Physics in the Laboratory - · PDF filePhysics in the Laboratory ... Object with displacement...

Physicsin the LaboratoryApplied Physics Second EditionFall Quarter1997

Robert Kingman

The author expresses appreciation to the Physics faculty and many students who havecontributed to the development of the laboratory program and this manual. Specialrecognition is acknowledged to professor Bruce Lee for the many years that he taught theGeneral Physics course and the introductory laboratories. Professors Margarita Mattingly andMickey Kutzner have made substantial contributions to the laboratory and to individualexperiment instructions. This manual has become a reality because of the efforts of JosephSoo and Tiffany Karr for rewriting, editing and taking and including the photographs and theoutstanding editorial assistance of Anita Hubin.

Copyright © 1996 by Robert Kingman

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Preface

It is the purpose of the science of Physics to explain natural phenomena. It is in thelaboratory where new discoveries are being made. This is where the physicist is makingobservations for the purpose of identifying the patterns which may later be fit to mathematicalequations. Theories are constructed to describe patterns observed and are tested by furtherexperiment. Therefore, it is imperative that students in an introductory Physics course areintroduced to both the existing theories in the classroom and to the ways of recognizingnatural patterns in the laboratory. In addition, the effort put into the laboratory experimentswill ultimately reward the student with a better understanding of the concepts presented inthe classroom.

This manual is intended for use in an introductory Physics course. Prior experience in Physicsis not a requirement for understanding the concepts outlined within. The book was writtenwith this in mind and therefore every experiment contains a Physical Principles section whichoutlines the basic ideas used. The analysis of the data may be done on the computer with theaid of Science Workshop and Graphical Analysis programs . These tools are very useful inthe generation of graphs and curve fitting. . It is important to keep in mind that the best wayto become familiar with new software is to use it a lot, trying more than the minimumrequired for the completion of the laboratory.

The student is required to keep a laboratory journal in which the raw data will be recordedas well as the analysis, any graphs and calculations. The lab write-ups need to be done in ink.A good report should contain the date and time the experiment was performed, the title of theexperiment, the name(s) of the partners, the objective(s), a sketch of the apparatus properlylabeled, a brief summary of the procedures, all the performed analysis and a conclusion. Thepurpose of the conclusion is to allow the student to comment on the experiment. In addition,a discussion of the errors and where they might have been introduced, suggestions formodifying the experiment to reduce the possibility of errors and overall suggestions forimproving the experiment need to be addressed here.

In conclusion, we hope that the experiments in this manual will enhance your understandingof the concepts presented in class and will add pleasure to your journey through this excitingfield of Physics. Any comments you may have about the laboratories presented in this bookare welcomed and encouraged. Since this is a first edition, we hope that you will overlookany missspellings, omis ions, errors and inconsistencies and report such to the author.

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Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Experiment 1 Uniform Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Experiment 2 Uniform Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Experiment 3 Vector Addition of Forces . . . . . . . . . . . . . . . . . . .. . . . . . . . . 13

Experiment 4 Force and Acceleration - Newton’s Second Law . . . .. . . . . . . 23

Experiment 5 Conservation of Mechanical Energy . . . . . . . . . . . . . . . . . . . . 29

Experiment 6 Inelastic and Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . 33

Experiment 7 Torque and Angular Acceleration . . . . . . . . . . . . . . . . . . . . . . 39

Experiment 8 Conservation of Energy of a Rolling Object . . . . . . . . . . . . . . 45

Experiment 9 Rotational Equilibrium - Torques . . . . . . . . . . . . . . . . . . . . . 49

Experiment 1 Page 1

Figure 1: Object with displacement +2 from origin.

x ' v @ t%xo (1)

Applied Physics Experiment 1

Part I: Uniform Motion - Graphing and Analyzing theMotion

Objectives:

< To observe the distance-time relation for motion at constant velocity.< To make a straight line fit to the distance-time data.< To interpret the slope as the velocity of the motion.< To observe that the average mean square error is smallest for the closest fit.

Equipment:

< Motion sensor< Pasco 1.2 m track and dynamics cart< Computer with Signal Interface, Science Workshop and Vernier Graphical Analysis

software

Physical Principles:

The position of an objectmoving along a line is indicatedby its displacement. Thedisplacement is ±1 times thedistance of the object from areference point called the origin,the numbers being positive on one side of the origin and negative on the other side. Denotingthe displacement as x and the time as t

In a graph of x (on vertical axis) versus time (on horizontal axis) the velocity of the motionv is equal to the slope of the line. The initial position, the location at the beginning when timeis zero, is xo . This value is where the line crosses the vertical axis and is called the intercept.The best fit of a straight line to a data set is the one with the smallest value of the averagesquare deviation.

Experiment 1 Page 2

Figure 2 Slope, v, and intercept, xo.

v ' slope ' riserun

')x)t

'x2&x1

t2& t1(2)

The slope is given by

It is often possible and convenient to take x1 and t1

to be zero.

Predictions:

Draw a rough graph in your journal of what youthink the motion will be. Plot the displacement xversus the time t. Do this when the cart starts at aninitial position of 50 cm and travels for a time of 2seconds at a speed of 50 cm/s. Will the curve be a straight line or a curved line? If it isstraight will it slope up or down. If it is a curved line will it curve up or down? Explain whyyou think it will behave this way. Do this for two cases and label the graph for each. Thetwo cases are: 1. Motion toward the origin, 2. Motion away from the origin.

Procedure:

Setup:Plug the motion sensor’s phone plugs into digital channels 1 and 2 with the yellow bandedplug into channel 1. Place the motion sensor about 40 cm from the end of the track oppositethe bumper with the center of the sensor about 12 cm above the track. Align the sensor sothat the sound waves will travel directly along the track. Place the cart on the track at the endnear the sensor.

Data Collection:Double click the left mouse button on the physics labs folder to open it if necessary (it isusually open). Double click on the scwkshp icon in the folder to open Science Workshop.See Figure 3 below.Click and drag the phone plug icon to digital channel 1, choose Motion Sensor. Click on the REC button and at the same time push the cart away. Wait until data collectionstops.Drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2.Click on the rescale icon (fourth from the left in the lower left of the graph window).Drag the Table icon onto the Motion Sensor icon.

Experiment 1 Page 3

Figure 3 Science Workshop window.

Click on the clock to the right of the E at the upper left of the Table window to display thetimes. Click just above the time-distance data columns to select all of the data or click and drag toselect the portion of the data that is valid.Under the Edit menu, choose copy to store the data temporarily in the Window’s “clipboard”.

Graphing Data:Double click on the VernierGA icon in the physics folder to open the graphing analysisprogram, click on OK and click on the restore (upper right center icon).Click on the row 1, x data position. Under the Edit menu option choose paste data to copyyour data from temporary storage in the “clipboard”.

Analyzing Data:Note that the displacement is plotted vertically (y-axis) and the time data is plottedhorizontally (x-axis).Click on the graph of your data on the right to select the graph.Choose Analyze from the main menu and click on Manual Curve Fit.Select the Stock function M*x + B to select a linear (straight-line) model. (According to Eq.(1), the “x” here corresponds to your time values, the M corresponds to your velocity values,and B corresponds to your beginning location xo ).

Experiment 1 Page 4

Figure 4 Graphical Analysis window.

Change the values in the intercept box B = at the lower left and the slope box M = to vary theintercept and slope of the model line.Note values of the Mean Square Error at the lower right of the graph for each value of slopeM and intercept B.Do this until the model line visually fits most closely to the data and then make furtheradjustments until the Mean Square Error is as small as possible.Record the values of the slope M, intercept B and the Mean Square Error.Click on OK-Keep Fit.In the Main menu click on Analyze and choose Automatic Curve Fit, click on the Stockfunction M*x + B and click on OK. Click on OK-Keep Fit and record the values of the slopeM, intercept B and Mean Square Error.Click on the Linear Regression icon (the rightmost icon under the Data menu) to obtain againa linear fit to the data. Record the regression coefficient. A value close to one indicates aclose fit to the line.Compare these values with those obtained in your manual fitClick on the graph title and change the title to Displacement versus Time.Click in the text window and enter your name, experiment name, date and experiment details,ie motion away from detector.Choose File in the main menu, then Print, click on Selected Display and click on OK. How does your observed curve compare with your predicted curve? What is the speed of thecart? How far from the detector is the cart when the detector begins measuring its motion.What does the value of the Mean Square Error indicate?

Motion in opposite direction:

Experiment 1 Page 5

Return to the Science Workshop window and repeat the experiment placing the cart on theend opposite the motion sensor and pushing it toward the sensor. At the same time click onthe REC button. Repeat the analysis above.

Part II: Walking Motion - Distance Versus Time Graphs

Objectives:

< To observe the distance-time relations for a variety of walking motions.< To determine from the slope of the distance versus time graph the velocity of the

motion at various points.

Equipment:

< Motion sensor and reflector board< Computer with Signal Interface, Science Workshop and Vernier Graphical Analysis

software

Predictions:

A General Physics student stands in front of a motion detector for three seconds. Then thestudent backs slowly away from the detector at constant speed for a time of four secondsreaching a distance of two meters from the detector. The student stops for five seconds andthen walks toward the detector with constant speed for six seconds reaching a distance of .6meters from the detector. Finally the student stands at this point for two seconds. Draw arough graph of what the distance versus time and velocity versus time graphs would look like.Title the graph and label the axes indicating distances in meters and times in seconds.

Procedure:

Mount the motion sensor on a rod so that it is about five feet above the floor. Align thesensor so that you can walk away from the sensor to a distance of two meters using thereflector to send the ultrasound waves back toward the sensor. Direct the sensor along thepath that you will walk.

Select the Science Workshop window to activate it.Click on the REC button while your lab partner tries to duplicate the motion that you drewin the prediction section. Click and drag the Graph icon onto the Motion Sensor icon belowdigital channels 1 and 2. Click on Position and then click on Velocity to display these graphs.Click on the rescale icon (fourth from the left in the lower left of the graph window).

Experiment 1 Page 6

Click and drag the Table icon onto the Motion Sensor icon. Click on the clock to the rightof the E at the upper left of the Table window to display the times. Click just above the time-distance data columns to select all of the data or click and drag to select the portion of thedata that is valid. Choose Edit and Copy to copy this data to the “clipboard.” Click on the Vernier Graphing window to activate it. Click on the row 1, x data position,and Paste the data from the “clipboard.” To change the label for the X axis to Time click onGraph, click on Column Appearance and double click on X. Type Time in the New Namebox, click on the New Units box and enter seconds. Click on OK to accept these labels. Inthe same way change the Y label to Displacement with the units of meters.

Click and drag (right on the graph!) to select a region where the velocity is approximatelyconstant. Click on the Linear Regression icon (seventh from the left in the tool bar) todetermine the speed in that region. Record the velocity from the slope listed as M = ....Repeat this for other regions of interest. To observe the slope at each point click on the slopeicon (fifth from the left on the tool bar). Move the mouse along the curve to observe thechanging instantaneous velocity (slope at a point).

Click in the text window and enter your name, experiment name, date and experiment details,ie motion away from detector.Print the Selected Display.Compare your actual walk with what you had drawn in the prediction section.If you have time make your own walk plan, draw a rough graph of it and repeat theexperiment. You might wish to try moving with a constant acceleration as well. What canyou conclude about how the sensor is responding? Why is the velocity curve not smooth?

Experiment 2 Page 7

x ' xo%vo@t%12

a @ t 2(1)


Part I: Constantly Accelerated Motion - Distance,Velocity, and Acceleration Versus Time

Objectives:

< To observe the distance-time, velocity-time, and acceleration-time relations for a cartmoving up and down an inclined track.

< To determine from the slope of the distance versus time graphs the velocity of the cartat various points.

< To determine from the slope of the velocity versus time graphs the acceleration of thecart at various points.

< To compare distance traveled by the cart with the area under the distance-time graph.

Equipment:

< Motion sensor and reflector 2x4< Pasco dynamics track and cart< Lab jack< Computer with Signal Interface, Science Workshop and Vernier Graphical Analysis

software


The position of an object moving along a line is indicated by its displacement. Thedisplacement, x, is ±1 times the distance of the object from a reference point called the origin,the numbers being positive on one side of the origin and negative on the other side.

When the velocity changes in time (acceleration) the graph of x versus t is no longer a straightline. However the instantaneous velocity of the motion, v, is equal to the slope of the tangentline at that time. The initial position, the location at the beginning when time is zero, is xo .For a constant acceleration the relation between x and t is

where a is the acceleration and vo is the initial velocity.

The relation between velocity, v, and time is

Experiment 2 Page 8

v ' vo%a @ t (2)

Figure 1 Graph of velocity vs. time.

which is the equation of a straight line with a slope, a, and intercept, vo.

Predictions:

A physics student pushes the dynamics cartup the inclined track and observes itsdistance-time motion with a motion sensor.Draw a rough graph of what the distanceversus time and velocity versus time graphswould look like. Title the graph and label theaxes indicating distances in meters and timesin seconds. Is the velocity zero at any point?Is the acceleration zero at any point?

Procedure:

Setup:

Plug the motion sensor’s phone plugs into digital channels 1 and 2 with the yellow bandedplug into channel 1. Place the 2x4 reflector upright on the dynamics cart and secure it witha rubber band. Elevate one end of the track by placing the lab jack under one end. Set thelab jack to its lowest position. Place the cart on the low end of the track. Mount the motionsensor on a stand so that it is about eight inches above the table top. Align the sensor so thatit is directed down along the track and toward the 2x4 reflector on the cart.

Data Collection:

Double click the left mouse button on the physics labs folder to open it if necessary (it isusually open).Open Science Workshop.Click and drag the phone plug icon to digital channel 1, click on Motion Sensor and then OK.Click on Sampling Options, set the sampling rate to 10,000 Hz and the sampling time to 3seconds.Press RETURN and click on OK to accept these values.

Click on the REC button while your lab partner gives the cart a quick thrust up the track. BECAREFUL NOT TO SEND THE CART OFF THE TOP END OF THE TRACK. Click and drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2.

Experiment 2 Page 9

Click on Position, then click on Velocity and click on Acceleration to display these graphs.Click on the rescale icon (fourth from the left in the lower left of the graph window).Click on the Restore icon in the upper right of the Science Workshop window to fill yourcomputer screen with the Science Workshop window.

Acceleration from a quadratic fit to the distance-time data

Click and drag to select a region where the distance-time data is in the shape of a parabola.Do not include the points near the top where the curve begins to flatten. Click on the E tothe right of this graph and drag the mouse down to Curve Fit. Then click on Polynomial Fit.The constant a3 should be ½ of the acceleration as you can see from Eq. (2). Record thisvalueof a3 and multiply it by 2 to obtain the acceleration from this fit.

Acceleration from the slope of the velocity-time data

Click and drag to select a region of the velocity-time graph where the velocity isapproximately constant. Click on the E to the right of this graph and drag the mouse downto Curve Fit. Then click on Linear Fit. Record the acceleration from the slope listed as a2

= ... and note that it is from the slope of the velocity-time graph.

Acceleration from the mean of the acceleration-time data

Click and drag to select a region of the acceleration-time graph where the acceleration isapproximately constant. Click on the E to the right of this graph and drag the mouse downto Mean. Record the acceleration from the mean of the y data and note that it is from theslope of the velocity-time graph. Compare the three values for the acceleration that youobtained from the three graphs. Compare the graphs that you obtained with those that youdrew in your predictions. Click on the lower left icon in the graphing window and change thetitle of the graph to Distance, Velocity, and Acceleration vs Time by (enter your name) andpress RETURN. Click on File and then Print to print the Graph window. What are theanswers to the questions in the predictions section?

Comparison of the velocity to the slope of the distance-time graph

Click and drag to select about three data points on the distance-time graph. Click on the Eto the right of this graph and drag the mouse down to Curve Fit. Then click on Linear Fit. Record the velocity from the slope listed as a2 = ... and the time at the midpoint of the smalltime interval. This is an approximation to the slope of the distance-time curve at themidpoint. Click on the exam icon nest to the E at the lower left of the Graph Displaywindow and move the cross hair so that it is on the velocity-time graph at the midpoint time.Record the velocity and time values and compare the velocity value with that obtained fromthe slope of the distance-time graph at that time.

Experiment 2 Page 10

Comparison of distance traveled to the area under the velocity-time graph

Click on the zoom icon (bottom left) then click and drag on the velocity-time graph to zoomin on a region that includes all the positive velocity data. Click and drag on the positivevelocity data, then click on the E at the right and click on Integration. The area under thisregion of the velocity-time graph is displayed. Click on the cross-hair examine icon at thebottom left and move the mouse so that the cursor is at the left edge of the gray shadedregion in the velocity-time graph and the cross is on the distance-time graph. Read the initialposition from the distance (y) axis and record this value in your journal. Repeat this processat the right edge of the gray shaded region. Compare the difference of these two values, thedistance traveled with the area under the curve displayed as area = ... .

Applied Physics Experiment 2 Part II

Uniform Acceleration - The Acceleration of Gravity

Objectives:

< To test the hypothesis that the acceleration of gravity is approximately constant andto measure its value.

Equipment:

< Timer photogate< Motion Sensor< Computer with Science Workshop and Vernier Graphical Analysis software< Ball

Physical principles:

A specific case of the equation (1) is the free fall of a body initially at rest. Since the body isinitially at rest, vo becomes zero. If we assume that do is also zero, the equation becomes

d' 12@g @ t 2

(3)

where the acceleration is now symbolized by g, the acceleration of gravity.

Velocity can be found as a function of time according to equation (2).


%Err '|g&gstandard|

gstandard

×100% .

v'vo%g t (4)

It is obvious from equation (4) that on a graph of velocity vs. time, the slope represents theacceleration. In this experiment such a method is utilized to estimate the value of g, theacceleration of gravity.

Procedure:

The Acceleration of Gravity with a Motion Sensor and Falling Ball

Plug the motion sensor into channels 1 (plug with yellow band) and 2 on the signal interfacebox. The sensor should be attached to a stand 1b meters above the floor. On the screen,click on the digital plug icon and drag it over digital channel 1 of the signal interface box,select motion sensor and click on OK. Enlarge the graph to the desired size. Now click onthe recording options button, make sure that periodic sampling rate is set to 10,000 Hz andstop condition to time-2seconds, then click on OK. Hold the ball directly beneath the motionsensor so that it is almost touching it. Release the ball and as it is falling click on the recordbutton. In the graph display window click on the rightmost button of the upper row ofbuttons in the bottom left hand corner of the window to make the graph zoom in on the data.You will notice that there are a few regions where the graph increases steadily in a smooth,sloped line. Highlight one of these regions by clicking at one end of it and dragging to theother, creating a box around it. Next click on the button with the E symbol on it. A new boxwill appear on the right side of the graph window. Click on the E button in this window,then select curve fit, then linear fit. The acceleration of gravity will be given by this curve fit’sslope (a2). Record your value for the acceleration due to gravity and compute the percentageof error using g = 9.81 m/s2 as the standard:



Figure 1 The sides of a right triangle.


Vector Addition of Forces

Objective:

< To test the hypothesis that forces combine by the rules of vector addition and that thenet force acting on an object at rest is zero.

Equipment:

< Five spring balances< Five 1-2 kg weights used for anchors< Small washer< Large sheet of paper< Ruler, protractor, right triangle< Scientific calculator (with sin, cos & tan functions) or Mathcad

Physical principles:

Definitions of Sine, Cosine, and Tangent of an Angle

Consider one of the acute (less than 90E) angles, a, of the right triangle shown in figure 1.As a result of where they reside, thethree sides of the triangle are calledthe opposite side, adjacent side andhypotenuse. The two sides that makeup the right angle (exactly 90E) arealways the adjacent side and theopposite side. As a result, the lengthof the hypotenuse is always greaterthan the length of each of the othertwo sides but less than the sum of thelengths of the other two sides. Thesize of the angle a can be related tothe length of the three sides of theright triangle by the use of thetrigonometric functions Sine, Cosine and Tangent, abbreviated sin, cos and tan, respectively.They are defined as shown below.


Figure 2 Vector addition by the polygonmethod.

Figure 3 Finding the two perpendicularcomponents of a vector.

sin(a)' oppositehypotenuse

cos(a)'adjacent

hypotenuse

tan(a)'oppositeadjacent

(1)

Vector Addition

Polygon method - Vectors may beadded graphically by repositioningeach one so that its tail coincideswith the head of the previous one(see figure. 2). The resultant (sum ofthe forces) is the vector drawn fromthe tail of the first vector to the headof the last. The magnitude (length)and angle of the resultant ismeasured with a ruler and aprotractor, respectively. Note: Inorder to measure the angle, a set ofaxes must first be defined.

Component method - Vectors maybe added by selecting twoperpendicular directions called the Xand Y axes, and projecting eachvector on to these axes. This processis called the resolution of a vectorinto components in these directions.If the angle a that the vector makesfrom the positive X axis, is used (seefigure 3), these components are givenby

Fx'F @cos(a)

Fy'F @sin(a)(2)

The X component of the resultant is the sum of the X components of the vectors being added,and similarly for the Y component.


Figure 4 Sample setup of three forcesacting on a small washer at equilibrium.

Rx'j Fx

Ry'j Fy

(3)

The angle that the resultant makes with the X axis is given by

a'arctanRy

Rx(4)

and the magnitude is given by

R' R 2x %R 2

y (5)

Equilibrium Conditions

Newton's second law predicts that a body will not accelerate when the net force acting on itis zero. So, for an object to be at rest, the resultant force acting on it must be zero. Inequation form, the above statement can be written

j PF'0 (6)

Thus, if four forces act on an object at rest, the following relationship has to be satisfied.

PF1%PF2%

PF3%PF4'0 (7)

An equivalent statement is

PF4'&( PF1%PF2%

PF3 ) (8)

so that is equal in magnitude and opposite in direction to the resultant of the other threePF4

forces.

Procedure:

Set up the following situations so that ineach case the magnitudes of the forces areunequal.

1. Attach three strings about 12 cm longto the small washer and connect the otherend to spring balances, to the endconnected directly to the center forceindicating shaft. Connect string loops,about 8 cm long, to the other end of the


Figure 5 Procedure 1.a) setup.

Figure 6 Procedure 2 setup.

spring balances and wrap these loops around the 1-2 kg weights (see figure 4). You will needto make sure that when there is no load on the spring balance the scale reads zero. If it doesnot, you will need to adjust it by sliding the metal tab at the top of the device.

a) Move the weights so that the anglebetween forces F1 and F2 is 90E (seefigure 5). On a paper (as large as .3by .3 m, if possible) draw linesparallel to its edges and intersectingnear its center. These lines will act asthe X and Y axes, described in thePhysical Principles section. Positionthe paper so that the origin of theaxes is right under the small washer,with the forces F1 and F2 along thetwo lines. Tape the sheet of paper tothe table. Use a pencil to mark twopoints at opposite ends of the string supplying the force F1. By connecting these two points,draw a line below the string showing the direction of the force. Following the sameprocedure, draw the direction of the other two forces. Record the weight on each string inNewtons. For those spring balances calibrated in grams, convert the scale readings bymultiplying by 9.80*10-3 N/g. Place arrows on your lines in the direction of the force exertedby the spring balances. Select your X axis to be along the line of force F1. Add the vectorsfor F1 and F2 both graphically (polygon method) and with trigonometry (component method).Compare the magnitude of the resultant with that of the force, F3 for both solutions. Usinga protractor, measure a3 and compare it with the similarly measured angle of your graphicaladdition and your trigonometrically computed angle. Do your measurements satisfy therequirements of Newton's second law?

b) Repeat as outlined in part (a) using the component method only, but with the anglebetween F1 and F2 at about 120E. Do your measurements satisfy the requirements ofNewton's second law?

2. Repeat step 1a, using only thecomponent addition method with 4spring balances (see figure 6). Drawthe forces F1, F2, F3, and F4

approximately as illustrated. Find andadd the components of F1, F2, and F3.Compute the magnitude and directionof the sum of these forces andcompare your result with a4 and F4.Do your measurements satisfy the


Figure 7 Procedure 3 setup (extra credit).

requirements of Newton's second law?

3. If time permits, for extra credit,repeat as in step 2 using 5 forcesextended approximately as illustratedin figure 7. Do your measurementssatisfy the requirements of Newton'ssecond law?



Recording data:

Part 1a. Table 1 Polygon Method

Force Magnitude (N) Angle (E)

Force 1

Force 2

Force 3

Resultant of 1 & 2

Table 2 Component Method

Direction Force 1 Force 2 Resultant

X

Y

Magnitude of resultant = _______________________

Angle of resultant = ___________________________

Part 1b.Table 3 Component Method

Direction Force 1 Force 2 Resultant

X

Y





Part 2Table 4 Component Method

Direction Force 1 Force 2 Force 3 Resultant

X

Y



Part 3 (Optional - Extra credit)Table 4 Component Method

Direction Force 1 Force 2 Force 3 Force 4 Resultant

X

Y





Figure 1 Free body diagram of the hangingmass.


Force and Acceleration - Newton's Second Law

Objective:

< To observe the relationship between force and acceleration and to test the hypothesisthat the force is equal to the mass times the acceleration.

Equipment: < Track with cart, accessory weights< Smart pulley timer< Table clamp< Triple beam balance < Slotted weights, one 10 g, and three 20 g Physical principles: A net force, F, applied to an objectwith a mass, M, will cause the mass toaccelerate with an acceleration, a.Newton's law of motion asserts thatthe net force is directly proportional tothe acceleration produced. Theproportionality constant is denoted bythe inertial mass, M. In equation form,this law can be written as

F'M @a (1)

When an object with a mass M, on a smooth horizontal surface, is connected by a string overa pulley to another mass m, a tension is created in the string. This tension is the force thataccelerates the object on the surface. From the free body diagram shown in figure 1, it can bededuced that the total force acting on the mass is the tension in the string minus the force ofgravity. Assuming that the mass of the hanging weight is m, and its acceleration is a, thefollowing equation can be written. The acceleration of gravity is symbolized by g.

m @g&T'm @a (2)


Figure 2 Free body diagram of the cart.

Figure 3 Smart pulley.

Figure 4 Cart on the air track.

Equation (1) can be solved for tensionto yield the following equality.

T'm @ (g&a ) (3)

Since the tension is constant in thestring, the object on the surface and themass hanging on the string have thesame acceleration. Thus, Newton’slaw of motion for the object on thesurface is

T'M @a (4)

Procedure: Place the cart on the track and level the track sothat the cart does not accelerate in either direction.With a table clamp, position the smart pulley (seefigure 3) at the aisle end of the track. Measure andrecord the mass of the cart, Mcart . Measure andadd the masses of the two blocks to the mass ofthe cart. Record the total mass, Mtotal. Connect a string to the paper clip or wire loop onthe front of the cart and place it over the pulley atthe end of the track. Make a loop on the other endof the string and slip a 10 g slotted weight into it.The length of the string should keep the massabout 5 cm above the floor when the cart is at thetrack bumper. Following the procedure below,obtain the value of the acceleration.

Run Science Workshop.Plug in the smart pulley on the screen by clickingon the digital plug icon, dragging it over digitalchannel 1 and selecting smart pulley (linear). Clickon OK.Click on the recording options button, set periodicsamples to 10,000 Hz, and click on OK.Make a graph of velocity versus time by clicking


on the graph icon, dragging it over the smart pulley icon, selecting velocity, and clicking onOK.Start statistics by clicking on the E button in the graph window, then click on the E button inthe new window annd select curve fit and then linear fit.Position the cart so that the small slotted weight is just below the smart pulley.Release the cart, click on the REC button, when click on Stop just before it reaches the endof the track.

Record the value of the slope (a2) from the statistics section of the graph window in thecolumn entitled a in Table 1. This value represents the acceleration of the cart system. Takea series of seven (7) more measurements each time increasing the mass at the end of the stringby 10 g. Be sure to delete the previous run between measurements by clicking on Run #1,pressing the delete key on the keyboard and clicking on OK. Record the acceleration for eachmass in Table 1.

Analysis of Data:

Complete column 3 of both tables by calculating the values for g-a. For the value of g use theaccepted value of 9.81 m/s2. Compute the values for T by using equation (3). Plot a graphof tension T vs. acceleration a using Mathcad. Use the slope() function of Mathcad tocompute the slope of the best fitting line. Use the corr() function to find the closeness of thefit. As equation (4) predicts, this slope should be very close to the mass of the cart (Mcart) inthe case of Table 1. In the case of Table 2 the slope should be very close to the total mass(Mtotal). Calculate the percent error for both cases by using

%Err' *slope&M*M

×100 (5)

In your conclusions you should:

Discuss the percent error that you calculated for both graphs.Interpret the value of the Correlation Coefficient.Examine how the presence of constant frictional force would affect the results of theexperiment.Speculate on the origin(s) of error.Mention what you learned in this experiment.Include any additional comments that you think are essential.



Recording data:

Mcart = __________________ Mtotal = Mcart + Mblocks = _______________________

Table 1 Cart without additional mass data

m a g-a T

10g

20g

30g

40g

50g

60g

70g

80g

Slope of the Tension vs. Acceleration line = __________________ %Err = _________________

Table 2 Cart with two blocks data

m a g-a T

10g

20g

30g

40g

50g

60g

70g

80g

Slope of the Tension vs. Acceleration line = __________________ %Err = _________________




Conservation of Mechanical Energy

Objective:

< To measure kinetic and potential energies of a mass suspended vertically on a spring,and to test the hypothesis that mechanical energy is conserved for a system free offrictional forces.

Equipment:

< Table clamp, 2m rod, 30 cm rod, right angle clamp< Triple beam balance< Computer with Science Workshop, and Pasco interface box< Ultrasound motion sensor< Hooked masses, 1 kg, .1 kg< Tapered spring< Force transducer< IP 18 power supply with red and black 1.5 m long banana leads


Kinetic Energy

A body which has a mass m and moves with a speed v has energy by virtue of its motion. Thisis energy is called kinetic energy and is given by

KE'12

m @v 2(1)

Potential Energy

A body moving in a force field has energy by virtue of its position. This energy is calledpotential energy. The potential energy of an object at a point B with respect to a point A isthe work (component of force along displacement multiplied by the displacement) which mustbe done to move the object from A to B.

In the case of a spring, the force produced is the product of a constant k and the amount bywhich the spring is stretched. Mathematically, this relation can be written as


Figure 1 Plot of spring force versusdistance stretched.

F'k @x (2)

The work done by a force is equal to thearea under the force versus distance graph.For a spring, this is a straight line throughthe origin (see figure 1) so that the workdone is the area of the shaded triangle. Thiswork done represents an energy called thepotential energy PE that is stored in thespring. Mathematically, the PE of a springcan be written as

PE'12

x @F'12

x(k@x)

PE'12

k @x 2(3)

Energy Balance

When no frictional forces are present

KE%PE'E'constant (4)

where E is a constant called the total mechanical energy. When an object is supported by avertical spring, it will come to rest at an equilibrium position. If it is moved up or down fromthis position, it will oscillate. At the top and bottom of its motion, it will be at rest and willhave zero kinetic energy. At these two points its energy is all in the form of potential energy.At the equilibrium point x = 0 and the potential energy is zero. The object is moving fastestthere and all its energy is in the form of kinetic energy. As the object moves the spring doespositive or negative work causing the kinetic energy to change. The sum of the kinetic andpotential energies of the object is constant and is equal to the total energy of the object. Theeffect of frictional forces is to cause the total energy to gradually decrease.

Procedure:

Use a triple beam balance to measure and record the mass of the spring ms and the mass of theobject m (this mass should be 1 kg). Record the effective mass, me = m + ms/3.

Mount the force transducer at the top of the 2 m rod using a right angle clamp. Mount the 2m rod on the table near one of the aisle corners using a table clamp. Rotate the force


Figure 2 Setup of theapparatus.

transducer so that the transducer beam is horizontal. Run Science Workshop and plug theforce transducer into analog channel A on the signal interface box. Plug it in on the screen aswell by clicking on the analog plug icon, dragging it over analog channel A, selecting voltagesensor, and clicking on OK.

Calibrate the force transducer. To do this, double click onthe force transducer icon. Set the low value to 0 and clickon the Read button. Hang a 1 kg hooked mass (weight of9.8N) on the transducer, set the high value to 9.8, click onthe Read button, then click on OK. Remove the 1 kgmass from the force transducer. Raise the 2 m rod as highas possible. Place the hook at the small end of the taperedspring over the S hook of the force transducer and supportthe 1 kg mass from the hook at the large end of thespring. Lower the mass to a point near the equilibriumposition of the spring and release it. Directly under themass position the ultrasound motion sensor (see figure 2). Plug the motion sensor into channels 1 (yellow bandedplug) and 2 on the signal interface box. Also plug it in onthe screen by clicking on the digital plug, dragging it overdigital channel 1, selecting motion sensor, and clicking onOK. Click on the recording options button, set the periodicsamples rate to 1,000 Hz, the stop condition to time: 5seconds and click on OK. Displace the 1 kg mass about .2m, and release it. After several oscillations, click on REC to begin data collection.

After data collection has stopped, make graphs of distance versus time and velocity versus timeby clicking on the graph icon, dragging it over the motion sensor icon, selecting position orvelocity and clicking on OK. Make a force versus time graph by clicking on the graph iconand dragging it over the voltage sensor icon. Then make a force versus distance graph by firstmaking a force versus time graph and then clicking on the clock below the graph and selectingdigital 1 and then position.

Next calculate the average displacement of the mass. To do this, click on the position versustime graph and click on the E button. Record the value given for mean (y =) as dave.

Next find the value of k by finding the absolute value of the slope of the force versus distancegraph. Click on the E key and then the newE key. Select curve fit and linear fit. The slopewill be displayed as a2.

Now open up the calculator window which can be found in the Experiment menu. Enter theequation for kinetic energy using the mass and velocities that you measured. To get thevelocities for the calculation, click on the input button and select digital 1 and velocity. Enter


kinetic energy for the calculation name, KE for the short name, and click on the = button.Click on the NEW button and enter the equation for potential energy, using the values formass and distance that you measured. To get distance, click on the input button and selectdigital 1 and position. Be sure when you enter distance that you factor in the averagedisplacement by entering it as (position - dave). Enter potential energy for the calculation name,PE for the short name and click on = and NEW. Finally, enter the equation for total energywhich is KE + PE. To get the KE and the PE, click on the input button and select calculationand KE or PE. Click on the = button.

Finally, create graphs of kinetic, potential, and total energy. To do this, first make a graph ofposition versus time by clicking on the graph icon and dragging it over the motion sensor iconand selecting position. Then, click on the downward pointing triangle on the left side of thegraph window and select calculation and the type of energy you are graphing. In the totalenergy graph window, click on the statistics E button and then calculate percent error fromthe equation

%Err' stdev(E)mean(E)

×100 (5)

Analysis:

What is the period of the oscillation? How does the variation of the total energy compare withthe variation of the kinetic energy? How does the variation of the total energy compare withthe variation of the potential energy? Read the maximum velocity from the velocity versustime graph, compute the maximum kinetic energy, and compare it with the total energy. Readthe maximum displacement from equilibrium from the displacement versus time graph,compute the maximum potential energy, and compare it with the total energy. Discuss eachone of these in your conclusions.



Inelastic and Elastic Collisions

Objectives:

< To observe the conservation of momentum during collision processes.< To test that in elastic collisions the kinetic energy is conserved.< To test that in inelastic collisions the kinetic energy is not conserved.

Equipment:

< Two Pasco photogate timers < Pasco interface and personal computer< Two carts and blocks< Triple beam balance


It can be said that the impulse acting on an object is equal to the change in momentum of theobject. In mathematical form, this can be written as

PF @)t')Pp' PI (1)

where I is the vector impulse, the product of the force and the time that the force acts on thesystem. When the force is varying in time, this expression gives the impulse imparted in ashort time, and the total impulse is just the vector sum of these or the area under the forceversus time graph. When the system consists of several parts, the force in equation (1) is thevector sum of the individual forces and the momentum is the vector sum of the moments ofall parts of the system.

From equation (1) it can be deduced that if there is no force acting on the system (constantvelocity), the initial and final momenta must be equal, to make the change ()) in momentumzero. When two objects collide with no external force acting on the system and the totalkinetic energy KE of the setup is conserved, it is said that an elastic collision has occurred.The total KE of the system is the sum of the KE of all the moving parts. An inelastic collisionis defined as a collision when the total KE is not conserved. In general, an inelastic collisionoccurs when the objects attach to each other. In the case of one dimensional motion, that isall motions occur along a line, and with no net external force acting on the system, the initialand final momenta for the case of inelastic collision, can be equated.


m1 @v1%m2 @v2' ( m1%m2 ) @vf (2)

If one of the carts is initially at rest (say m2), then the equatin (2) can be rewritten as

m1 @v1' (m1%m2 ) @vf (3)

The initial KE of the system consists of only the initial KE of m1 and is

KEi'12

m1 @v21 (4)

The final KE can be related to the initial KE by a series of steps involving equations (3) and(4), as follows.

KEf'( m1%m2 ) @v 2

f

2' (m1%m2 ) @

m 21 @v

21

2( m1%m2 )2'

12

(m1

m1%m2

) @m1 @v21 '

m1

m1%m2

@KEi

(5)

It can be seen that the initial and final KE are not equal.

Procedure:

Place a cart on the track and level the track so that the cart does not roll in either direction.Place the photogates at 70cm and 140 cm along the track and plug them into slots 1 and 2 onthe Pasco computer interface. The left photogate should be connected to slot 1. Also,connect the two photogates to the interface on the screen by dragging the digital plug icon tochannels 1 and 2 and selecting photogate and solid object. Adjust the photogate heights sothat the beam is blocked by the blocks on the cart when the blocks are placed on their sidesso that they are taller. Measure and record the lengths of the blocks on the carts. Make surethat you place 1 block on the right cart and record it as L2 and 2 blocks on the left cart andrecord it as L 1. Under recording options, set the periodic sample rate at 10,000 Hz. Finally,make a table for each gate by dragging the table icon over its respective icon and selecting timeto clear gate.

In each of the following collisions, make sure that the carts are moving freely before they enterthe photogates and that the collisions occur when the carts are entirely between the two


photogates.

Use the triple beam balance to determine the masses of the carts, including blocks (see figure1) and record these values as m1 and m2 (m1>m2).

1. Inelastic collisions, m1 > m2 , cart 2 at rest:

Place cart 2 at rest and midway between the photogates with the velcro end to the left. Clickon record, then push cart 1 towards the photogate and cart 2. Let the combined setup (cart1 and 2) go all the way through the second photogate and then click on stop. Make sure thecollision occurs between the two photogates. On the table for the first photogate, there is onevalue displayed. This is the time it took cart 1 to pass through the first photogate. Record thisvalue as t I . On the second table there are two values displayed. The first one is the time ittook cart 2 to pass through he second photogate and should be recorded as t f . Now click onRun #1 and delete it by pressing the delete key on the keyboard.

2. Elastic collisions, unequal masses m1>m2 , cart 2 at rest:

Turn cart 2 around so that the non velcro end faces cart 1. Place cart 2 between thephotogates, click on record, and then send in cart 1 from the left. Cart 2 will pass through theright photogate (stop it and remove it as soon as it passes completely through the photogate)followed a short time later by cart 1. Click on stop after cart 1 passes through the secondphotograte. Record the one value displayed in the table for photogate 1 as t1, the first valuein the table for photogate 2 as t2 and the second value in this table as t3.



Recording Data:

L1 = ______________ L2 = ______________

m1 = ______________ m2 = ______________

Inelastic collision

Compute and record the initial and final kinetic energies in joules by filling in the table below.Compare the ratio of Kef /KEi from the table to the prediction of equation (5). Compute,record, and compare the initial and final momenta of the carts in Newton seconds.

t1 tf v1 vf pi pf KEi KEf

Elastic collision

Complete the following tables. Compare the initial and final momenta in Newton seconds andthe initial and final kinetic energies in joules for the collision.

t1 t2 t3 v1 v2 v3

p1=pi p'1 p'2 p'f KE1=KEi

KE'1 KE'2 KEf

Compare the total initial momentum with the total final momentum and also the total initialkinetic energy with the final total kinetic energy.

For the elastic collision of two equal masses qualitatively analyze what happens. Observe thefinal velocity of cart 2 and compare that with the initial velocity of cart 1. Does cart 1 keepmoving after the collision? Explain by using theory the observations that you saw.



Figure 1 Main setup for theexperiment.


Torque and Angular Acceleration

Objective:

< To observe the relationships between the torque and angular acceleration and the angularimpulse and angular momentum.

Equipment:

< Rotating table< Ring and disk< Slotted and hooked weight sets< Pulleys< Pasco Photogate timer< Vernier caliper


Consider the setup shown in figure 1. A net forceof acts on the mass m hanging on the stringm@g&Twith a tension T. According the Newton’s SecondLaw equation (1) Can be written.

m @g&T'm @a (1)

Solving equation (1) for T yields

T'm @ (g&a ) (2)

This tension acting tangent to the rotating table drum with radius r produces a torque

J'T @r'm @r @ ( g&a ) (3)

There is a frictional torque where mo is the mass on the string required to keep theJf'mo@g@rtable rotating without acceleration. Newton's law for rotational motion asserts that

Jnet'T @r&m0 @g @r' I @" (4)


where I is the moment of inertia and " is the angular acceleration. When the mass m falls fromrest through a distance d in a time t the acceleration is determined from

a' 2d

t 2 (5)

The angular acceleration " is related to the acceleration a by where r is the table drum"'a/rradius. The moment of inertia is the sum of all the where m is the mass and r is them@r 2

perpendicular distance of the mass to the axis of rotation and the sum includes all the smallpieces of the object (an infinity of them). For example, the moment of inertia of a systemcomposed of two masses is the sum of the moments of inertia of each piece

Itot' I1% I2 (6)

The following expressions describe the moments of inertia about the axis of symmetry ofseveral familiar objects that have a mass M.

Idisk'M @R 2

2(7)

Iring'M @ ( R0

2%Ri2 )

2(8)

where R is the radius of the disk, R0 = outer radius of ring and Ri = inner radius of ring.

Procedure:

General Measurements

Without additional mass on the rotating platform determine the frictional force. This is doneby determining the mass of an object mo that, when hung by a string connected to the platformover a pulley, does not cause the system to accelerate. Attach a 10 g mass to the loose endof the string that is wrapped around the rotating platform drum and drape the string over thesmart pulley timer. The frictional force is determined by trial and error so keep varying themass until the system does not accelerate. Record the value of mo. Using the vernier calipermeasure the diameter, ddrum, of the drum on which the string is wrapped. Compute the radiusof the drum, rdrum, and record these values. Measure the diameter, Ddisk, of the disk and the inner and outer diameters, Di, Do, of the thickring. Compute and record the corresponding radii, Rdisk, Ri, and Ro.


Read and record the mass values stamped on the disk and thick ring.

Inertia of the rotating platform

Run Science Workshop. Plug the smart pulley into the digital slot #1 on the Pasco computerinterface and do the same on the screen by clicking on the digital plug, dragging it over digitalchannel 1, selecting smart pulley (linear) and clicking on OK. Take care that the rotatingplatform does not rotate except when taking acceleration data! Keep the string tautwith care that it does not become wrapped around the axle!

Rotate the platform so that the mass is just below the top of the smart pulley. Release theplatform and click on Record after it has started to rotate. Be sure to click on stop before theweight hits the floor.

After the run, create a graph of velocity vs time by dragging the graph icon over the smartpulley icon and selecting velocity. Click on the statistics buttons E and select curve fit andlinear fit. The acceleration will be displayed as a2. Be sure that between runs you delete theprevious run by clicking on Run #1 and pressing delete on the keyboard. Take two readings of the acceleration, a, for masses on the string of about 50 g, and 100 gwith the platform empty. Record the results.

Inertia of the disk

Place the disk on the rotating platform and by following the procedure of the previousexperiment take a series of readings of the acceleration, a, for masses on the string of about50, 100, 150, 200, 250, and 300 grams. Record the results. Also, determine a new value forthe torque caused by frictional forces.

Inertia of the ring

With the ring on the platform, determine the moment of inertia of the system as done for thedisk. A new frictional torque also needs to be obtained.

Inertia of the ring and disk together

With both the ring and the disk on the platform determine the moment of inertia as done in theprevious experiments. Take two readings of the acceleration with masses of 500 and 600grams hanging on the string. A new value for the frictional torque is also needed. Record thevalues.


Recording Data:

General measurements

mo = _________ ddrum = _________ rdrum = = _____________ddrum

2

Ddisk = _______________ Di = _________________ Do = ________________

Rdisk = = __________ Ri = = ____________ Ro = = ___________Ddisk

2

Di

2

Do

2

Mring = _________________ Mdisk __________________

Inertia of the rotating platform

Jf = _____________

m (kg) a (m/s2) J=m(g-a)rdrum

Jnet=J-Jf "=a/rdrum Io=Jnet/"

Take the average of the two values for the moment of inertia from the table above and use thisfor subsequent calculations that involve Io.

Io = __________________

Inertia of the disk

Jf = ____________

m (kg) a (m/s2) J=m(g-a)rdrum Jnet=J-Jf "=a/rdrum

m (kg) a (m/s2) J=m(g-a)rdrum Jnet=J-Jf "=a/rdrum


Graph the net torque vs. angular acceleration using the Graphical Analysis program. Comparethe slope with the theoretical value predicted by equations (6) and (7).

Inertia of the ring

Jf = ____________

m a J=m(g-a)rdrum Jnet = J - Jf "=a/rdrum

With Mathcad make a graph of the net torque, Jnet , versus the angular acceleration, ".Compare the slope, Islope, with the predicted value of equations (6) and (8).

Inertia of the ring and the disk together

Jf = ___________

m a J=m(g-a)rdrum Jnet=J-Jf "=a/rdrum Itorque=Jnet/"

Take the average of the two inertias. Compare this value with that predicted by equation (6).




Conservation of Energy of a Rolling Object

Objective:

< To test the principle of conservation of energy in the case of rolling motion for threeobjects with different moments of inertia.

Equipment:

< Racquetball sphere< Superball< PVC tube< Track< Motion sensor connected to a PC through a data acquisition board< Triple beam balance< Vernier caliper< Lab jack


A very important concept in physics is the Law of Conservation of Energy. According to thisprinciple, the difference between the final and initial energy of an object is the work done onthe object itself. Recall that the work done on an object is given by the product of the forceacting on the mass and the distance it moved. If there is no net force acting on the body, andfriction is neglected, the total work done on the object is zero and, as a result, the final andinitial energy of the object must be equal. In other words, the total energy of a system staysconstant.

At any given time, total energy is the sum of two components. One of the components, causedby motion, is the kinetic energy KE, and the other is a result of position and is called thepotential energy PE. It follows that the law of conservation of energy can be writtenmathematically as

KEi%PEi'KEf%PEf (1)

The PE of the object is given by . However, it must be recognized that the KEm@g@h

consists of two parts. The first is given by


Incline

Reference PointObject

a

Figure 1 Diagram of the inclineused.

Iball'2 @M @R 2

5Isphere'

2 @M @R 2

3Iring'M @R 2 (4)

Motion Sensor

Track

Figure 2 Setup of the equipment.

. Picture an object spinning in one place,12

m@v 2

with no net movement. The velocity v is thereforezero, but it has a KE caused by rotation. It hasbeen determined that the KE caused by rotation is

given by where I is the inertia of the12

I @T2

object and T is the rotational speed.Consequently, the total KE is the sum of the two.

In the case of an incline, the PE can be related tod, the distance measured from a reference pointabove the object (see figure 1) by

PE'&m @g @d @ sin(a) (2)

The total energy is given by

Etot'&m @g @d @sin(a)% 12

m @v 2%12

I @T2(3)

The inertia of the object must be known in order to calculate the KE. The three inertias usedin this experiment are given below.

Procedure:

Set up the equipment as shown in figure2. Make sure that the angle of the motionsensor is adjusted in such a way that it ispointing down parallel with the track.Run Science Workshop. Connect the twoconnectors of the motion sensor to theSignal Interface box, the one with theyellow band to the digital input 1 and theother to digital input 2. Place the cursor


on the phone plug icon, hold the left mouse button down and drag the cursor to the digitalinput 1 on the Science Workshop window. Click on the Motion Sensor option and click onOK. Click on the REC OPT button, set the periodic samples rate to 50,000 Hz and set thestop condition to 2 seconds. Place the PVC tube between the two edges of the track, closeto the top, so that it could roll down the incline. Move the motion sensor or the tube so thatthe distance between them is at least 40 cm. At this point you are ready to take data. Let goof the tube and click on the REC button very shortly thereafter to start recording the data. Thevelocity graph should be straight line and the distance graph should be parabolic. If they arenot, the angle of the motion sensor needs to be adjusted and the experiment redone. It willtake you several trials to get a good set of data. Once you have a good set, record the massand radius of the tube and complete the analysis of this data. Repeat this using a sphere(racquetball) and a ball.

Analysis:

Open the calculator window which can be found in the experiment window. Enter theequation for kinetic energy using the mass, radius, and velocity that you measured. To enterthe velocity values from the experiment click on the input button and select velocity. Enterkinetic energy for the calculation name, KE for the short name, then click on the equals button.Now click on new and enter the equation for the potential energy using the mass, angle anddistance that you measured. Get the symbol for distance in the same manner as you did forkinetic energy. Name the calculation potential energy, the short name PE and click on theequals button. Finally enter the equation for the total energy, the sum of the kinetic andpotential energies. To reference these values click on the input button and select calculatorand then KE or PE. Enter energy for the calculation name and E for the short name, then clickon equals. Graph the kinetic energy, potential energy and total energy versus time by clickingon the down arrow to the left of the graph in the graph display window and selectingcalculations. The total energy should be constant with time. Calculate the percent error bythe formula.

%Err' *stdev(Etot)*max(KE)

×100 (5)



Figure 2 Torque generation.

Figure 3 Generation of a counterclockwisetorque.


Rotational Equilibrium - Torques

Objective: < To test the hypothesis that a body in rotational equilibrium is subject to a net zero torque

and to determine the typical tension force that the biceps must produce. Equipment:

< Arm assembly < Spring scale, 10 N < Rod, .8 m < Table clamp < Swivel clamp (or 2 right angle clamps and a short rod) < Hooked weight set Physical Principles: A torque is produced about an originwhen a force acts at a point of a body ina direction other than the direction ofthe origin from this point. Torquestend to make a body rotate about anaxis. Those torques that rotate a bodyin a clockwise direction are calledclockwise and are usually described bynegative numbers. Those that rotate abody in a counterclockwise directionare called counterclockwise. The torque generated by a force Facting at a point which is a distance, d,from the origin is defined as

J'Fz @d (1)

Fz is the component of the forceperpendicular to the line from theorigin to the point of action of theforce and therefore is given by


Figure 4 Diagram of Setup 1.

L arm

b

d m

scale

lower forearm

load



. In figure 1 a is the angleFz'F@sin(a)between this line and the direction of theforce. The distance d is called the length ofthe lever arm. If a body is in rotationalequilibrium the net torque acting on it mustbe zero.

Procedure:

Referring to the figures for furtherassistance, perform the measurements andcalculations shown in the General measurementssection of the Recording data section of thisexperiment.

Setup 1

With the upper arm approximately vertical adjustits position in the clamp so that the lower arm ishorizontal (see figure 3). You may wish to sightalong the horizontal wall joints to level theforearm. Record the necessary measurements andperform the calculations as outlined by theformulas given. Setup 2 Position the upper arm at an angle so that theangle b is about 40E and the lower arm is againhorizontal. Repeat the observations andcalculations in setup 1.

Setup 3

Position the upper arm vertically and the lowerarm at an angle of about 40E above thehorizontal. Repeat the calculations of setup 1noting that the clockwise torque calculationsmust include the additional factor of cos a wherea is the angle of the forearm above thehorizontal.

Estimating the tension in a muscle


Estimate the muscle tension when you are raising a weight of about 20 N in your hand whenthe upper arm and forearm are nearly horizontal. Make reasonable estimates of the angles andlengths and neglect the weight of your forearm.



Recording data:

General measurements

Mass of wooden "forearm" = marm = ___________________

Weight of wooden forearm = = _________________marm@g

Length of wooden forearm = larm = ____________________

Distance of forearm center to "elbow" (joint) = dcg = _________________ Distance of attachment point to the joint = dm = ____________________ The length of the forearm lever arm larm should be taken to be the length from the center ofthe joint screw to the center of the hole on the opposite end, that contains the threadsupporting the load. The distance of the forearm center to elbow dcg should be taken to bethe length from the middle of the forearm to the center of the joint screw. The distance ofmuscle attachment dm should be from the center of the hole with the upper support string tothe center of the joint screw.

Setup 1

b = ____________________

T = ____________________

mload = _________________

Wload = _________________

Compute the clockwise torques: Load: = ___________________________________ Wload @darm Forearm weight: = ___________________________Warm @dcg Total: ___________________________

Compute the counterclockwise torque: Support string: = __________________________T @dm @sin(b)



Compare the total clockwise torque with the counterclockwise torque by computing thepercent error. This is done by dividing the difference of the two torques (clockwise andcounterclockwise) by their average and multiplying by 100.

Setup 3

a = __________________

b = __________________

T = __________________

mload = ________________

Wload = ________________

Compute the clockwise torque Load: = ____________________Wload @darm @cos(a) Forearm weight: =_____________Warm @dcg @cos(a) Total: _________________________________ Compute the counterclockwise torque:

Support string: = ________________T @dm @sin(b) Compare the total clockwise torque with the counterclockwise torque by computing thepercent error.

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