PHY161 Miami University Pendulum Experiment Lab Questions

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the lab handout is attached and my result of the lab is also attached. do like always and if you have any questions let me know.

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PHY161 Miami University Pendulum Experiment Lab Questions
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PHY161 Miami University Pendulum Experiment Lab Questions
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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: 𝑇 = 2πœ‹ 𝑙 𝑔 (1) where 𝑙 is the length of the string in meters and 𝑔 is the gravitational acceleration in π‘š/𝑠 ! . A physical pendulum is an extended body which oscillates about a fixed point. The period of oscillation for a physical pendulum is given as: 𝑇 = 2πœ‹ 𝐼 π‘š! 𝑔𝐻 (2) where π‘š! is the mass of the extended object, 𝐻 is the distance from the center 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 CM is given by: 𝐼!"!!"# = 1 π‘šπ‘™ ! 12 (3) where 𝑙 is the length of the rod and π‘š is the mass of the rod. For an oscillating rod, the distance from the pivot point to the CM of the rod is β„Ž, therefore, the moment of inertia of the system can be written as: 𝐼!"# = 1 π‘šπ‘™ ! + π‘šβ„Ž! 12 (4) where β„Ž is the CM from the rod to the pivot point. The moment of inertia of the hanging mass 𝑀 from the pivot point is: 𝐼!"## = 𝑀𝑑 ! (5) where 𝑑 is the distance of the mass 𝑀 to the pivot point. The total moment of inertia of the oscillator is the combination of the two moments of inertia: 𝐼 = 𝐼!"# + 𝐼!"## = 1 π‘šπ‘™ ! + π‘šβ„Ž! + 𝑀𝑑 ! 12 (6) The CM of the extended object can be found using the following equation: π‘š! 𝐻 = π‘šβ„Ž + 𝑀𝑑 (7) Equation (2) can now be re-written as: 1 π‘šπ‘™ ! + π‘šβ„Ž! + 𝑀𝑑 ! 𝐼 12 𝑇 = 2πœ‹ = 2πœ‹ π‘š! 𝑔𝐻 π‘šβ„Ž + 𝑀𝑑 𝑔 (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 β€˜blocked’. We will measure the pendulum’s 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 pendulum’s 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 β€œBuild your own Timer”. Click on β€œNext” button. Under β€œTimer Sequence Devices”, click on the photo-gate, Ch. 1 and select blocked. Do this three times. Click β€œNext” and rename the timer measurement name to β€œPhoto gate Period” 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 β€œLinear Accessory” 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 𝑦-axis in the upper graph to β€œAngle (rad) and in the lower graph to β€œPhoto gate Period”, 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, 𝐻, 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 𝐻 instead of 𝑙. To calculate the uncertainty in 𝑔, solve Equation (1) for 𝑔, and propagate the uncertainties in length, mass and period, using the partial derivatives in quadrate that we introduced last week. Does the value of 𝑔 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 𝑔 fall within 1 to 2 standard deviations of the accepted value? Make sure to write down your formula for the propagated uncertainty for 𝑔 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 𝑔, using the period of oscillations you measured. Method Period T and g (m/s2) using Simple Pendulum Data 𝑇 𝑔 Period T and g (m/s2) using Physical Pendulum (mass and rod) 𝑇 𝑔 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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TeacherSethGreg
School: Cornell University

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Abstract
The lab experiment was carried out to calculate gravitational acceleration by studying the
motion of a pendulum. The pendulum consisted of a mass attached to a rod using a string. A
weightless string attached to a mass forms the simple pendulum. Gravitational acceleration is a
factor of the period of oscillations provided that the pendulum oscillates at an angle of 100 or
less. A physical pendulum has some mass. It is made of a rod, with some weight and a mass
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