween the magnitudes?

2. Why must the graphical solution of the force required to produce equilibriumform a closed polygon?

3. Consider the three vectors a, b, and c such that c = a + b. Let us choose the x-direction in the same direction as vector a. Lets designate the angle between b andthe x-direction as θ so that bx = b·cosθ and by = b·sinθ.

a. What are the algebraic expressions for cx and cY?

b. Use the Pythagorean theorem, c2 = cx2 + cy

2 to showthat c2 = a2 + b2 + 2·a·b·cosθ.

Experiment 4

PROJECTILE MOTION

EQUIPMENTBallistic Pendulum ApparatusWooden Launching FramePaper, Tape , and GloveBox (to catch projectile)Meter Stick

Figure 1: Projectile Fired Horizontally

OBJECTIVESUpon completion of this experiment you will understand the basic concepts that describe projectile motion. You will have measured the range of a projectile for a specified initial elevation and angle of launch, and from this predicted the range for a second initial configuration.

CONCEPTSThe effects of air resistance will be negligible. This approximation is valid because air resistance effects depend strongly on the relative velocity through the air, and the size and mass of the projectile. Although the details vary from one apparatus to the next, the projectile used will be moving in low speed and have large mass to area ratios.

The gravitational acceleration g is taken as constant over the range of motion, 9.8 m/s2, down. The projectile’s motion will be described by x for the horizontal coordinate and y for the vertical coordinate, with positive y chosen as up. The projectile is launched at an angle θ relative to the horizontal and from an initial elevation y0 = H, and initial x-component x0 = 0. The acceleration, velocity, and position components are:


(1a) x-acceleration: ax = 0, (1b) y-acceleration: ay = -g

(2a) vx = v0cosθ, (2b) vy = v0sinθ – g·t,

(3a) x(t) = (v0cosθ) ·t, (3b) y(t) = H + (v0sinθ) ·t – g·t2/2.

For Projectile Fired Horizontally, as shown in Figure 1, you have conditions for the horizontal distance, R1, the height H1, the initial angle θ=0, and the position on the ground y = 0. The relation you will use to determine the launch velocity, v0, is deduced by combining the above equations:

v0 = R1·[g/(2 H1)]1/2 (4)

Figure 2: Projectile fired on an Incline

For Projectile fired on an Incline , as shown in Figure 2, you have conditions for the launch velocity, v0, the height H2, the initial angle θ, and y = 0. The relation you will use to determine the horizontal distance, R2, can be determined by solving the quadratic equation (3b) for the time, t, when the projectile strikes the ground, y = 0. This value of time is substituted into (3a) to evaluate R2 = x(t).

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Equipment:

Ballistic Pendulum SystemMetric ScaleTriple Beam BalanceWhite Paper and Tape45 deg. LauncherGlove

Figure 3: Push the Projectile with Glove on

PROCEDURE

Part A: Determining the Launch Velocity, v0. Projectile Fired Horizontally.

1. Arrange the ballistic pendulum spring gun device at the edge of the lab bench (or on a lab stool) such that the projectile will have about 3 meters of floor space in front of it. Position a cardboard box to catch the projectile after its rebound from the floor.

Note: Please wear the glove to push the projectile, as shown in Figure 3.2. Measure and record the elevation H1 from the floor to the bottom of the projectile. 3. Horizontally fire the projectile in a trial run to determine the approximate distance

R1. Tape paper on the floor.4. Fire the projectile and note the impact point on the paper. Make adjustments as

necessary. Measure the distance R1 along the floor from directly below the launch point to the impact point. Be careful not to move the gun between trials, and record 4 separate trials in the data table 1.

5. Calculate the average value of R1 and the uncertainty △R1. △R1 is the average absolute difference between each R1 value and the average R1 value.

6. Calculate v0 using equation (4).

Note: Calculate the initial speed of the projectile, v0, before proceeding.

Part B: Determining the Horizontal Distance, R2. Projectile Fired on an Incline.

1. Place the spring gun on a launch frame on floor placed such that the projectile can travel freely perhaps 5m before impacting the floor. Measure the incline angle of the launch and the elevation, H2, from the floor to the projectile at the launch point.

2. Place a cardboard box in a reasonable place to catch the projectile. Make a trial run to insure that you know the approximate position the projectile will impact the floor.


3. Tape paper to the floor to determine the projectile’s impact as you did in Part A of this experiment. Record the distance R2 for 4 trial runs in the data table 2. Be careful not to move the gun between trials. The measuring variations in the R2 are important since examination of measurement limitations is part of the experiment.

4. Calculate the average value of R2 and the uncertainty △R2. △R2 is the average absolute difference between each R2 value and the average R2 value.

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PRE-LAB FORMPROJECTILE MOTION

1. What is projectile motion?

2. A projectile is fired horizontally (θ = 0) from an elevation H1 = 1.00 m. It hits the floor at a distance R1 = 2.50 m. Show the calculation for

(a) the time of flight, t, and

(b) the initial velocity of the projectile, v0.


3. A projectile is launched with an initial speed v0 = 5.53 m/s, but now the elevation and angle are changed to H2 = 1.15 m, and θ = 30o. Show the calculation for

(a) the time of flight, t, and

(b) the horizontal landing distance, R2, for this configuration.

4. At what angle do you launch a ball to achieve longest time in the air for a fixed launching speed?

5. At what angle do you launch a ball to achieve longest range (horizontal distance) for a fixed launching speed?

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LAB REPORT FORM PROJECTILE MOTION

Data Table 1: Projectile Fired HorizontallyH1 = m.

Trail i 1 2 3 4 averageR1i (m) R1av

=|∆R1i| =

|R1i – R1av|∆R1

=

R1 = R1av ± ∆R1 = ± m. v0 = ___________m/s

Data Table 2: Projectile Fired on an InclineH2 = m. Incline Angle θ =___ o

Trail i 1 2 3 4 averageR2i (m) R2av

=|∆ R2i | =

| R2i - R2av |∆R2

=

R2 = R2av ± ∆ R2 = ± m.

QUESTIONS1. Show the calculation of the expected horizontal distance R2 for the incline firing

experiment. Use the value of the launch velocity, v0, from Data Table 1.

2. What is the % difference between the expected distance R2 and the measured value from Data Table 2?


3. Derive equation (4).

4. Show the calculation of the initial velocity, v0, of the projectile from the average of the horizontal launch data.

5. Based on information from the data table 1, show the calculation of the relative uncertainty in the projectile velocity. Does the uncertainty in R1 dominate the uncertainty of this calculation?

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Experiment 5

DYNAMICS OF A ROLLING CART

EQUIPMENTSmart PulleyPASCOL Computer InterfaceWeight hanger and 5 g massesPASCO cart and trackWeighing scale

M2

M1

Figure 1: Dynamic Track Equipments

OBJECTIVESUpon completion of this experiment, you will be able to:Properly set up and operate Smart Pulley and Dynamic Track Equipments.Understand the concept of Newton’s 2nd law. Measure the free fall acceleration g androlling friction f by measuring the dynamics of a rolling cart.

CONCEPTSNewton’s second law of motion states that when a net force Fnet acts on an object, theacceleration of the object equals the net force divided by the mass m of the object. That is

a = Fnet /m

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In this experiment the PASCO dynamic track will be used to demonstrate the Newton’ssecond law of motion and to measure the acceleration of gravity. Referring to Figure 1, acart having a mass M1 is pulled on a horizontal track by a string that passes over theSmart Pulley and is attached to a mass M2 . We will designate the rolling friction force ofthe cart as f, and the acceleration of each mass as a. Newton’s laws are used to deducethe dynamics relation

(M1 + M2) · a = g ·M2 – f

where g is the acceleration of gravity. In this lab a computer controlled Smart Pulley willbe used to determine the accelerations, a, of the total mass (M1 + M2) for a few values ofM2. You will plot the product (M1 + M2) · a (y-axis) against M2 (x-axis) to obtain astraight line. The standard straight line form is y = m·x + b where m (measured g) and b(friction f) are the slope and the intercept of the line.

PROCEDURE

HARDWARE ARRANGEMENT:

1. Set up the PASCO track and cart with a 500 g cart weight. The feet on the track are tobe adjusted until the track is as level as possible. You do this by giving the cart a littlemotion to the left, then comparing this with motion to the right.

2. Weigh the cart and the weight on the cart. Record this sum as M1 above the DATAtable.

3. Prepare a pulley string that is about 1.2 m long. One end is to be tied to a ‘weighthanger’, and the other end is to be attached to a 10/32 bolt on the cart.

4. Determine the mass of the ‘weight hanger’, and record this in the DATA Table. Setaside about six of the 5g/or 10g masses that you will hang on the ‘weight hanger’.

5. Attach the smart pulley at the edge of the bench. Make sure that the mass M2 canfreely fall off the lab bench. The pulley should turn very freely and be aligned withthe motion of the string.

6. Put a pad where the ‘weight hanger’ will land on the floor.


COMPUTER SETUP AND MEASUREMENTS:

1. Turn on the PASCO signal interface. Turn on computer. Plug the phone jack of theSmart Pulley into Digital Channel 1 of the signal interface.

2. Run DataStudio. Click on Create Experiment.

3. Click on “Channel 1” of interface image on your computer screen. Choose “SmartPulley” and click on OK.

4. Click on the “Graph” icon under the Display Window. Drag and drop the graph iconto the “Velocity” icon under the Data window. You should see a graph of velocityversus time.

5. Hold the cart and have the ‘weight hanger’ with one extra 5g mass suspended justbelow the Smart Pulley. Click the Start Button on your computer. Release the cart soit moves along the track. Whenever the hanging mass hits the pad on the floor, clickon the Stop Button.

6. Use the mouse to choose the linear region of data (Hold the mouse and high light thedata.) Click the “Fit” button on the graphic menu bar ( the 8th button). Choose theLinear Fit.

7. Record the value of the slope (The slope represents the acceleration.) in the DataTable.

8. Repeat steps 5 to 7, increasing M2 in 5g increments.

9. Compute the force, (M1 + M2) · a, in column 4 of the Data Table.

10. Plot (M1 + M2) · a against M2. Calculate g as the slope and determine the rollingfriction force from the intercept.

52

PRE-LAB FORM


A cart (mass M1) is pulled on a horizontal track by a string that passes over a SmartPulley and is attached to a weight (mass M2) as shown in the sketch. The rollingfriction force of the car t is f, and the resulting acceleration of each mass is a.

QUESTIONS

1. Apply Newton’s laws to each mass and combine these to deduce the formula

(M1 + M2) · a = M2 · g – f

2. The experimental data will be the measurements of accelerations a of the totalmass for few values of M2. You will plot the product (M1 + M2) · a (y-axis)against M2 (x-axis) to obtain a straight line. Explain how the free fall accelerationg and friction force f can be obtained from this straight line graph.


LAB REPORT FORM


Data Table

Mass of cart: kg. Mass of bar: kg.M1 = Mass of cart + Mass of bar : kg

Mass of ‘weight hanger’: kg.Note: M2 = mass of ‘weight hanger’+ added masses

Trial # M2

(kg)Acceleration a

(m/s2)(M1 + M2) • a

(N )

1

2

3

4

5

6

7

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Graph: Plot (M1 + M2) · a versus M2 . Label the graph appropriately.


QUESTIONS

1. Show the slope calculation of g (using proper units) and calculate the % error ofyour experimental value.

2. (a) Use your graph to find the cart’s rolling friction: f =-Intercept b=__________N

(b) Find coefficient of friction: µ = f/(M1·g):____________

3. If the track is not set up properly assume it is an inclined plane with angle θ, then the equation (M1 + M2) • a could be different. Derive the equation

(M1 + M2) · a = M2 · g – (M1 · g sinθ + f)

where θ is the angle of the track relative to the horizontal. This error would not affectthe experimental g!

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4. State in your own words what you have accomplished in this experiment.

Experiment 6

CENTRIPETAL ACCELERATIONAND THE MEASUREMENT OF g

EQUIPMENTWelch centripetal force apparatusScale (ruler)StopwatchWeights and Weighing scaleSafety Glasses

Figure: Centripetal Force Apparatus

OBJECTIVES: Upon completion of this lab you will be able to use the concept of centripetal force in circular motion applications. The free fall acceleration g is to be determined using the Welch centripetal force apparatus.

CONCEPTS: The apparatus is illustrated above. The mass m1 is rotated in a horizontal circle at a frequency f with a radius r. Applying Newton’s second law, the centripetal force, Fc, is given by

Fc = m1·ac = m1 · vt2 /r (1)

Since the tangential velocity, vt, is related with f, the centripetal force is also given by

Fc = m1 · [2π∙f]2 · r (2)


For this dynamic configuration, the spring force, Fs, provides the centripetal force. In the static configuration, as shown in the figure, the spring is stretched the same amount, r, by the weight M2· g. The spring force is

Fs = M2· g (3)

The spring force will be the same in both the static and dynamic configurations. The relation you will use to determine the free fall acceleration, g, is deduced by combining these equations:

g = m1· [2π∙f]2·r/M2. (4)

PROCEDURE1. Unhook and measure the swing mass m1. Record value in the data table.2. Set the vertical pointer for the smallest orbital radius and record this r value in the

data table.3. Attach the swing mass m1 to the suspension thread without attaching to the spring.

Adjust the bar on top so that the m1 hangs directly overt the pointer. The suspension thread must be vertical so that its tension does not contribute a horizontal force.

4. Connect the spring between m1 and the spindle. Attach a thread and weight hanger on the side opposite the spring. Add mass as necessary until its weight just balances the spring force. Mass m1 should hang over the pointer. Record the mass (including the weight hanger) as M2 in the data table.

5. Rotate the spindle at a rate such that the tip of mass m1 just lines up with the pointer. The spring force will be the same as in step 4. The experimenter spinning the mass should view the rotation in a plane perpendicular the apparatus to avoid viewing errors.Note: Please wear the safety glasses!

6. Adjust the rotation rate from moment to moment to keep the tip of m1 lining up with the pointer. Measure the time, t50, for 50 revolutions. Record this time in the data table. A large number of rotations is used in order to accurately determine an average rotation frequency.

7. Calculate frequency using f = (number of revolutions/time )=50/ t50 and g value using equation (4). Record values in the data table.

8. Compute the % error of g using 9.8 m/s2 as the standard value. The % error is negative if your value is less than the standard value. Record value in the data table.

9. Increase r about 1 cm. Repeat the steps 3 through 8 above for five sets of data. Each time adjust the beam so that the tip of m1 hangs over the new pointer position.

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PRE-LAB FORMCENTRIPETAL ACCELERATION AND THE MEASUREMENT OF g

1. Consider a point moving at a constant speed v on a circle of radius r. What is the relation between v, r, and the rotation frequency, f ?

2. Show that the centripetal acceleration relation, ac = v2/r, can be put into the form ac = [2π∙f]2·r.

3. In this experiment you will determine g from experimental values of m1, M2, r, and f. The frequency f is found by dividing the number of revolutions by the time for those revolutions. Assume g = 9.80 m/s2, take m1 = 500 g, M2 = 800 g, and r = 18 cm, and calculate the frequency f, and then the time for 50 cycles.


4. Referring to the equation (4), what do you expect the frequency to vary with the radius of the rotational motion?

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LAB REPORT FORMCENTRIPETAL ACCELERATION AND THE MEASUREMENT OF g

Data TableSwing mass m1 = kg

Trailr

(m)M2 (kg)

t50 (s)

f (Hz)

g (m/s2) % Error

1

2

3

4

5

QUESTIONS

1. Show one sample calculation of g value.

2. What are the average g value and its % error?


3. Which below statement is correct description of the dynamics in this experiment?

(a) The centripetal force is balanced by the spring.(b) The spring force is the centripetal force.

4. Show that the free fall acceleration g for this lab is given byg = m1· [2π∙f]2·r/M2.

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Experiment 7

KINETIC & POTENTIAL ENERGYEQUIPMENTDynamics Cart Track, 2.2 meterDynamics Cart (ME-9430) with PlungerForce SensorPhotogateFive-pattern Picket FenceRuler, Two Brackets, and Rubber Band

Figure: Experimental Apparatus

OBJECTIVES

Upon completion of this lab you will be able to calculate the potential energy of a rubber band, and use the energy theorem to compute the velocity of an object. In this experiment you will compare the velocities of a dynamic cart computed from the work-energy theorem with those measured using elementary kinematics.

CONCEPTS

When a spring (or rubber band) is pulled x distance away from its relaxed position (x=0), there is a force from the spring. This force, called spring force, is given by

F = kx (1)


where k is the spring constant. The value of k represents the stiffness of the spring and is measured in force for per unit length of change (unit in N/m).

A stretched or compressed spring stores elastic potential energy. This elastic potential energy can be expressed as

U = ½ kx2 (2)

Where x is stretched or compressed length in terms of the relaxed position and U has units in joules.

If a cart (or object) with mass M is attached to the spring and is launched by the spring, as shown in the Figure, the cart will gain speed v and kinetic energy K. If the speed of the cart is v at the relaxed position, the kinetic energy is given by

K = ½ Mυ2 (3)

This is because the stored elastic potential energy will be released in the form of kinetic energy resulting a launch velocity, v. Based on the theory of conservation of energy:

U = K (4)

Thus, the speed of the glider can be calculated by

υ = MU⋅2 (5)

PROCEDURE

Part A: Calibrating the Force Sensor

1. Turn on the Pasco Signal Interface first! Then turn on the computer.2. Connect DIN plug of the Force Sensor into the analog channel A of the PASCO

interface.3. Run “DataStudio”. Click on Create Experiment. You will see the interface window on

the computer screen.4. Click on the analog channel A of interface icon. Choose “Force Sensor” from the

sensor menu and click OK.5. Hold the force sensor in your hand. Click on Calibration Sensor icon from the bar

menu. A calibration window appears. Press the TARE button on the side of the Sensor. Type in 0 in the Calibration Point 1 box as standard value and click on Read from Sensor.

64

6. Hang 500 g mass (4.9N in weight) on the Sensor. Type in –4.9N in the Calibration Point 2 box as standard value and click on Read from Sensor.

7. Click on OK at the bottom of the calibration window. You are ready to measure the force.

Part B: Measuring the Force

1. Click on Table icon under the Display window. Drag and drop the table icon on Force icon under Data window.

2. Place the PASCO track on the table.3. Mount two brackets on the track. Twist a rubber band to form a single line and hang it

horizontally on the two brackets (Reference to the Photo picture). Adjust the height of the rubber band to be against the top part of the Dynamics Cart (just under the notch).

4. Use the Force Sensor to pull the notch of the cart. Let the cart move forward by x = 1 cm. Hold the Force Sensor steady at this position.

5. Click on “Start” button for 3 seconds. Then click “Stop”.6. Click on the Statistic “∑” icon from the table menu bar. Record the absolute mean

value of force in your Data Table 1.7. Repeat steps 3 to 5 for x = 2 cm, 3 cm, 4 cm, and 5 cm.8. Plot the force versus x using DataStudio. Rerun DataStudio. Click “Enter Data”.

Type in your data: X-displacement and Y-force.9. Click the “Fit” button on the bar menu and choose “Linear Fit”. The slope is value k.

Record k under the Data Table 1.10. Unplug the Force Sensor and close the DataStudio. Do not move the rubber band.

You will use it to launch the cart.

Part C: Launching the Cart and Measuring Its Speed

1. Record the mass of the Cart in Data Table 2.2. Adjust the Track to make sure that it is leveled.3. Connect the photogate timer’s phone plug into the Digital Channel 1 of the interface.4. Run the DataStudio. Click on “Create Experiment” . The experimental setup window

will appear. 5. Click on “Channel 1” of interface image. Then choose “Photogate” as your digital

sensor. Choose “Time in Gate, Ch1” box from Experimental Setup window.6. Click on the “Table” icon under the Display window. Drag and drop it into “Time in

Gate, Ch 1” under the Data window.7. Mount the Five-pattern Picket Fence on the cart. Align the photogate with the top

solid band of the Fence. Record the transit length l = 0.025m on Data Table 2.8. Using your hand, push the cart (against the rubber band) forward by x = 3cm, as the

same way as in Part B for the Measuring Force. Click Start button. Launch the cart. Click the Stop button and record the Elapsed time on Data Table 2.

9. Repeat above step for x = 4 and 5 cm. Complete all calculations for Table 2.


PRE-LAB FORMKINETIC & POTENTIAL ENERGY

1. Write a brief statement of the Work-Energy Theorem or Kinetic Energy Theorem. (This is not the definition of work.)

2. Consider the force-displacement graph for a spring shown. Determine (a) the spring constant, (b) the potential energy stored when the spring is stretched from

x = 0 to x = 4.0 cm, (c) the change in the potential energy stored in stretching the spring from x = 1.0 cm to x = 4.0 cm.

3. A cart having a mass M = 180 g on a friction free horizontal surface is accelerated from rest by the launching spring of problem 2. What is the cart’s final speed if the spring’s potential energy with a compression of 4.0 cm is completely transferred to the cart?

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LAB REPORT FORMKINETIC AND POTENTIAL ENERGY

Data Table 1: Determination of Spring Constant

x(cm) 1 2 3 4 5

F(N)

Slope k: N/m

Data Table 2 Mass of Cart M: kg. Transit length: l = 0.025m

x(cm)

Elapsed Time t(s)

V’ = l/t(m/s)

U = ½ kx2

(J)υ = (2U/M)1/2

(m/s)% Difference

in velocity

3

4

5


QUESTIONS

1. Show one sample calculation of the energy U = ½ kx2.

2. Explain which do you think is most significant source of difference in the two sets of velocity values.

3. Not all the energy stored in the rubber band can be completely transferred to the cart because of losses in the transfer process. Explain which velocity you expect to be larger, that from the energy calculation or that from the elapsed time calculation.

4. We assumed a linear force-distance relation for the launch device. Consider the two F-x graphs shown. Explain which would produce a larger launching velocity.

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Experiment 8

IMPULSE AND MOMENTUM

EQUIPMENTDynamics Cart Track, 2.2 meterDynamics Cart (ME-9430) with Plunger500 g Bar MassForce Sensor, 500g Mass

OBJECTIVESUpon completion of this lab you will have verified the relation between the changes in momentum of an object with the impulse on the object. You will be able to use conservation of energy in basic applications.

CONCEPTSThe momentum p of an object is simply the product of its mass m and its velocity v, p = mv. The impulse I of net force Fnet acting on an object over the time interval ∆t = t2 – t1is defined by

∫2

1

t

t


I ≡ Fnetdt (1)

The impulse of an object is also given by the change of its momentum

I = p2 - p1

The impulse causes the change in the momentum of the particle. In this lab you will use a force sensor and a photogate to measure the force as a function of time and calculate the impulse. At the same time you will also measure the change of momentum, p2 - p1. In an ideal situation the results should be the same.

PROCEDUREPart A: Measuring the Angle of the Inclined Plane

1. Place a book, such as your physics book, under the dynamic track to form an inclined plane.

2. Measure the height at the both ends of the track, recorded as h1 and h2. The length of

the track is L = 227cm. Calculate the angle with θ = sin-1( Lhh )( 12 − ) and record θ on

the data table.

Part B: Calibrating the Force Sensor

1. Turn on the Pasco Signal Interface first! Then turn on the computer.2. Connect DIN plug of the Force Sensor into the analog channel A of the PASCO

interface.3. Run “DataStudio”. Click on Create Experiment. You will see the experimental setup

window with interface image.4. Click on the analog channel A on the interface image. Choose “Force Sensor” from

the sensor menu and click OK. The experimental setup window for the force sensor opens.

5. Hold the force sensor vertically in your hand. Click on Calibration Sensor icon from the bar menu. A Calibrate Sensor window appears. Press the TARE button on the side of the Force Sensor. Type in 0 in the Calibration Point 1 box as standard value and click on Read from Sensor.

6. Hang 500 g mass (4.9N in weight) on the Sensor. Type in –4.9N in the Calibration Point 2 box as standard value and click on Read from Sensor.

7. Click on OK at the bottom of the calibration window. You are ready to measure the force.

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Part C: Measuring Force and Impulse

1. Go to Sample Rate box under Experimental Setup window. Choose the sample rate to be 1000 Hz.

2. Click on Graph icon under the Display window. Drag and drop Graph icon into Force Sensor under the Data window. Graph 1 window will appear. You are ready to take collision data.

3. Place the force sensor so that its back is firmly against the blocker at the end of the track. The spring plunger of the Cart should be out. When you release the cart, it will strike the force sensor.Note: An alternative way to do it is to replace the hook by a rubber bumper on the force sensor.

4. Place the 500g bar mass on the Cart and measure the total mass. Record the total mass in the data table 1.

5. On the incline, when the spring plunger of the cart makes contact with the force sensor, record the position as x0 (which is the front edge of the cart) in the data table.

6. Raise the cart to a higher position for about 150cm on the incline and record this position as x1 (use the front edge of the cart as a reference line.)

7. Click on Start button. Release the cart and observe how far up the track when it rebounds. Record the highest bouncing back position as x2. Click on Stop button just after the collision.

8. Click the Autoscale button on the Graph menu bar (It is the first button.) to fit the data. Use your mouse to high light the impulse area, which is the area under the force versus time plot. You need this area for impulse calculation.

9. Click the Statistic (Σ) button and choose Area. Record the value of area in the data table 1. This represents the impulse value.

10. Calculate the momentum change from the collision data using

I = ∆p = p2 – p1 = M[(2g∆x2sinθ)1/2 + (2g∆x1sinθ)1/2]Where M is the mass of the cart, ∆x2 = x2 – x0, ∆x1 = x1 – x0 , and θ is the angle of the inclined plane. Calculate the % difference between the impulse values of steps (9) and (10). Record values in data table 2.

11.Repeat steps 5 to 9 for three more trials (Don’t have to be the same starting point.).


PRE-LAB FORM IMPULSE AND MOMENTUM

1. Show that the relation below is true for a fixed mass by substituting Fnet = mdv/dt.

∫2

1

t

tFnetdt = p2 - p1

2. Consider two masses in a closed system. Show that Newton's third law results in conservation of momentum

3. A mass M drops from a height h1 and rebounds to a height h2. Show that the impulse is given by M[(2gh1)½ + (2gh2)½]. Note that the velocity just after rebound is opposite in direction to the velocity just before rebound.

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LAB REPORT FORM IMPULSE AND MOMENTUM

DATA TABLE 1

θ = sin-1( Lhh )12( −

) = ; Mass of (Cart + Bar): M = kg

Trial

Collision position

X0

(m)

Release position

X1

(m)

Rebound position

X2

(m)

ImpulseI

(Area under the impulse graph)

(N·s)

1

2

3

4

Data Table 2

Trial ∆x1 = x1 – x0 ∆x2 = x2 – x0

I=∆p = M[(2g∆x2sinθ)1/2 +

(2g∆x1sinθ)1/2]%

Difference

1

2

3

4


QUESTIONS:

1. Prove that the equation of momentum change for the cart is given by ∆p = p2 – p1 = M[(2g∆x2sinθ)1/2 + (2g∆x1sinθ)1/2]

where M is the mass of the cart (with bar mass), ∆x2 = x2 – x0, ∆x1 = x1 – x0 , and θ is the angle of the inclined plane.

2. What experimental errors are significant for this experiment?

3. If the initial height of a ball is 1.80 m, a ball of mass 0.22 kg is dropped from rest. It rebounds back from the floor to a height of 1.50 m. Determine the impulse on the ball delivered by the floor.

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Experiment 9

DYNAMIC CARTS COLLISIONS:CONSERVATION OF LINEAR

MOMENTUM

EQUIPMENTDynamic Cart Track, 2.2 meterTwo Dynamic Carts (ME-9430) with PlungerOne Collision Cart (ME-9454)Two Photogates with mounting bracketsTwo Five-pattern Picket FencesOne 500g bar mass

Figure: Experimental Apparatus

OBJECTIVES:

Upon completion of this experiment you will understand the concepts of linear momentumand energy conservation. In this experiment you will examine the motions of the elasticand inelastic collisions between two carts.

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CONCEPTS:

If an object is in motion with mass m and velocity v, the linear momentum p of the objectis defined as

p = mv

When two carts collide each other, the sum of two carts linear momentum before thecollision equals the total linear momentum after the collision regardless of type ofcollisions. That is

Total momentum before the collision = Total momentum after the collision

Or

Pi = Pf

If there is no loss of kinetic energy after the collision, it is called an elastic collision.Otherwise it is an inelastic collision. If the two carts stick together after the collision, it iscalled a perfectly inelastic collision.

PROCEDURE

Experimental Setup

1. Level the dynamics cart track. If a cart stays stationary on the track, it indicates aleveled track.

2. Select two Dynamics carts. Place the 500g bar mass on A cart. Call the lighter one Bcart. Two carts with spring plungers will be used for nearly elastic collisions. Selecttwo carts with the Velcro strip on one end for inelastic collision. Place the PicketFences on the top of each cart where the 2.5 cm opaque is at the top.

3. Place two photogates on the track with a space about 2 carts length between them. Setup the photogates so that they can measure the interruption times only for the topband of the strips (The bandwidth of the strip is 2.5 cm.).

4. Measure masses of the two carts. Record values in the DATA TABLE FOR CARTS alongwith the width of strips (transit length) lA = lB = 2.5cm = 0.025 m for each cart.

5. Turn on the signal interface first, then the computer. Insert two photogate plugs intochannel one and two of the interface unit. Run the DataStudio. Click on CreateExperiment.

6. Click on Channel 1 of the interface image. Choose “Photogate” from the ChooseSensor window. Click OK. Experiment Setup window opens.

7. On the Experimental Setup window, fill one check mark into the box of Time in Gate,Ch1. Undo the check mark on Velocity in Gate, Ch1.


8. Repeat steps 6 and 7 on Channel 2 for the 2nd photogate sensor. Be sure that the checkmark is only left on Time in Gate, Ch2. Click on 2nd measurement tab under the samewindow. Take the check mark out of the box of Velocity Between Any Gates, Ch2.

9. Click on Table icon under the Display window. Drag and drop the Table onto Time inGate Ch. 1 under the Data window.

10. Click on Table 1 icon under the Display window. Drag and drop onto Time in Gate,Ch. 2. Now you are ready to measure the time.

CASE 1: PERFECLY INELASTIC COLLISION

1. Prepare the two carts for a perfectly inelastic collision, velcro to velcro configuration.Place lighter cart B in between two photogates at rest.

2. Click on Start button on DataStudio. Gently push cart A (releasing it before photogateI). Determine the times for the A cart’s strip to pass through photogate I, tA, and thetime, t’B, for the B cart’s strip (with cart A stuck to it) to pass through photogate II.The prime sign t’B indicates that it is a time after the collision.

3. Click on Stop button on DtaStudio. Record the Elapsed time on Data Table. You willcalculate the velocities, linear momentum, and kinetic energy before and after collisionusing

vA = lA/tA = 0.025/tA , v’AB =lB/t’B = 0.025/tB

pA = mA vA, pAB’ = (mA + mB) v’AB.KA = mA vA

2/2, KAB = (mA + mB) v’AB2/2

4. Calculate the change in total momentum and in total kinetic energy before and aftercollision. Calculate the percentage loss for total momentum and total kinetic energy.Record values in the case 1 data table.

CASE 2: NEARLY ELASTIC COLLISION:

Each of these collisions involves 3 velocities -- the velocity of the colliding cart before thecollision, and the two velocities of the two carts after the collision. One of the photogatesmust be used twice!

Part A: Heavy cart mA collides with the light cart mB.

1. Prepare two carts for elastic collision, plunger to plunger configuration (or plungerto hard bumper). Place cart B at rest in between two photogates.

2. Add the 500g bar mass on cart A and place the cart before photogate I.3. Click on Start button on DataStudio. Gently push cart A (releasing it before

photogate I). It moves through photogate I and collides with cart B.4. Click on Stop button on DtaStudio. Record the Elapsed times tA and t’B in DataTable

of Part A, where t’B is the time for cart B to pass through the photogate II after thecollision. After B passes through the gate, photogate II will record a second time forcart A to pass through. Designate this third time as t’A.

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5. Compute velocities vA = lA/tA, v’B = lB/t’B, and vA'= lA/ t’A . Calculate the linear

momentum and kinetic energy before and after collision. Calculate the change in totalmomentum and in total kinetic energy before and after collision. Calculate thepercentage loss for total momentum and total kinetic energy. Record values in thecase 2 part A data table.

Part B: Light cart mB collides with the heavy cart mA .

1. Use the same two carts. Place cart A at rest in between two photogates.2. Place cart B before photogate II.3. Click on Start button on DataStudio. Gently push cart B (releasing it before

photogate II). It moves through photogate II and collides with cart A.4. Click on Stop button on DtaStudio. Since the Light cart B will bounce back,

photogate II will record two elapsed times tB and t’B. The velocity vB' = - lB/ t’B . (B's

velocity after collision will be negative since it is opposite to its initial motion.)5. Record data and finish all required calculation in the case 2 part B data table.


PRE-LAB FORM

DYNAMIC TRACK COLLISIONS: CONSERVATION OFLINEAR MOMENTUM

1. What is an elastic collision?

2. What is an inelastic collision?

3. What is a perfectly inelastic collision?

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4. If a mass ma = 2m has initial velocity 4 m/s and mass mb = m is initially at rest,they undergo perfectly inelastic collision. Calculate:

(a) the final velocity and

(b) % loss of kinetic energy.


LAB REPORT FORM

CONSERVATION OF LINEAR MOMENTUM

DATA TABLE FOR CARTS

MA (Kg) lA(M) MB (kg) lB(M)

Data Table for Case 1: Perfectly Inelastic Collisions (Velcro to Velcro Configuration)tA (S) t’B (S) vA (m/s) v'AB (m/s)

Pi =pA

(kgm/s)Pf =p'AB

(kgm/s)ΔP= Pf

– Pi

% Loss=

ΔP /Pi•100

Ki = KA

(J)Kf =K’AB

(J)ΔK =Kf

– Ki

%Loss=

ΔK/Ki•100

Case 2: Nearly Elastic Collision (Plunger to Plunger/or Hard Bumper Configuration)Part A: Heavy Cart mA Collides with the Light Cart mB.

tA (S) t’B (S) t’A (S) vA (m/s) v'B (m/s) v'A (m/s)

Pi =pA (kgm/s) p'A (kgm/s) p'B (kgm/s) Pf =p'A + p'B % Loss= ΔP /Pi•100

Ki = KA(J) K’A (J) K’B (J) Kf =K’A + K’B % Loss= ΔK/Ki•100

Part B: Light Cart mB Collides with the Heavy Cart mA.tB (S) t’A (S) t’B (S) vB (m/s) v'A (m/s) v'B (m/s)

Pi =pB (kgm/s) p'A (kgm/s) p'B (kgm/s) Pf =p'A - |p'B| % Loss= ΔP /Pi•100

Ki = KB(J) K’A (J) K’B (J) Kf =K’A + K’B % Loss= ΔK/Ki•100

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QUESTIONS:

1. Show that the kinetic energy of an object having a mass m and momentum p is givenby K = p2/2m.

2. A mass m1 traveling down the x-axis with a speed v undergoes a perfectly inelasticcollision with a second mass m2 that was initially at rest. Show that the total kineticenergy just after the collision, Kf, is [m1/(m1 + m2)]Ki, where Ki is the initial kinetic of thesystem.

3. If a mass ma = 2m has initial velocity 4 m/s and mass mb = m is initially at rest, theyundergo elastic collision. Calculate their final velocities after the collision.

4. If a mass ma = 2m has initial velocity 4 m/s and mass mb = m has initially velocity-6 m/s, they undergo elastic collision. Calculate the final velocities.

Experiment 10

THE BALLISTIC PENDULUM

EQUIPMENT:Ballistic PendulumPaper and TapeCardboard BoxTriple Beam BalanceGlove

M1

M2

h1

h2

Figure 1: Ballistic Pendulum

OBJECTIVES:

Upon completion of this laboratory you will have used the concepts of momentum and energy conservation to determine the speed of a spring gun’s projectile and you will have compared this velocity with that determined from the range of the projectile.


CONCEPTS:

The ballistic pendulum experiment, as shown in Figure 1, is based on the classic method to measure the speed of a gun’s projectile. The projectile, mass m1 with a horizontal velocity v, is fired into a pendulum, mass M2, that catches it. The pendulum and projectile swing as one unit with velocity vf. This collision process is perfectly inelastic collision. Conservation of momentum yields

m1•v = (m1 + M2)•vf (1)

After the collision they swing to a highest point h and remain stopped at the highest point due to a ratchet mechanism on the pendulum. If elevations h1 and h2 are measured to the center of mass of pendulum with the ball inside, h is h2 – h1. Based on conservation of energy, vf is given by

vf = hg ⋅⋅2 (2)

Combining equations (1) and (2), the calculation formula for the speed v in terms of the masses m1 and M2, and the rise in the center of mass, h, is

v = {(m1 + M2)/m1} • hg ⋅⋅2 (3)

Figure 2: Projectile Motion

An alternative method to determine the initial speed, the projectile motion method, uses basic kinematics. The projectile is fired horizontally from an initial elevation H above the lab floor and impacts the floor after traveling a horizontal distance R, as shown in Figure 2. The initial velocity of the projectile is given by

v = HgR⋅2

(4)

You are asked to derive this formula in the Pre-Lab form.

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PROCEDURE

Part A: Ballistic Pendulum Method

1. Measure and record the projectile mass, m1. Take the pendulum apart. Measure and record the pendulum mass, M2, in Data Table 1.

2. Place the projectile inside the pendulum for one unit. Determine the position of the center of mass of the system by balancing it, say on your finger tip. Mark this position on the pendulum.

3. Assemble the pendulum securely so that it does not wobble out of the plane when it swings. Measure h1, the center of mass elevation above the table, when the pendulum hangs vertically. Record value on Data Table 1.

4. Fire the projectile into the pendulum. The pendulum system reaches to a higher position. Measure the stopping position of the pendulum, h2, the center of mass elevation above the table.Note: Please wear the glove to load the projectile.

5. Repeat step 4 for four times. Record h2 values on Data Table 1.6. Calculate the average value of h2 and the uncertainty △h2. △h2 is the average

absolute difference between each h2 value and the average h2 value.7. Calculate h = h2av – h1. Assume △h = △h2.

Part B: PROJECTILE MOTION METHOD

1. Place the projectile-gum system on a lab stool. Make sure that the system is leveled.

2. Measure and record the elevation H from the floor to the bottom of the initial position of the projectile in the gun.

3. Put a cardboard box on the floor about 3 m from the gun to catch the projectile after it rebounds from the floor.

4. Make a trial run to find the approximate range. Tape paper on the floor at the approximate impact point in order to mark the projectile’s impact. Fire the projectile horizontally, and then measure the range R as the horizontal distance from the launch point to the impact point.

5. Repeat and record a total of four measurements of R. Finish all required calculations and record values in the data table 2.


PRE-LAB FORMBALLISTIC PENDULUM

1. What is an elastic collision? Is the ballistic pendulum collision elastic?

2. Show the derivation the ballistic pendulum calculation formula, Equation (3).

3. Referring to the following figure, if a projectile, m1 = 65g, is horizontally fired into the ballistic pendulum, M2 = 190g, the projectile is caught by the pendulum and the center of mass the system is raised by h = h2 – h1 = 10.5 cm. What is the initial speed of the projectile?

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4. Use basic kinematics principle to derive the range method calculation formula, Equation (4).


LAB REPORT FORMTHE BALLISTIC PENDULUM

Data Table 1: Ballistic Pendulum Methodm1 = kg, M2 = kg, h1 = m.

Trail i 1 2 3 4 averageh2i (m) h2av

=|∆h2i| =

|h2i - h2av|∆h2

=

h2 = h2av ± ∆h2 = ± m; h = (h2 - h1) = ± m.

Data Table 2: Projectile Motion MethodH = m.

Trail i 1 2 3 4 averageRi (m) Rav

=|∆Ri| =

|Ri - Rav|∆R

=

R = Rav ± ∆R = ± m.

CALCULATIONS1. Ballistic Pendulum (a) Calculate the projectile velocity, v, using equation (3) and the data from the data

table 1.

(b) Determine the uncertainty of this velocity.

2. Projectile Motion Method(a) Calculate the projectile velocity, v, using equation (4) and the data from the data table 2.

(b) Determine the uncertainty of this velocity. It is assumed that the uncertainty in the Projectile Motion Method measurements is dominated by the determination of the range, R.

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QUESTIONS

1. Why are the positions of the center of mass of the pendulum with the projectile measured, rather than the empty pendulum?

2. Compute the % difference between the ballistic pendulum method and projectile motion method velocities. Is the velocity difference reasonable in view of the experimental limitations?

3. If the elevation change, h, in the Ballistic Pendulum Method were increased by 1%, then by what would be % change in the resulting velocity v? (Please use an approximation technique to answer this question.)


4. If the projectile’s velocity were doubled, then by what factor would its momentum be changed? By what factor would its kinetic energy be changed?

Momentum change factor:___ Kinetic energy change factor:____

5. Referring the following figure, if m1 = 65g and M2 = 190g, what is the percent kinetic energy loss during the perfectly inelastic collision? (Hint: Kf =[m1/(m1 + M2)]Ki, where Ki is the initial kinetic of the system.)

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Experiment 11

ROTATIONAL INERTIA OF A WHEEL

EQUIPMENT:Centripetal Force ApparatusPASCO Smart PulleySmall MassesString and Scotch tapeCaliper and RulerWheelRing stand and Table Clamp

m

Figure 1: Axle-wheel Setup. Measuring IWA.

OBJECTIVES:

Upon completion of this lab you will understand the concepts of the moment of inertia(rotational inertia) and Newton’s second law for rotational motion. In this experiment youwill measure the rotational inertias of an axle and a wheel using Smart Pulley andcomputer.

CONCEPTS:

In this experiment there is a wheel mounted on an axle. One end of a string is rounded onthe wheel (or axle) and the other end is attached to a mass (m), as shown in Figure 1. Thestring passes over a Smart Pulley. If the system is released, there is nonzero net torque onthe wheel and net force on the falling mass, m.

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The net forces on m are its weight and the tension Ft. Applying Newton’s second law,

m•g – Ft = m•a (1)

Applying Newton’s second law for rotational motion and using the relation between theangular acceleration of the wheel and translational acceleration a, = a/R, the nettorque on the wheel is given by equation (2).

R•Ft = IWA •a/R (2)

The relation you will use to determine the rotational inertia is deduced by combiningthese two equations:

IWA = m • R2 • (g-a)/a (3)

where IWA is the rotational inertia of the wheel and axle, R is the radius of thewheel, m is the hanging mass, and a is the translational acceleration of m.

2r

m

Figure 2: Measuring IA

Using the similar way we can derive the equation for axle only without the wheel:

IA = m • r2 • (g-a)/a (4)

where IA is the rotational inertia of axle, and r is the radius of the axle, and m is thehanging mass.

The rotational inertia of the wheel, IW, can be calculated from the following equation:

IW = IWA - IA (5)


PROCEDURE

Part A: Computer Setup

1. Turn on the PASCO scientific interface box, and the computer. Plug the SmartPulley phone jack into the digital Channel one.

2. Open DataStudio and click in Create Experiment. Click on channel one of theinterface image on Experimental Setup window. Choose “Smart Pulley” underChoose Sensor menu. Click OK. The experimental setup window for smartpulley will open. Be sure that the Velocity, Ch. 1 has check mark undermeasurement.

3. Click on the graph icon under Display window. Drag and drop the graph icononto “Velocity, Ch. 1” under Data window. The velocity-time graph appears.

Part B: Measuring IA

1. Set up the apparatus as illustrated on the Figure 2. You will do this experimentfirst without the wheel. Measure the radius r of the Axle using caliper. Attachone 5 g or 10 g on the ‘weight hanger’ (the mass of ‘weight hanger’, mh). Recordr, mh, and hanging mass m = mh + 5g (or 10g) in Data Table 1.Note: Clamp the ring stand at the back edge of the lab table using the tableclamp, as shown in Figure 2.

2. The Smart Pulley is held in place with a clamp on a ring stand, as shown in Fig. 2.

3. Take a string, about one meter long, and make a loop at one end to hold the‘weight hanger’ and the extra mass.

4. Fasten the other end of the string to the axle with scotch tape, wind it around theaxle several turns, then pass the loop end over a Smart Pulley. Be sure that thestring is parallel with the table and pulley is aligned so that the mass falls freely.

5. Click Start button. Release the hanging mass. Data recording begins on velocity-time graph. Click Stop button just after the string is completely stretched out.

6. High light the linear section of the graph with the mouse. Click on Fit button onGraph Tool Bar (8th button). Choose the linear fit. Record the slope (It is thetangential acceleration, a.) on Data Table 1.Note: If you can not find a good “linear section” on the graph, you should repeatsteps 1 to 7 with larger mass.

7. Use equation (4) to calculate the rotational inertia of axle, IA and record it in DataTable 1.

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8. Take the string off the axle.

Part C: Measuring IWA

1. Measure the mass M and radius R of a wheel. Record these values in Data Table 2.

2. Mount the wheel on the axle. Fasten the string to the wheel with scotch tape.

3. Wind the string around the wheel several turns, then pass the loop end over aSmart Pulley. Be sure that the string is parallel with the table and the pulley isaligned so that the mass falls freely.

4. Repeat steps 5 and 6 in Part B for three different hanging masses (5g or 10 g asincrement). Record the slope on Data Table 2.

5. Use equation (3) to calculate the rotational inertia of the wheel and axle, IWA. Useequation (5) to calculate the rotational inertia of the wheel, IW. Record these datain Data Table 2.


PRE-LAB FORM

ROTATIONAL INERTIA OF A WHEEL

1. Write a brief statement to describe the physical meaning of the moment of inertia.

2. A washer is constructed from soft steel (density ρ = 7.9•103 kg/m3). Its inside andoutside dimensions are 2.00 cm and 3.30 cm, and its thickness is 0.50 cm. Determine itsmass.

3. Calculate the moment of inertia of the washer.

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4. Referring to the following figure, the hanging mass has a mass, m = 19g, thewheel has a radius, R = 0.10m, and the wheel and axle has a moment of inertia,IWA = 4.70x10-3 kg·m2. If the pulley’s moment of inertia and friction on the axleare negligible, determine (a) the tension in the string and (b) the acceleration ofthe hanging mass.


LAB REPORT FORM

ROTATIONAL INERTIA

TABLE 1: DETERMINATION OF THE AXLE’S MOMENT OF INERTIA

r (m) mh (kg) m* (kg) a (m/s2) IA (kg●m2)

m* is the hanging mass, m* = mh + 5g (or 10g).

TABLE 2: AXLE-WHEEL SYSTEM MOMENT OF INERTIA

Wheel Mass M: (kg), Wheel Radius R: (m)

Trial m* (kg) a (m/s2) IWA (kg●m2) IW (kg●m2)

1

2

3

m* is the hanging mass, m* = mh + 5g (or 10g).

Calculations:1. Show the calculations of the axle’s moment of inertia, IA.

2. Show one sample calculation for moment of inertia of the axle-wheel system, IWA.

3. Show one sample calculation, IW.

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QUESTIONS

1. Suppose there is a constant friction in the bearings of the axle.a. Would this tend to make your experimental value of the axle, IA; too small or too

large?

b. If the friction drag is unchanged for the measurement of the axle-wheelcombination, IWA, would it affect the result of the wheel alone? Explain.

2. Derive: IA = m • r2 • (g-a)/a .

3. Referring to your data of Data Table 1, determine (a) the tension in the string and(b) the net torque on the axle.

4. Assuming that the wheel is a solid disk, calculate the IW using the lab data.Explain why there is difference between the calculated value and measured one.


Experiment 12

SIMPLE HARMONIC MOTION:THE SPRING-MASS SYSTEM

EQUIPMENT:Motion Sensor, Metal GuardRingstand, Clamp, Table ClampBoxed weightsMeter stick and Spring

Figure 1: Experimental Setup

OBJECTIVES:

Upon completion of this lab you will understand the time dependent descriptions ofposition, velocity, and acceleration for an object in Simple Harmonic Motion. You willunderstand the physical properties such as spring constant, amplitude, period, frequency,and phase angle. A Motion Sensor computer probe is used in the investigation. Theconstant, k, of a spring, and the free fall acceleration, g, are to be measured.

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CONCEPTS:

For an object moving with simple harmonic motion, the acceleration, a, is oppositelydirected to its displacement, x, measured from equilibrium:

a(t) = -ω2·x(t) (1)

The solution of equation (1) is given by

x(t) = A· cos(ω·t - Φ ) = A· cos(2πf·t - Φ) (2)

ω = 2πf = 2π/T (3)

where A is the amplitude, ω and f are the angular and linear frequencies, Φ is the phaseangle, and T is the period. The velocity and acceleration of an object undergoing simpleharmonic motion can be obtained using equation (2).

V = dx/dt = - Vmaxsin(ω·t - Φ ) (4)

a = d V /dt = - amax cos(ω·t - Φ ) (5)

where Vmax = Aω is the maximum speed and amax = Aω2 is the maximum acceleration.

Consider the spring-mass system illustrated on Figure 1. There is a restoring force, Fr,acting on the oscillating object. It is oppositely directed to the displacement x. ApplyingNewton’s second law,

Fr = m·a = - k·x (6)

Dividing by m results in the form for simple harmonic motion, equation (6):

a = - (k/m)·x (7)

Comparing equations (7) and (1), we see that ω2 = k/m. The relations for the linearfrequency f (or just frequency) and period T of the motion are:

f = ω/2π = (1/2π) · [k/m]1/2 (8)

T = 1/f = 2π· [m/k]1/2 (9)

Then the maximum speed Vmax and acceleration amax of the motion are given by

Vmax = Aω = A· [k/m]1/2 (10)

amax = Aω2 = A· [k/m] (11)


PROCEDURES:

Part A: Measuring A, T, f, Φ, Vmax, and amax.

1. Clamp the ring stand on the back corner edge of the lab table, as shown in Figure 1.Make sure it is secure.

2. Place motion sensor on stool.Note: Your instructor might ask you to cover the sensor with a metal guard.

3. Hang a spring on a ring stand and a 50 g mass at the other end of the spring. Don’tdrop the mass from the spring. Hold the mass slowly going down till it hangs atrest when released. Use scotch tape to stick a small card at the bottom of the mass,as shown in Figure 1.

4. Adjust the distance between the mass and the motion sensor in the range of 20 cmto 50cm. Be sure that the mass is vertically above the motion sensor.

5. Turn on PASCO Signal Interface first. Then turn on the computer.6. Insert the yellow phone jack of the motion sensor into the Digital Ch. 1 of the

interface and another one goes to Ch. 2.7. Run DataStudio. Click on Create Experiment. Experimental Setup window opens.8. Click the channel one on interface image. Choose Motion Sensor from the list.

Click OK. Experiment Setup window for Motion Sensor opens. Be sure that thereare check marks in front of Position, Velocity, and Acceleration, Ch1 & 2.

9. Click the “Graph” icon under Display window. Drag and drop it onto Positionch1&2 under Data window.

10. Click the Graph 1 icon under Display window. Drag and drop it onto Velocitych1&2 under Data window. Repeat the same operation for Acceleration. When thegraph 1 window for Position, Velocity, and Acceleration appears, you are ready forthe observation of simple harmonic motion.

11. Gently pull down the hanging mass about 2cm and release it to set it in oscillatorymotion.

12. When a good oscillation pattern is formed, click Start button. Wait for few secondsand click Stop button. If the graphs do not look right, adjust the height of hangingmass to achieve the best graphs. (Note: After each practice trial you should deletethe Run Data. Keep only the best graph to analyze your data.)

13. Click anywhere on Position graph. Click the Scale to Fit button, the first button, onthe tool bar menu. Repeat the scale to fit process for Velocity and Acceleration.

14. Click anywhere on Position graph. Click the Smart Tool button, 6th button, on theTool Bar menu. The Smart Tool crosshair appears on the position graph. Thiscrosshair acts as a xy line that allows you to read the coordinates of any point on thegraph.

15. Measuring the amplitude A: Drag the crosshair to measure the maximum andminimum positions (displacements) of the motion. Record them as ymax and ymin onData Table 1. Calculate the amplitude using A = (ymax - ymin )/2.

16. Measuring the period T, frequency f, and phase angle Φp of the oscillation ofposition: Drag the crosshair to measure times t1 and tN for the first peak and lastpeak (as Nth peak) on the position graph. Calculate the average period using T =

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(tN - t1 )/(N-1), frequency f using f = 1/T, and phase angle using Φp = 2π·t1/T.Record these values on Data Table 1.

17. Measuring the phase angles of velocity Φv and acceleration Φa, the maximum speedVmax and acceleration amax: Click anywhere on Velocity graph. Click the Smart Toolbutton, 5th button, on the Tool Bar menu. Drag the crosshair to measure the time t1v

and maximum speed Vmax of the first peak. Calculate phase angle of velocity Φv

using Φv = 2π·t1v/T (T is the same period as position graph.). Repeat the same stepsfor Acceleration Graph to measure the time t1a and maximum acceleration amax ofthe first peak. Calculate phase angle of acceleration Φa using Φa = 2π·t1a/T.

18. Remove the 50 g mass from the spring. Delete the Data of Run #1 on the computer.Complete the required calculations on Data Table 3.

Part B: Measuring Spring Constant k and Free Fall Acceleration g

1. Measure the bottom position of the unstressed spring with respect to floor using ameterstick. Record this position as ho on Data Table 4.

2. Hang 40 g mass on the spring. Don’t drop the mass from the spring. Hold themass slowly going down till equilibrium position. Measure the bottom position ofthe stressed spring as h. Calculate the displacement of the spring yo = (ho - h ).Record values on Data Table 4.

3. Use scotch tape to stick a small card at the bottom of the mass.4. Gently push the hanging mass up about 2cm and release it to set it in oscillatory

motion.5. Click Start button. Wait for about 10 s and click Stop button. If the graphs do not

look right, adjust the height of hanging mass to achieve the best graphs. (Note:After each practice trial you should delete the Run Data. Keep only the best graphon computer.)

6. Click anywhere on Position graph. Click the Scale to Fit button, the first button,on the tool bar menu. Don’t need the graphs for Velocity and Acceleration.

7. Click anywhere on Position graph. Click the Smart Tool button, 5th button, on theTool Bar menu. The Smart Tool crosshair appears on the position graph.

8. Measure the period T of the oscillation of position: Drag the crosshair to measuretimes t1 and tN for the first peak and last peak (as Nth peak) on the positiongraph. Calculate the average period using T = (tN - t1 )/(N-1) . Calculate theexperimental free fall acceleration using gexp = yo·(2π/T)2 and the reciprocal ofmass 1/M. Record these values on Data Table 4.

9. Repeat steps 2 to 8 for 50g, 70g, 80g, and 90g masses.10. Plot the weight Mg (on the y-axis) against the displacement yo of spring (on x-

axis). Draw one best fitting straight line on the graph. The spring constant k is theslope of this line. Calculate the slope and record it under your graph.

11. Plot the gexp (on the y-axis) against 1/M (on the x-axis). Draw one best fittingstraight line on the graph. The best experimental value gexp is the “y-interceptvalue”. Determine the “y-intercept value” of gexp and record it under your graph.


PRE-LAB FORM

SIMPLE HARMONIC MOTION: SPRING-MASS SYSTEM

1. In this experiment the Motion Sensor is employed to measure the simple harmonicmotion of a mass-spring system. Graphs of position, velocity, and acceleration are shownabove. Perform basic measurement on above graphs. Obtain physical properties such asamplitude A, period T, frequency f, and maximum speed Vmax and acceleration amax andrecord them on Table 1 and 2.

Table 1: Position GraphMeasurement of Amplitude A, Period T, and Frequency f

ymax(m) ymin(m) A(m) t1(s) tN(s) N T(s) f(Hz)

Where ymax and ymin are the maximum and minimum positions (displacements) of themotion, amplitude is decided by A = (ymax - ymin )/2, t1 and tN are times for the first peakand last peak (as Nth peak) on the position graph, the average period T is calculatedwith T = (tN - t1 )/(N-1), and frequency f is 1/T.

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Table 2: Velocity and Acceleration GraphsMeasurement of Maximum Speed Vmax and Acceleration amax

Vmax(m/s)measured

V*max =A2π/T

%Difference

amax(m/s2)measured

a*max =A(2π/T)2

%Difference

Where Vmax and amax are measured from Velocity and Acceleration Graphs, V*max anda*max are calculated using the values of Table 1.

2. If a mass M = 700 g is suspended from the end of a spring, its length is stretchedby yo = 40.0 cm at the equilibrium.

(a) What is the spring constant k?

(b) What is the frequency f of this mass-spring system?


LAB REPORT FORM

SIMPLE HARMONIC MOTION: SPRING-MASS SYSTEM

Part A: Measuring A, T, f, Φ, Vmax, and amax.

Data Table 1: Position GraphMeasurement of Amplitude A, Period T, Frequency f, and Position Phase Angle Φp

ymax(m) ymin(m) A(m) t1(s) tN(s) N T(s) f(Hz) Φp(rad)

Data Table 2: Velocity and Acceleration GraphsMeasurement of Phase Angles of Velocity Φv and Acceleration Φa, Maximum Speed Vmax

and Acceleration amax

t1v(s) Φv(rad) t1a(s) Φa(rad) Vmax(m/s) amax(m/s2)

Data Table 3: Calculation of Maximum Speed Vmax and Acceleration amax

V*max = A2π/T % Difference** a*max = A(2π/T)2 % Difference**

*A and T are from Data Table 1. ** % Difference is between Data Table 2 and 3.

Part B: Measuring Spring Constant k and Free Fall Acceleration g

Data Table 4: Position Graphs with Varying MassesBottom position of unstressed spring with respect to the floor: ho = m

M(kg) h(m) yo (m) t1(s) tN(s) N T(s) gexp (m/s2) 1/M(1/kg)

0.040

0.050

0.070

0.080

0.090

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Mg(N)

yo(m)

Spring constant k from the slope: N/m


gexp (m/s2)

1/M(1/kg)

“y-intercept value” of gexp: g = m/s2

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QUESTIONS

1. Calculate the phase angle differences between position (displacement) andvelocity(∆Φpv), position and acceleration(∆Φpa), and velocity andacceleration(∆Φva) using the data from Data Table 1 and 2.

∆Φpv = Φp – Φv ∆Φpa = Φp – Φa ∆Φva = Φv – Φa

2. If the displacement x(t) of a mass-spring system is given by x(t) = A· cos(ω·t - Φ )= A· cos(2πf·t - Φ), show that Vmax = Aω = A· [k/m]1/2 and amax = Aω2 = A·[k/m].

3. Show that the free fall acceleration g is given by g = yo·(2π/T)2

where yo is the stretched length of the spring caused by a suspended mass m, T isthe period, and ω2 = (2π/T)2 = (2πf)2 = k/m.

4. Explain (a) if we ignore the spring’s mass, why equation k = mg/yo = m·ω2 resultsthat the experimental value k is less than actual value of k and (b) why equation g= yo·(2π/T)2 is a better approximation for large mass m.


Experiment 13

LANTENT HEAT AND SPECIFIC HEAT

EQUIPMENTThermometerStyrofoam cupMetal blocks (aluminum and iron)Triple beam balanceGlass beakerHot plateCrucible tongsIce chips (Pick it up when needed!)

OBJECTIVESWhen you have completed this experiment, you will be able to properly operate thethermometer, calorimeter, and hot plate. You will be able to determine the latent heat offusion for water and determine the specific heat for two metals.

Figure 1: Experimental Apparatus

CONCEPTSHeat is thermal energy in the process of transfer between a system and its surroundingsor between two systems with a different temperature. When two substances with differenttemperature are placed in thermal contact, there is heat exchange until they are in thermalequilibrium at the same temperature. The heat-exchange processes can cause eitherchanging the temperatures of the substance or changing the phase of the substance.

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As heat is added to a substance, the temperature usually rises in direct proportion, when asubstance does not undergo a change of phase. Heat gained (or loss ), ΔQ, is proportionalto the mass m of the substance and the temperature change T.

ΔQ = m•c•ΔT (1)

The proportionality constant c is called specific heat. For a same change of temperature,different substances require different amount of heat. The specific heat c is the quantity tomeasure this property of the matter. The specific heat c is the amount of heat per unitmass needed to change the temperature by one degree. The units of specific heat used inthis experiment are cal/g-C0 ( calories/gram- C0).

As heat is added to a substance, its temperature does not change during the phase changeof a substance until the phase change is completed. The three states of matters, e.g., solid,liquid, and gas, are also called phases. If enough heat is added to a solid, it melts intoliquid. If enough heat is added to liquid, it vaporizes into a gas. These are called phasechanges. If the latent heat, L, is the heat required to affect the phase change per unit ofmass, the heat for a substance to change phase with total mass m, Qp, is given by

Qp = m•L (2)

For a given substance, there are different latent-heat values characterizing different phasechanges. The latent heat associated with melting a solid is called the latent heat offusion, denoted by Lf. The latent heat associated with vaporizing a liquid is called thelatent heat of vaporization, denoted by Lv.

In this experiment we use the calorimeter to measure the latent heat of fusion for waterand the specific heats of metals. A calorimeter is a device used to produce a thermallyisolated environment to insure all heat exchanges are inside the system. In this lab, wewill use a Styrofoam cup as a calorimeter. We will ignore the heat capacities of the cupand the thermometer. If the data is accumulated fast enough, the heat exchange with thesurroundings could be neglected as well. The law of conservation of energy at thethermal equilibrium requires that the amount of thermal energy gain of one substanceequals the amount of energy loss of other sample:

Qgain = Qloss (3)

The SI unit of heat is Joule, but unit of heat based on this experiment is the calorie. Thedefinition of the calorie unit is based on the properties of water. A calorie is the heatrequired to raise the temperature of one gram of water by one Celsius degree.

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PROCEDURESPart A: Latent Heat of Fusion for Water

You will add some ice to a pre-measured amount of warm water and measure the drop inthe temperature of water. Heat from the water melts the ice and warms the melt up to thefinal common temperature. The latent heat of fusion is calculated by equating the heatexchanges between the ice and the water (see Eq. 3).

1. Weigh the empty Styrofoam cup with the triple-beam balance and record the result asMc in Data Table 1.

2. Fill the glass beaker with water until it is about half full, and place it on the hot plate.Turn the hot plate to Medium. Constantly monitor the temperature of water using thethermometer, and when the temperature of water is within the range 35-45oC, then fill theStyrofoam cup about 2/3 full with the warm water.

3. Weigh the Styrofoam cup filled with the warm water, and record the measured mass asMcw. The mass of water alone Mw can be calculated by subtracting the mass of the emptycup Mc from the mass of cup with water Mcw ( ccww MMM ). Then record the mass of

the water Mw in Data Table 1.

4. Measure the temperature of the water in the Styrofoam cup just before adding ice, andrecord this as T1 in Data Table 1.Note: Carefully handle the thermometer to avoid punching hole on the Styrofoamcup.

5. Water adhering to the ice is a major source of error. Dump about five teaspoons of icechips into the cup as quickly as possible. Swirl (instead of stir) the cup with thethermometer until all the ice has been melted.

6. Measure the final temperature of the mixture, and record this as T2.

7. Weight and record the mass of the cup with water and melted ice as Mcwi. Record themass of the ice as Mi , (Mi = Mcwi - Mcw ).

8. Calculate the latent heat of fusion for water using the following expression (which canbe deduced by combining the Eq. 1, 2, and 3)

i

wiwwf

M

TcMTTcML 221

(4)

where the specific heat of water, cw, is 1 cal/goC. You might compare the measuredvalue, Lf , with the accepted value 79.5 cal/g.

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Part B: Specific Heats of Metals

Metal block (iron or aluminum) will be heated in water to nearly 60oC. The hot metalblock will be then added to colder water in the Styrofoam calorimeter cup, and thecommon final temperature of the water and the metal block in the calorimeter is recorded.The specific heat of the metal will be calculated from the heat exchange between themetal bock and the water.

1. Weigh the empty Styrofoam cup with the triple-beam balance and record the result asMc in Data Table 2.

2. Fill the Styrofoam cup about half full with tap water at about 20oC. Then weigh thewater filled Styrofoam cup, and record the measured mass as Mcw. Find the mass of wateralone Mw by subtracting the mass of the empty cup Mc from the mass of water filled cupMcw ( ccww MMM ). Then record the mass of the water Mw in Data Table 2.

3. Measure the temperature of the water in the Styrofoam cup, and record this as Tw inData Table 2.Note: Carefully handle the thermometer to avoid punching hole on the Styrofoamcup.

4. Weigh the iron metal block and record its mass as Mb in Data Table 2.

5. Fill water in the glass beaker, about half full, and place it on the hot plate. Turn the hotplate to Medium.

6. Constantly monitor the rising temperature of the water in the glass beaker. When thetemperature reaches values in the range of 50-60oC, turn off the hot plate and leave thebeaker on the hot plate.

7. Using the crucible tongs, carefully place the metal block into the hot water in thebeaker. Leave the block in the hot water for about 5 min.Note: Occasionally stir with the thermometer.

8. Read the temperature of the hot water in the beaker and record this as Tb in DataTable2. Take the metal block out of the beaker using the crucible tongs, and put it in thecolder water in the Styrofoam cup, as quickly as practically possible.

9. Leave the metal block in the Styrofoam calorimeter cup for a few minutes, andconstantly monitor the temperature of the water in the cup. Occasionally, swirl (instead ofstir) with the thermometer until the temperature has settled to a maximum value. Recordthis as Tf in Data Table 2.Note: Do not puncture the Styrofoam cup when you handle the mixture.

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10. The energy conservation law requires that the heat lost by the hot metal block is equalto the heat gained by the colder water in the calorimeter

wfwwfbbbwaterblock TTcMTTcMQQ .

This leads to the following expression for the specific heat of the metal block

fb

wf

b

wwb

TT

TT

M

Mcc

. (5)

Use the expression above to calculate the specific heat of the metal, and record thecalculated value in Data Table 3.

11. Repeat the steps 2 to 10 for the second metal block (aluminum).

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PRE-LAB FORM

LATENT HEAT AND SPECIFIC HEAT

1. What is your explanation of heat?

2. What is a calorimeter?

3. Define the specific heat.

4. Define the latent heat of fusion and the latent heat of vaporization.

5. In an experiment 70 gram of ice at 0 0C is dropped into 300 gram of water at50 0C. Assuming that the calorimeter prevents heat flow to the surroundings,what is the final temperature of thermal equilibrium? (Ignore the calorimeter.)

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LAB REPORT FORM

LATENT HEAT AND SPECIFIC HEAT

Data Table 1: Latent Heat of Fusion for Water

Initial Temperature of Water BeforeAdding Ice, T1 [oC]

Final Temperature, T2 [oC]

Mass ofEmpty Cup

Mc [g]

Mass of Cupwith Water

Mcw [g]

Mass of Water

Mw=McwMc [g]

Mass of Cupwith Water andIce

Mcwi [g]

Mass of Ice

Mi=McwiMcw

[g]

Calculate the latent heat of fusion for water using the expression

_cal/g__________221

i

wiwwf

M

TcMTTcML ,

where the specific heat of water, cw, is 1 cal/goC.

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Data Table 2: Specific Heats of Metals

Initial Temperature ofMetal Block

Tb [oC]

Initial Temperature ofWater

Tw [oC]

Final Temperature

Tf [oC]

Mass of Block

Mb [g]

Mass of EmptyCup

Mc [g]


Mcw [g]

Mass of Water

Mw=McwMc [g]

Iron

Initial Temperature ofMetal Block

Tb [oC]

Initial Temperature ofWater

Tw [oC]

Final Temperature

Tf [oC]

Mass of Block

Mb [g]

Mass of EmptyCup

Mc [g]


Mcw [g]

Mass of Water

Mw=McwMc [g]

Aluminum

Compute the specific heat of metal using the expression

fb

wf

b

wwb

TT

TT

M

Mcc

,

where the specific heat of water, cw, is 1 cal/goC. Record the calculated value in Table 3.

Data Table 3

Metal Iron Aluminum

Standard Specific Heat [cal/goC] 0.11 0.22

Experimental Specific Heat [cal/goC]

% Error

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QUESTIONS

1. Deduce equation (4) using equations (1) to (3).

2. The standard value of Lf for water is 79.6 cal/g. What is the percent error of yourmeasurement? A major error in the determination of Lf arises because the ice isnot “dry ice”, i.e. water adheres to the ice making Mi more than the true ice mass.Explain how this makes the calculated Lf value too large or too small.

3. Referring to Data Table 3, analysis the major sources of error. How could youimprove the result? Give your suggestions.

4. Specific heat of Al is twice of that Fe.(a) If Al and Fe have same mass and the same amount of heat is added to Al and

Fe individually, discuss what will happen.

(b) If the same amount of heat is individually added to Al and Fe results in thesame temperatures increase, discuss how this can be done.

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5. A 60 gram iron at 60 0C is dropped into 200 gram of water at 20 0C. What is thefinal equilibrium temperature of the system? (Ignore the heat loss to thesurroundings.)


Experiment 14

ELECTRICAL METHOD:MECHANICAL EQUIVALENT OF HEAT

EQUIPMENTElectrical Calorimeter Heater and StirrerCalorimeter CupAmmeter and VoltmeterDC Power SupplyStopwatch, ThermometerLead Wires (5)

A

V

Heater

Figure: Photo: Experimental Setup. Insert: Circuit Diagram

OBJECTIVEUpon completion of this lab, you will be able to determine the conversion factor,mechanical equivalent of heat KJ, between the mechanical energy (in units of joules) andenergy in the form of heat (in units of calories). Upon completion of this experiment youwill be familiar with a basic electric circuit and some of the units used in those circuits.

CONCEPTSWhenever an electric current flows through a resistive material, a part of the electricalenergy will convert to heat. The electrical energy is measured with a unit of joule (J) in

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the SI system. The heat is often measured with a unit called the calorie. The conversionfactor between calories and joules, for purely historical reason, is known as themechanical equivalent of heat:

1 cal (calorie) = 4.186 J (joules)Or it can be written as

KJ = 4.186 J/calIn this electrical experiment you will reexamine the value of KJ. There are few otherbasic electrical SI units involved in this experiment:

Coulomb (C): unit for electric chargeAmpere (A) : unit for electric current, flow rate of charge, A = C/sVolt (V) : unit for electric potential, energy/coulomb, V = J/CWatt (W) : unit for power, energy/time, W = J/s

In this experiment charge flows through a resistive heater driven by a power supply, asshown in the figure, and heat energy is deposited in a heater submerged in water. Youwill measure the voltage V across and the current I through the heater for a time t. Theelectrical power, P, is

P = I·V (1)

The electrical energy expended (or the work done) in the circuit in time t is

WJ = P·t = I·V·t (2)

where WJ is measured in joules and time t is in seconds.

This energy, WJ , will become heat energy, WH, in the water. The heat generated by anelectric device is commonly called joule heat. The heat energy, WH, will also bemeasured in heat units from the heat capacity of the cup and water, C, and its temperaturechange, ∆T:

WH = C·∆T (3)

where WH is measured in calories, C is in calories per Celsius degree, and ∆T is in Celsius degree. The heat capacity of the cup and water is the sum of that for the cup(mass MC) and that for the water (mass MW):

C = MC·cc + MW· cw (4)

where cc is the specific heat of cup and cw = 1 cal/g C0 is the specific of water. Since aStyrofoam cup is used, we can neglect its heat capacity and take

C ≈ MW·1 cal/g C0 (5)

The heat capacity of the water is numerically equal to the mass of the water in the cup.The conversion factor, mechanical equivalent of heat KH, can be decided using equations(2) and (3):

KH = WJ/WH (6)


PROCEDURE

1. Record the room temperature. Record the thermometer’s least count (the smallestscale division). Calculate the uncertainty = 0.5 • (least count).

2. Weigh the calorimeter cup and record as MC.3. Fill this cup about 2/3 full of water cooled to about 10oC by adding about two

teaspoon of ice. Weigh the cup and water, and record this mass as MCW.4. Record the mass of water, MW = MCW - MC.5. Calculate the heat capacity of the cup and water using equation (5), and record

this in Data Table 1.6. Insert the heater into the cold water.7. Wire the electrical circuit, as shown in the Figure. Begin with the power supply

turned off and the black power control knob is turned to zero. Connect the redterminal of the power supply to the top red terminal of the ammeter and theammeter’s black terminal to one connector of the heater. Connect the other sideconnector of the heater to the black terminal of the power supply. Connect thevoltmeter across the power supply: red to red and black to black.

THE HEATER MUST BE SUBMERGED BEFORE APPLYING POWER

8. Set the voltmeter’s scale to the 25 V scale and the ammeter to the 10 A scale. Todetermine the current in amps, multiply the ammeter’s needle indication by 10.

Note: Please let your instructor check your circuit before turn on the powersupply.9. Turn on the power supply, and slowly crank up the power knob until the product

of the voltage and current readings is about 30 W (The voltmeter will indicateabout 10 V.).

10. Start your Stopwatch when the voltage is about 10V.11. Immediately measure the temperature inside the calorimeter cup. Record the

starting temperature in Data Table 2. Note: If it has risen above 15 oC, then youmust prepare it again with cold water and reweigh it.

12. When power is applied, remember to stir the water continuously to distribute theheat in the water. Record the voltage and current. The voltage and current shouldremain constant from the starting temperature to the ending temperature.

13. Continue stirring till the temperature has reached about 10 oC above roomtemperature (the End Temperature). Stop your Stopwatch and record the time as itreaches this temperature.

14. Turn off the power supply. Put the lab equipments back into their proper places.15. Complete all of the calculations.

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PRE-LAB FORM


1. When the electrical energy in joules is completely converted to heat in calories,(a) what is the ratio between these two?

(b) what is the designed name for this ratio?

2. An ammeter is a device to measure the electrical current in a circuit. A voltmeteris a device to measure the electrical potential difference between two points in acircuit. What are units for electrical current and electrical potential?

3. Explain the difference between heat capacity and specific heat.


4. If the electric current passing a heater is 3.0 A, the voltage cross the heater is10.0V, and the heater is turned on for 15 minutes, calculate

(a) the dissipating power and

(b) the energy consumed by the heater.

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LAB REPORT FORM


Data Table 1: Cup and Water DataMC

(g)MCW

(g)MW = MCW - MC

(g)cW

(cal/gC0)C= MW ∙cW

(cal/C0)

1

Data Table 2: Temperature DataRoom Temp

(C0)Least Count ofThermo. (C0)

UncertaintyδT (C0)

Starting Temp.Ti (C0)

Ending Temp.Tf (C0)

Data Table 3: Voltage, Current, and Time DataVoltageV (V)

CurrentI (A)

PowerP=I·V (W)

Timet (s)

CALCULATIONS

1. Show the calculation of the energy WJ =P∙t in joules.

2. Show the calculation of the energy WH = Mw∙cw∙∆T =C∙∆T in calories.


3. Show the calculations of(a) the conversion constant of calories to joules, KH = WJ/WH and

(b) the % error from the standard value, 4.186 J/cal.

QUESTIONS

1. The experiment was performed using a temperature range that started and endedabout 10 oC below and 10 oC above room temperature. Explain how thistechnique minimizes the net transfer of heat energy to or from the room.

2. What are the major sources of error in the experiment?

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3. If the cost of electricity is $0.106/(kWh), determine the cost of electricity for thisexperiment.

4. Assuming that you use the same lab equipment, but electric current is 2.8A,voltage is 11V, starting temperature is 10 oC, ending temperature is 40 oC, andmass of water is 320g. How much time does it take to complete this experiment?

Date post:	22-Oct-2015
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DCCCD Phys2425 Lab Manual Fall

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