Measurement and Calculations

Murphy’s Law0Captain Edward A. Murphy Jr. was an engineer in the Air Force.

0 In 1949, officers were conducting tests to determine once and for all how many Gs -- the force of gravity -- a human being could withstand.

0The project team used a rocket sled dubbed the "Gee Whiz" to simulate the force of an airplane crash.

0The sled traveled more than 200 miles per hour down a half-mile track, coming to an abrupt stop in less than a second.

0The the team needed an actual person to experience it. Enter Colonel John Paul Stapp.

0A set of sensors that could be applied to the harness that held Dr. Stapp to the rocket sled. These sensors were capable of measuring the exact amount of G-force applied when the rocket sled came to a sudden stop, making the data more reliable.

0The first test after Murphy hooked up his sensors to the harness produced a reading of zero -- each one was installed the wrong way.

0During his ride he was subjected to broken bones, concussions and broken blood vessels in his eyes all in the name of science.

0So became Murphy’s Law.

What did we learn?

0Having accurate calculations and measurements is essential in conducting experiments, especially when conducting experiments that could prove to be dangerous.

Certainty0When communicating in science you should express how

certain you are about your measurements.

0This Degree of Certainty(or uncertainty) is expressed as: record all those digits that are certain plus one uncertain digit, and no more.

0These “certain-plus-one” digits are called significant digits.

0The certainty of a measurement is determined by how many certain digits (plus one) are obtained by the measuring instrument.

The measurement of the distance from the centre of point A to the centre of point B is 2.05 cm. with a certainty of 3 significant digits.

0The greater the number of significant digits, the greater the certainty of the measurements.

0Rule: All digits included in a stated value (except leading zeros) are significant digits.

0The position of the decimal point is not important when counting significant digits; ignore the decimal point!

0Ex: Adam has a vertical leap of 15.75 inches. This measurement contains 4 significant digits.

Certainty of Measurement Chart

Measurement Certainty

307.0 cm 4 significant digits

61 m/s 2 significant digits

0.03 m 1 significant digits

0.5060 km 4 significant digits

3.00 x 10³ m/s 3 significant digits

Counted or Defined Values0When you directly count the pairs of shoes in your closet, this is

an exact number.

0When you find the average height in the class you can get an exact value.

0Exact(counted) and defined values are thought to have an infinite number of significant digits.

0Defined values include such examples as 100 cm/m and 60 s/min.

Exact Values

Counted Values Defined Values

4 cars 1000 m/km

60 DVDs 10 mm/cm

9 snowballs 1 h/ 60 min

Certainty Rule for Multiplying and Dividing0Rule: When multiplying or dividing, the answer has the

same number of significant digits as the measurement with the fewest number of significant digits.

0Ex:

= 32 cm²

0 If you use your calculator to solve the above problem you would get 32.32cm²

0For us we see that 3.2 has 2 significant digits and 10.1 has three significant digits, therefore, the answer must only have 2 significant digits.

Rounding0To obtain the correct certainty we need a general rule for

rounding answers.

0Rule: If the digit after the digit to be retained as significant is a 5 or greater, round up.

0Ex: Rounding 9.147 cm to three significant digits would yield 9.15 cm.

0Rounding 7.23 g to two significant digits would yield 7.2 g.

Precision Rule for Adding and Subtracting

0 Precision is defined as the place value of the last digit obtained from a measurement or calculation.

0 Rule: When adding and subtracting measured values of known precision, the answer has the same number of decimal places as the measured value with the fewest decimal places.

0 Ex: When adding 1.2 mm, 3.05 mm and 7.60 mm the answer can be no more precise then the least precise value of 1.2

0 Therefore the answer (11.85 mm) is rounded to 11.9 mm.

Did you Know?

0The first interplanetary spacecraft launched by the United States, Mariner 1, never reached its target, Venus, because of one missing hyphen in its software.

Conventions of Communication

0The previous rules are generalizations, however, they provide a set of principles that allow the science community to communicate with each other without much confusion.

0Despite these guidelines many people understand that their are limitations to these rules.

0The scientific community agree to the SI conventions that include quantity and symbols.

0Ex) km not mi (miles) and h not hr (hours)

Solving Equations

0Rearrange defining equations to help solve for unknown variables.

0A Defining equation is the definition of a quality expressed in quantity symbols .

0Ex. Solve for b0A= bh

0 b =

Converting Units

0Some people have an easy time converting units in their heads while other need to write it down.

0We should all practice writing down conversions as they tend to get quite difficult as we progress into more advanced work.

0Ex. An athlete competed in a 5 km race in 19.5 min. Convert this time to hours.

0 t = 19.5 min x 0= 0.325 h

Converting Units

0Ex2: 0A train is travelling at 95 km/h. Convert 95 km/h into

metres per sec.

0v = 95 x x x

= 26 m/s