Module 2
Planning Guide

Lab 1.6

Lesson 1.5
Lesson 1.6
Lesson 1.7
Lesson 1.8
Lab 1.5
Lab 1.6
Lab 1.7
Lab 1.8
Project 2


Physics Lab 1.6 The Scientific Method: Time, Pendulums & Frequency

Measurements & Accuracy

When we carry out experiments, we need to know how accurate our measurements are. It is impossible to measure anything with absolute accuracy. Whenever we measure something, the accuracy of our result will depend on what we use to take the measurement and also our skill in using the measuring device. There is always a degree of error in any result and it is often taken for granted that this is acceptable. The conclusions drawn from any experiment or set of experiments must take into account the uncertainties in measurement.

Accurate measurements are always those that are sufficiently accurate for the task. There are margins of error in any measurement and we need to know how potential errors can be compounded by mathematically combining different values – each with its own degree of inaccuracy.

Significant Figures

Measurements are always reported with a significant number of figures. The number of figures usually implies the degree of accuracy of the measurement. For example, a length reported as 3 centimeters could be anything between 2.5 cm and 3.5 cm - or worse. A length reported as 3.002 cm is more accurate. The first value has 1 significant figure. The second has 4. If we were to multiply the two figures together, the overall accuracy would be that of the least-accurate of the two. 9.006 cm would be a misrepresentation of the accuracy of the result. The result should only contain 1 significant figure.

When we refer to significant figures, leading zeros (and trailing zeros) are ignored.

For example, the numbers: 12345.000 and 0.000012345 both contain five significant figures.

The number of significant figures in the number: 0.003004506 is seven.

Experiment 1.1.1 Pendulum Time

Test an Hypothesis
We can illustrate the the use of the scientific method by using a pendulum to test a pendulum equation.
According to this equation (which started as an hypothesis), the period - or time taken to complete a full cycle of its swing motion - depends only on its length of the pendulum and is independent of the mass of the bob.

The following equation has been proposed:

t = 2 p (L/g)

where: t = period of the pendulum (seconds) (i.e. The time taken for pendulum to complete one cycle)

L = length of the pendulum in meters

g = Acceleration due to gravity. At the earth’s surface. g = 9.81 m/s2

p is a constant

Materials & Equipment
To perform an initial test of the equation, we will need the following materials and items of equipment: (We in Lab 1.0 - Units and measurement.)

Stopwatch
Tape measure
String (1 to 2 meters)
Metal weight to act as a bob at the end of the pendulum
Suitable structure from which to suspend the pendulum.

Procedure
Construct the pendulum by attaching the metal weight to one end of the string and then suspending the pendulum from a suitable structure. (i.e.one that won't move when the pendulum swings backwards and forwards.)

Measure the length of the pendulum from the point at which it is suspended to the mid-point of the metal weight.
Draw the pendulum bob to one side and release it to allow the pendulum to swing backwards and forwards.
Using the stopwatch, measure the time taken for the pendulum to complete 10 cycles. (A cycle starts when the pendulum leaves a particular point and the cycle is completed when it next arrives at that point - moving in the same direction as it was traveling in when it left the point. For example: If we select the lowest point in a pendulum's swing as the starting point of a cycle - and it is moving from left to right, to complete a cycle it must move to the extreme at the right hand side, swing back to the extreme at the left hand side and back to the lowest point traveling from left to right. If a cycle starts at one of the extremes, a cycle is completed when the pendulum returns to that point.)
Divide the time for 10 cycles by 10 to obtaing the period of the pendulum.

Compare the measured period with that predicted by the equation.

Repeat the experiment after changing the length of the pendulum.

Questions
1. Can we conclude that the pendulum equation is valid as long as the effect of gravity remains constant?
2. Does the length of the swing (distance between two extremes) affect the period of the pendulum?

The purpose of this experiment is to determine the relationship between the length of a pendulum and its period. (the time taken to complete a full cycle as it swing.) The experiment is also intended to provide experience in estimating accuracy in time measurements..

Background

Typically, a clock consists of an oscillator (or "ticker") that oscillates at a constant rate and a mechanism to display the time based on a fixed number of oscillations. Prior to the "Quartz Age", oscillators were usually pendulums or rotating devices connected to a spring. Some electric clocks still use the cycle of the electric power lines to track the passing seconds but utility companies use accurate clocks to ensure that the frequency stays on track.

The period of a pendulum depends on the force of gravity on the pendulum and its length. This is only true if all of its mass is concentrated at the bottom of the pendulum. Pendulums consisting of a thin string with a heavy object at the end are quite close to this ideal pendulum. For the purposes of the experiment we will assume that the mass of the material connecting the bob to the point of suspension is not significant.

Equipment

1 stopwatch

1 meter of string

A metal ball, lead weight or socket wrench (to be used as the pendulum’s bob)

A sturdy object or point from which to suspend the pendulum.

Tape measure

Procedure

  1. Attach the heavy object to one end of the string.
  2. Attach the other end of the string to a suitable anchor point in such a way that the pendulum can swing freely.
  3. Define a point that is roughly at the center of the heavy object. (Place a mark on the outside of the object that is near this point.)
  4. Measure the distance from this point to the point at which the pendulum is suspended. This is the length of the pendulum.
  5. Pull the pendulum to one side and allow it to swing backwards and forwards.
  6. Using the stopwatch, measure the time taken for 10 cycles. (To complete a cycle, the bob must pass, reach or leave any point in its path going in opposite directions.
    For example, if the point from which the pendulum is initially released is chosen as the point at which each cycle is counted, a cycle is completed when the pendulum reaches that point coming from the other end of its swing. If the lowest point of the pendulum’s swing is selected, a cycle is completed when the pendulum returns to this point having traveled to both of the extremes.
  7. Divide the time for 10 cycles by 10 to determine the average time per cycle.
  8. Repeat this measurement twice (i.e. three measurements) and calculate the average of the three.
  9. Change the length of the pendulum and complete the following table:
  10. Length of Pendulum

    (meters)

    		 Time

    Measurement

    #

    		 Time for

    10 cycles

    (seconds)

    		 Average

    Period [Time per

    Cycle (s)]

    		 Average

    Period

    [Measured]

    (seconds)

    		 Average

    Period

    [Predicted

    from equation]
    Length

    =…………(m)

    		 1        
    2    
    3    
    Average =    
    Length

    =…………(m)

    		 1        
    2    
    3    
    Average =    
    Length

    =…………(m)

    		 1        
    2    
    3    
    Average =    
    Length

    =…………(m)

    		 1        
    2    
    3    
    Average =  

Analysis

It has been proposed that if the period of a pendulum depends on the force of gravity and the length of the pendulum, the relationship between the period and the length of the pendulum could be given by the following equation:

t = 2 p (L/g)

where: t = period of the pendulum (seconds) (i.e. The time taken for pendulum to complete one cycle)

L = length of the pendulum in meters

g = Acceleration due to gravity. At the earth’s surface. g = 9.81 m/s2

For each of the lengths chosen above, calculate the average period and compare this with the value predicted from the equation.

Accuracy

Stopwatch readings are affected by human reaction time and perception of when the pendulum reaches the reference point.

It may be safe to assume an accuracy of 1/10th of a second in measuring time. This would mean that for times between 1 and 9 seconds, two significant figures would be appropriate. (Between 10 and 99 seconds, 3 significant figures etc.)

After using a measurement in a calculation, we must assume that the number of significant figures in the result cannot be greater than the number of significant figures in the measurement.

Conclusion

Questions:

  1. How do the predicted values for the period of the pendulum (from the equation) compare with the measured values?
  2. Can you suggest any reasons for the variation (if any)?
  3. Why is the period of a pendulum not affected by the angle at which it starts swinging?

Experiment 1.1.2 Stroboscope and Frequency

The purpose of this experiment is to illustrate how something that is relatively easy to measure can be used to estimate a value for something that is more difficult to measure.

The experiment involves making a simple stroboscope and using it to estimate the speed of rotation of a hand-held, battery-operated fan.

The stroboscope consists of a circular piece of cardboard (disk) with a hole in the center and a number of evenly spaced slits around the edge of the disk.

Two people will be needed for this experiment. The first person will rotate the stroboscope while the second person used a stopwatch to measure the rate of rotation of the stroboscope.

We will use the stroboscope later in the program to study the behavior of waves.

Materials
Template for Stroboscope (Click here)
Cardboard
Scissors and sharp blade / knife
Stopwatch
Wooden dowel, screw & washer
Battery-operated fan

Important Safety Notice

  1. Be very careful when using sharp instruments to cut cardboard.
  2. Always cut in a direction away from yourself.
  3. Assume that the blade or sharp device could slip and ensure that any part of your body (finger, hand etc.) would not be in the path of the sharp instrument if it slipped.

Procedure

  1. Use the diagram provided to mark out the shape on the cardboard. If necessary, use a paper template.
  2. Cut the cardboard to the desired shape.
  3. Make a small hole at the center of the disk and a second (finger) hole slightly off-center. The second hole should be about 2 cm. in diameter.
  4. Place a dark mark/spot on the edge of the disk. This will be used to count the number of times that the disk rotates during a particular time.
  5. Place a dark mark on one of the fan’s blades.
  6. Start the fan.
  7. Use your finger to rotate the disk slowly at a constant rate.
  8. Hold the disk in such a way that the fan can only be seen through the slits of the disk while the disk is rotating.
  9. Increase the speed of rotation until the fan appears to stop turning. Make sure that the blade with the dark mark on it also appears to stay in the same place.
  10. Use a stopwatch to measure the time taken for the stroboscope to complete 10 rotations.

Results

The number of times (frames) per second that the fan is observed through the slits in the stroboscope is therefore equal to 10 times the number of slits in the stroboscope divided by the time taken for 10 rotations of the stroboscope.

This is the rotational speed of the fan in rotations per second (rps).

Carry out the experiment 3 or 4 times.

Record the results in the following table:

Number of slits in stroboscope = 12

Test Number

Time Taken for 10 Rotations

(seconds)

Fan’s Rotational Speed

[Rotations per second (rps)]

1

    

2

    

3

    

4

    

Average Speed of Rotation

  

Accuracy

 +/- ……………. rps.

Conclusion

Frequency is the number of times something repeats itself during a particular time.

If the system of measurement is 100% accurate, there would be no variation between the four results above. To calculate the accuracy as a percentage, take the measurement that is furthest from the average, calculate the absolute difference between it and the average and then express this difference as a percentage of the average.

Question:

Can you suggest any improvements to the equipment or system of measurement that could improve the overall accuracy?

STROBOSCOPE TEMPLATE
(Print the image and blow up until the diameter is about 12")