The moment of inertia of a rod rotating about its centre is given by:

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1 Pendulum Physics 161 Introduction This experiment is designed to study the motion of a pendulum consisting of a rod and a mass attached to it. The period of the pendulum will be measured using three different methods: 1) with a simple stopwatch, 2) using a photogate, and 3) with a rotary motion sensor. At the end, using the period of the pendulum, you will calculate the gravitational acceleration. Reference Freedman & Young, University Physics, 12 th 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 by: l T 2 (1) g where l is the length of the string in meters and g is the the gravitational acceleration in m/s/s. A physical pendulum is an extended body which oscillates back and forth about a fixed point. The period of oscillation for a physical pendulum is given by: I T 2 (2) m gh o where m o is the mass of the extended object, H is the distance of centre of mass (CM) of the extended object to the pivot point, g is the gravitational acceleration and I is the moment of Inertia of the extended object about the pivot point. The moment of inertia of a rod rotating about its centre is given by: 1 I ml 2 CM rod (3) 12 where l is the length of the rod. For an oscillating rod since the distance from the pivot point to the centre of mass of the rod is h, then the moment of inertia of the system is I rod ml mh (4) The moment of inertia of the hanging mass (M) from the pivot point is 2 Md I mass (5) Where d is the distance of the mass M to the pivot point.

2 The total moment of inertia of the oscillator is the combination of the two moments of inertia: I I rod I mass ml mh Md (6) The CM of the extended object can be found using the following equation: m0 H mh Md (7) Where m 0 is the mass of the extended object, H is the distance of CM of the extended object to the pivot point, m is the mass of the rod, h is the CM of the rod to the pivot point, M is the mass of attached mass to the rod and d is the distance of CM of the attached mass to the pivot point. Using the above equations, the period for the physical pendulum (in eq. 2) can be rewritten as: T ml mh Md I 2 12 mo gh ( mh Md) g 2 (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. Set up the equipment as displayed in Figure Data collection will begin using a hand-held stopwatch. The stopwatch is able to measure to 1/100 th 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 full periods. Repeat this 10 times. Again calculate the average and standard deviation of the measurements. Divide the average and the STD by 10 and compare the value to individual period measurements calculated in #3.

3 Figure 1 5. Next, we will use the computer to measure the period of the pendulum in two ways. Look at the pendulum set up in the picture on the first page of this lab. The pendulum will oscillate between the U-sides of the photogate sensor, and the pendulum will be 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. In today's lab you will collect data of the period of the pendulum through the use of the computer in two ways: the photogate sensor and the rotational motion sensor. 6. The photogate sensor must be setup to measure the period of the pendulum's oscillations. Find the photogate on the sensor list in the program and drag it to the correct port on the image of the interface box. Do the same for the rotational motion sensor. 7. The photogate sensor measures a period by the number of times the pendulum will 'block' the photogate's signal. The photogate works by maintaining a signal (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. Therefore, if you place the pendulum between these holes, the signal is stopped. For today's experiment, we will measure time between the 'blocking' of the signal by the pendulum.

4 v v v Start: 1st block of the sensor Beginning of Period Start: 2nd block of the sensor Half of a Period Figure 2 Start: 3rd block of the sensor One Full Period 8. To examine a full period though, we need to measure the time from the first pass through the photogate until the pendulum passes again in same direction. The pendulum will swing through the photogate (going to the right), then again (going to the left), then finally going right again. These three passes constitute one period (see Figure 2). Timer Icon Photogate Sensor Icon Photogate Pulldown Menu Figure 3 9. From the discussion above it is known that the sensor will be 'blocked' three times for one Period. Let's tell Data Studio this so it measures the period. Press the Timer button on the Experiment Setup window and a Timer Setup window will appear, as shown in Figure 3. Here, simply choose 'Blocked' three times on the photogate pull-down menu.

5 Rename the timing sequence 'Pendulum' and then choose Done. In the Data section a timer with the words Pendulum will appear. 10. The rotational motion needs to be set up to collect data of the angular position. This will graph the angle of the pendulum versus time. Inspecting this graph, we can measure the length of time for one period and compare it with the period measurement of the photogate. Double click on the rotational motion sensor. You will want to take only angular position data today so click on Measurements and make sure only Angular Position (in degrees) is selected. Finally, we need to let the rotational sensor know 'how' we are attaching the pendulum to the sensor. Go to the Rotational Motion Sensor tab by the Measurement tab, select Other under the Linear Calibration tab and click ok, see Figure Now you are ready to start. Make sure before you swing the pendulum you press the Start button. This will allow the rotational motion sensor to calibrate its 0. Then, give the pendulum a push trying to keep the angle under 10. Figure As soon as the start button is pushed, a 'Run 1' icon will appear for the Pendulum and the Angular Position. Drag these icons down to the Graph Icon in Displays and two graphs will pop up showing the Period (Time elapsed) vs. Time, and Angular Position vs. Time. 13. Look at the Period vs. Time graph. Let the motion settle until the period remains basically constant. If the angle was large, you will notice an initially strong downward trend.

6 14. Acquire data for at least 120 seconds. Examine the graphs. At one point compare the period given by the photogate sensor to the period (full sine wave) given by the rotational motion sensor. The two values should be close. 15. Now, let's interpret the Angular Position vs. Time graph. The pendulum's motion is oscillatory. If damping is negligible in the system over a short period of time the graph should appear sinusoidal with constant amplitude. To find the sine wave which best approximates the graph we can try to 'fit' a selected portion of the graph to a sine wave. Make sure the pendulum has settled into a nice motion and that the amplitude is as constant as possible so the above assumptions are met. 16. Once this section of the graph is found, drag your mouse over the data points you would like to fit. The points and curve will appear yellow. On the button menu above select the fit button, and choose Sine Fit. This will bring up a sine wave on the graph with set variables (see Figure 5). 17. Record the Period and its uncertainty from the Fit function in Data Studio. 18. Now let us turn our attention to the photogate sensor graph. This graph allows us to examine the periods of oscillation over time. What do you think should be true about the Period measurements over time? Manual Button Curve Fit Parameter Boxes Curve Fit Info Box Figure 5

7 19. We can now calculate the mean and the standard deviation of this data set. To calculate the mean and the standard deviation go back to the Elapsed Time vs. Time 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. 20. Calculate the centre of the mass of the rod/mass system using Equation Using the stop watch data calculate the acceleration of gravity with uncertainty using Equation 1 (where you substitute l with H). Does the value of g fall within 1 to 2 standard deviations of the accepted value? Note: to determine the standard deviation of g you need to propagate the uncertainties of T and H. 22. Now calculate the acceleration of gravity using Equation 8. Does the value of g fall within 1 to 2 standard deviations of the accepted value? Note: use again Equation 1 to determine the standard deviation of g.

8 For your in-class lab report: Make the following table in Excel and fill in all the data. Method g (m/s/s) using Simple Pendulum Data Stop Watch (single) Stop Watch (10 times) Photogate Rotary Motion Sensor %diff between theoretical and experimental g g (m/s/s) using Physical Pendulum (mass and rod) %diff between theoretical and experimental g Using the stop watch data, show a sample calculation with uncertainties of the gravitational acceleration for both simple and physical pendulums. Answer the questions in steps 20, 21, and 22. Which method gave you the best result? Explain.