Physics 161
Pendulum
Introduction
This experiment is designed to study the motion of a pendulum. The pendulum consists of a rod
and a mass attached to it. You will measure the period of the pendulum. Using the period of the
pendulum, you will calculate the gravitational acceleration and test whether your measurements
agree with the theoretical value.
Reference
Freedman & Young, University Physics, 12th Edition: Chapter 13, section 5, and 6.
Theory
A simple pendulum consists of a small mass attached to a massless string. For a small angle of
oscillation, less than 10 degrees, the period of the pendulum is given as:
π = 2π
π
π
(1)
where π is the length of the string in meters and π is the gravitational acceleration in π/π ! .
A physical pendulum is an extended body which oscillates about a fixed point. The period of
oscillation for a physical pendulum is given as:
π = 2π
πΌ
π! ππ»
(2)
where π! is the mass of the extended object, π» is the distance from the center of mass (CM) of
the extended object to the pivot point, π is the gravitational acceleration and πΌ is the moment of
inertia of the extended object about the pivot point.
The moment of inertia of a rod rotating about its CM is given by:
πΌ!"!!"# =
1
ππ !
12
(3)
where π is the length of the rod and π is the mass of the rod. For an oscillating rod, the distance
from the pivot point to the CM of the rod is β, therefore, the moment of inertia of the system can
be written as:
πΌ!"# =
1
ππ ! + πβ!
12
(4)
where β is the CM from the rod to the pivot point.
The moment of inertia of the hanging mass π from the pivot point is:
πΌ!"## = ππ !
(5)
where π is the distance of the mass π to the pivot point. The total moment of inertia of the
oscillator is the combination of the two moments of inertia:
πΌ = πΌ!"# + πΌ!"## =
1
ππ ! + πβ! + ππ !
12
(6)
The CM of the extended object can be found using the following equation:
π! π» = πβ + ππ
(7)
Equation (2) can now be re-written as:
1
ππ ! + πβ! + ππ !
πΌ
12
π = 2π
= 2π
π! ππ»
πβ + ππ π
(8)
Procedure
1. Remember in all data collecting today that we are assuming the period of the pendulum will not
change as long as the initial angle is smaller than 10 degrees. This means that each time the
pendulum is put into motion as long as the angle of oscillation is less than 10 degrees, the period
should remain constant. This assumption holds true when damping is ignored, which is a good
approximation during short time scales.
2. Data collection will begin using a handheld stopwatch. The stopwatch is able to measure to
1/100th of a second, but what about your reaction time to stop the data collection? Since our
hands and eyes are slower than the accuracy of the stopwatch, uncertainty exists in any
measurement using the stopwatch. Play around starting and stopping the watch, how long does
it take for you to simply press the stop button? Record this value and make an estimate on the
lag time measured between turning the stopwatch on and off.
3. Record the time it takes for the pendulum to complete one period. Repeat this process 10 times
so that you have 10 measurements of a single period of the pendulum. Calculate the average
period and the standard deviation of these 10 measurements.
4. Now, measure the time for 10 contiguous periods. Repeat this step 10 times. Calculate the
average and standard deviation of the measurements. Now, divide the average and the STDEV
by 10 and compare the value to individual period measurements calculated in Step 3.
5. Now you will collect data of the period of the pendulum using the computer. You will use the
photo-gate sensor and the rotational motion sensor. The pendulum will oscillate between the Usides of the photo-gate sensor, while the pendulum is attached to the rotational motion sensor.
The rotational motion sensor will measure angular position as the pendulum swings. The photogate sensor will measure the period of oscillation directly as it passes through the U-sides of the
sensor.
6. The photo-gate works by maintaining a signal, a red light, between the two U-sides of the
sensor. Look at the sensor; notice two small holes on each of the U-sides of the apparatus. A
signal is sent and received at these points. If you place the pendulum between these holes, the
signal is βblockedβ. We will measure the pendulumβs period using the 'blocking' of the signal by
the pendulum. To examine a full period of the pendulum, we need to measure the time from the
first pass through the photo-gate until the pendulum passes again the in same direction. Three
passes constitute one period. This process is shown in Figure 1.
Figure 1: Photo-gate process.
7. The photo-gate sensor must be oriented perpendicular to the pendulumβs path to function
correctly. Be sure that the screw, which fastens the weight to the rod, is facing one of the sensor
hole in order to avoid asymmetric blocking. Make sure the photo-gate sensor and the rotational
motion sensors are plugged into the PASCO interface box. Find the photo-gate on the sensor
list in the Hardware icon after clicking on port 1 on the left. Do the same for the rotational
motion sensor but put the yellow and black leads into ports 3 and 4 respectively, see Figure 2.
Figure 2: Photo-gate and rotary sensors in Capstone.
8. From the discussion above it is known that the sensor will be 'blocked' three times for one
period. In Capstone, under the Timer Setup tab, choose βBuild your own Timerβ. Click on
βNextβ button. Under βTimer Sequence Devicesβ, click on the photo-gate, Ch. 1 and select
blocked. Do this three times. Click βNextβ and rename the timer measurement name to βPhoto
gate Periodβ and then choose Save. See Figure 3.
Figure 3: Setup photo-gate sensor.
9. The rotational motion sensor will collect data of the angular position of the pendulum.
Inspecting the angular position versus time plot of the pendulum, we can measure the length of
time for one period. Click the Icon for the Rotary Motion Sensor. Then click on Properties on
the bottom right. Make sure that under the drop-down box labelled βLinear Accessoryβ the
option Rack and Pinion is selected.
10. Drag two graph icons into the workspace area assigning each sensor to one of the graphs. Set
the π¦-axis in the upper graph to βAngle (rad) and in the lower graph to βPhoto gate Periodβ,
from the selection list.
11. Make sure before you swing the pendulum you press the Record button. This will allow the
rotational motion sensor to calibrate its 0Β°. Start acquiring data for at least 15 seconds. Look at
the Photo Gate Period versus Time graph. If you start the swing with an angle larger than 10Β°,
you will notice an initially strong downward trend. Let the motion settle until the period remains
basically constant. This the portion of data you must use in your analysis.
12. Now, let us turn our attention to the photo-gate sensor graph. This graph allows us to examine
the periods of oscillation over time. Calculate the mean and the standard deviation of this data
set. To calculate the mean and the standard deviation go back to the Photo Gate Period graph
and click on the
icon. A drop down window will appear where you can select Mean and
Standard Deviation by checking them. Now, look at the graph legend box and you will see
these values, see Figure 4.
Figure 4: Plots of angular position vs time, and pendulum period vs time.
13. Calculate the CM of the rod/mass system, π», using Equation (7).
14. Using the stop watch data calculate the acceleration of gravity, with uncertainty, as if it were a
simple pendulum, use Equation (1) and use π» instead of π. To calculate the uncertainty in π,
solve Equation (1) for π, and propagate the uncertainties in length, mass and period, using the
partial derivatives in quadrate that we introduced last week. Does the value of π fall within 1 to
2 standard deviations of the accepted value?
15. Now calculate the acceleration of gravity using the Equation (8) for the physical pendulum. Use
Equation (2) to find the uncertainty, and check to see if the value of π fall within 1 to 2 standard
deviations of the accepted value? Make sure to write down your formula for the propagated
uncertainty for π in your notebook.
Repeat steps 14 and 15 using the average value for the period and its uncertainty measured using the
photogate.
Make the following table in Excel and fill in all the data. Use Equations (1) and (8) to calculate your
experimental value of π, using the period of oscillations you measured.
Method
Period T and g (m/s2) using
Simple Pendulum Data
π
π
Period T and g (m/s2) using
Physical Pendulum (mass and rod)
π
π
Stop Watch (single)
Stop Watch (10 times)
Photo-gate
Which timing method gave you the best result? Which formula (simple or physical pendulum) gave
you the best result? Explain.
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