Rotational Motion, Moment Of Inertia, And Motion Of A Simple Pendulum (815 words)

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/Rotational Motion, Moment of Inertia, and Torque Introduction Consider the system of a mass, rn, hanging by a light string which is wrapped (counterclockwise for the lab setup) around the smaller of two concentric, thin cylinders of masses M1 and Jvh, as shown in figure 13.1. Figure 6.1: The setup for this experiment. Notice how we are defining +x to be downward The equations of motion for the mass m and rotating cylinders M, + M2 are ma= rng -T an<l (6.1) (6.2) wlicre T R 1 is the torque on the inner wheel by the tension (T) in the string, l is the moment of inertia. and o is the angular acceleration. Note here that the positive direction is set to downward, 22 EXPERIMENT 6. ROTATIONAL MOTTON, MOMENT OF fNeRTIA, ANO TOHQUE Use the mechanical times and the apparatus on the wall to cletcrrni11c the tirnc it takl's th<: 1mll!f> to fall. Setup the distance between the paddles to !Jc as far as possible. Make a loop i11 tlH· string and fit it around the nub on the bakelite disk. RotF1tc the disk so Lha,t, tlw string winds urow1d disk. Setup both platform::;. You have to hold 011 Lhe sLri11g from Lhc Lop 011c. Tlw ti1r1 ·r should read zero. Place the mass hai1ger on the Lop platforlll so Lh;:i.t it string is still wound ar<Jund the disk. When you release the top platform, the timer will start nrnl the rnass will start to fall, causing the disk to rota.Le. If the mass suclclc1ily drops, thc11 you have to r start, we want all of the downward acceleration of the mass to go Lo rota.Ling tlic cli::;k. Repeat for five measurements. Distance So m Trial Time (::;) 1 \o 2 5 Mean Standard Deviation Analysis • Calculate the mean and standard deviation for the times of falling mass. • Recall from kinematics, that the acceleratio11 of an object starting from rest is given by 2d a=,;· (6.8) From the average time, calculate the acceleration of the hanging mass and then use the acceleration to find the moment of inertia of the wheel. • We want to get an idea of the uncertainty in our measurement. Use the following formula to propagate your standard deviation/error for your time measurements to an error on your implied moment of inertia (6.9) where tis the mean time and O-t is the standard deviation in the time measurement. • Calculate the theoretical moment of inertia of the wheel and compare it to your cxpcrirncntul value. Find the percent error. Does the theoretical value fall within the uncertainty of your experimental value? Motion of a Simple Pendulum Objective The simple pendulum is a device with a long history, and which has been studied in great detail along with its many, more complicated relatives (conical pendulum, spherical pendulum, etc.). This laboratory experiment focuses attention on the connection between the pendulum and its oscillatory characteristics, such as its period of oscillation. The primary objective of this laboratory experiment is to verify the theoretical relation between a simple pendulum's period (P) for small oscillations and its length (L) from the axis of rotation to the pendulum bob of mass (m). Further, the theory predicts that there will be no correlation between the period and the mass of the pendulum bob. Introduction The simple pendulum is described in the diagram below. e : L The equation of motion for the pendulum bob is most easily written in terms of the angular variable, 0, and the angular acceleration, a: mLa =-mgsin0, or, for small angles where sin 0 Re 0 , You can see that the mass is no longer a variable in the equation of motion. The solution for the angular position, 8(t), of the bob is 0(t) = 80 cos(wt+cp), where cp is just a constant to match the right value of the cosine term with the initial conditions at t = 0, and Ba is the maximum angle of oscillation. Finally, the angular frequency, w, is defined such that T his relationship between the angular frequency and the length of the string on the pendulum is clearly independent of mass. It states that tu oc I/ VI , so that the longer the pendulum, the lower the angular frequency (i.e. the slower the pendulum oscillates). We can connect this to the period of the pendulum through the definition for the period (P) and the pendulum frequency (f), which is measured in cycles per unit time (e.g. Hertz, or Hz, are units cycles per second). So, or P=!=2:n; / (l) ' p = 2:n; 􀀈. Experimental Procedure In this lab experiment, you are asked to measure the period of oscillation, using a stopwatch. You will measure the period five (5) times for each case of varied parameters m, L, and Ba. For each value of L, you should measure the period for each case of Ba, using the 2-inch steel bob. For one case of Ba, you are asked to measure again, with the 2-inch brass bob, and then with the 2-inch lead bob. You can, and should, measure the mass of each bob using the digital balance in the front of the lab. Be gentle with the balance, and make sure to follow the instructions your lab instructor has provided on how to zero the balance, and how to read it. It is important that you only change the length between sets of measurements for the other variables, so that you don't have to go back to a length you tested earlier, as it may be difficult to get exactly the right length again. Also, your measurements will go a lot faster if you minimize the time needed to adjust the length of the string. To measure the period, allow the pendulum to complete several complete cycles (e.g. backand- forth is one cycle), and time the entire set of cycles observed. Then divide your time by the number of cycles you timed to find the average period of one cycle. Note that friction will begin to slow the pendulum down. So, only time a few cycles (maybe 3 - 5) for each trial. Use the same number of cycles for all your timing measurements, once you've established how many you wish to use. You should report the number of cycles you timed, but you only need to tabulate the single-cycle period in your data tables. 1. Test the cases of three lengths: 10 cm, 20 cm, and 30 cm. 2. Test each length with initial angles (8a) of 10° , 20° , and 30° , using the 2-inch steel bob. 3. Test the case of 10° again with the 2-inch brass bob, and then the 2-inch lead bob. 4. Make data tables for all of your cases in Excel, or your favorite spreadsheet program. For each set of trials (5 trials for every case), you should calculate an average and a standard deviation. 5. Make the following plots, using the average values you determined in your spreadsheet: a) Plot measured period versus ,'I. Do this for all three values of 00 on your plot, so you can compare the resulting curves. b) Add a linear trendline to each of your sets of data (one set for each 0o), making sure to include the equation for the line on the plot. To do this, right-click on each set of data, and select "Add Trendline," making sure to select "linear" and "show equation on plot" in the dialog box which pops up. c) In your analysis, discuss whether the best-fit trendlines are similar, or if you see a trend in the resulting slopes. Also, compare the slopes for each to the remaining constants in the formula for the theoretical period: e.g. slopes should equal 2 TI/ sqrt(981. cm/s2), assuming that you're using your length measurements in centimeters. d) What can you say about the period you measured for the cases of increasing initial angle? Does the theoretical solution for small oscillations hold, and if so how well? If not, how badly does it seem to be wrong? 6. Repeat the procedure you did in (5) for the three cases of different masses, all starting from 80 = 10° . a) How do the best-fit trendlines compare for the three different masses over the range of pendulum lengths? b) For each bob mass, calculate the percent deviation of the measured period from the theoretical period, which equals 2 TI sqrt(L / 981. cm/s2 ). In your lab report discussion, address the question of whether there is a noticeable difference between these values for the different bob masses. If you see a difference, to what do you attribute it? 7. As you should always do, consider your sources of error, and variables you couldn't control in the experiment. At least mention what some of these are in your lab report. It is most useful to do this in the places where you report data that would be affected by these errors, but you should certainly mention the impact of these error sources in your final summary of your results.

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As the mass falls, the tension in the string exerts a torque on the lighter cylinder causing both of them to rotate. The text of the lab derives an equation for the moment of inertia of the entire apparatus based on measurable parameters. This is then compared to an approximation of the theoretical moment of inertia considering only the contribution of the heavier cylinder. Some minor calculation work is also included to generate an error in the implied moment of inertia from the standard deviation of the time trials.
Data Collection:
The first step is to time the fall of the mass over a distance of 1.50 meters for five separate trials. Then the mean and standard deviation of these trials...
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