Lab4 Rotational Dynamics 2017 PDF

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Lab 4 Rotational Dynamics Introduction Rotating systems involve concepts that are similar to that of translational motion. In this experiment, we will be studying various aspects of rotation about a fixed axis, including: • angular velocity, the relation between torque and angular acceleration; • the relation between rotational inertia and angular motion; • the conservation of mechanical energy; • the conservation of angular momentum when no external torque is applied. Experiment The equipment is simple and relies on the use of air bearings in order to reduce the effect of friction. It is thus the rotational equivalent of the air track which you may have used in other courses to study linear dynamics or an air hockey table you may have played as a child. In this experiment, the discs that we are using have black and white bars on the circumference, each of 1 mm wide. The reading on the digital display of the PASCO rotational dynamics model is a measure of the number of black bars which pass the optical detectors each second. Therefore one count on the digital display corresponds to a movement of 2 mm on the circumference. If the radius of the disc is known (you can find this information on the sheet inside the equipment box), we can calculate the angular velocity of the rotating system as follows: 0.002x v = rad/s, r r where ω is defined as the angular velocity, r is the radius of the disc in meters and x is the reading on the digital display and v, the velocity perpendicular to the radius, is v = x · 2 mm/s= 0.002x m/s. From there, the angular acceleration can be calculated. We can then verify that F·r = Iα, where F·r is the applied torque and I is the moment of inertia, both about the same fixed axis. We will also be verifying the energy conservation principles of a rotating system by determining whether the change in potential energy and kinetic energy are equal. The potential energy lost is mg∆h, where ∆h is the change in height from the initial starting point. The velocity of the mass at each measured height can be determined from v 2 = v 20 + 2a∆h, with 1 2 v0 = 0, a = 2h t2 (from h = v0 t + 2 at ), and the angular velocity of the rotating disc from v = ωr, where r is the radius of the small pulley. The kinetic energy of the rotating disc is 1 Iω 2 , where I is calculated as above, and the kinetic energy of the falling mass is 1 mv 2 . 2 2 ω=

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Lab 4. Rotational Dynamics

Finally, we will be investigating the angular momentum of the system. Angular momentum is a rotational analog of linear momentum, we will verify whether angular momentum is conserved before and after collision in a rotating system. Part 1 Procedure a) Determination of angular acceleration, α The following procedure will be performed on four different systems, the first of which (System 1) is the following: 1. Record the mass and radius of each steel disc, the aluminum disc, the accelerating mass, and the small and large pulley. (The values can be found on the inside cover of the equipment box). 2. Make sure that the air bearing table is sitting ‘flat’ using the level provided in the box and the adjustable feet of the apparatus. 3. It is possible that the frictional loss will be less when the discs are rotating in one direction than the other. Find out which is the best and use that direction as much as possible. One way to do this is to spin the steel disc by hand until the digital readout is about 400 and then let it continue on its own for 20 s and compare how much it has slowed down. 4. Open the tab “Part 1 a)”. Put 2 steel discs on the rotational model and slowly turn on the air pumper to set the pressure to 9 psi. 5. Make sure that the hose clamp (clamp around the tube underneath the white plate) is open so that the bottom disc rests firmly on the plate. 6. Attach the thread to the small pulley by winding the thread around it. Use the solid, black cap screw to secure the pulley at the center of the top disc. The thread should fit the slot of the pulley, run over the indentation on the cylinder and suspend a mass of 25 g. The top of the mass should be leveled with the top of the ruler. 7. Press “record” and hold the top disc stationary for a moment until the PASCO digital display reads 0. Then release it immediately . 8. Avoid taking the first and last readings of data due to their inaccuracy. You should have 5 to 8 readings Repeat the experiment with the different systems: System 1 No change; i.e. should be your default setup. System 2 Change the torque applied to the disc by changing the accelerating mass to 50 g. System 3 Change the torque applied to the disc by using the 25 g mass with a larger pulley. Page 2

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Lab 4. Rotational Dynamics

System 4 Change the moment of inertia of the disc. Close the hose clamp and replace the solid, black cap screw with the hollow, black cap screw. The top steel disc should now rest on the bottom steel disc and both discs should move as one mass.

b) Verify Conservation of Energy Only do the following for System 1: Open up the tab labeled “Part 1b)” and use the timer at the bottom of the page as a stopwatch. Starting the 25 g mass at the same point for each trial, record the time taken for the descending mass to drop several distances, each increased by 15 cm compared to the previous distance (15 cm, then 30 cm, 45 cm, etc.).

Part 2 Conservation of Angular Momentum Use an air pressure of 9 psi in this experiment - unless otherwise specified 1. Use the two steel discs. Make certain that the tube clamp underneath is closed so that the bottom disc rotates freely. 2. Place the ‘Drop Pin’ through the hole in the center of the top disc. The top plate should now rotate independently of the bottom disc. 3. Hold the bottom plate stationary and spin the top plate so that a reading of 300 to 400 is obtained. Once you have stopped accelerating the upper disc wait two seconds and take the next frequency reading. The switch should be set at TOP. Release the bottom disc. 4. Immediately pull the Drop Pin straight up. The two discs will collide. Wait two seconds and record the frequency measurement. If you take the measurement too soon, the discs may still be in the collision process and your results will be skewed. 5. Repeat the experiment with different initial velocities of the top disc. 6. Repeat the experiment using the aluminum top disc in place of the steel top disc. If the aluminum disc does not drop when the ‘Drop Pin’ is pulled, the air pressure should be slightly reduced.

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In the following section both discs will have an initial angular momentum. The two discs can rotate in the same, or opposite directions. You will have to decide which direction will be positive and which will be negative. Note: It is very important that you remain consistent with your definitions throughout this procedure. 1. Use the two steel discs, close the tube clamp under the display housing and insert the Drop Pin in its hole. Both discs should now rotate freely. Use an air pressure of 9 psi for this part of the experiment. 2. Give the top disc an initial velocity in one direction and the bottom plate an initial velocity in the other direction. In as short a time as possible measure the two angular velocities. Although the friction between the disks and the table is small, it still plays enough of a role to wear down the momentum of the discs. First measure the top disc frequency by setting the switch to TOP. As soon as that measurement is taken set the switch to BOT. Wait two seconds and take the next measurement. This will be the bottom disc frequency. 3. Once these two measurements are made IMMEDIATELY pull out the Drop Pin and let the discs collide. Wait two seconds and measure the frequency of the two discs together. 4. Repeat the experiment with different initial velocities and directions of rotation. Use the aluminum top disc for some of these measurements. Tabulate all data and results. Analysis Part 1a): Verify Γ = F r = Iα • Calculate the angular acceleration that corresponds to each of the 4 systems as well as the moment of inertia of the disc considering it as a cylinder. Record your results as shown in table 1.

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Lab 4. Rotational Dynamics

Time sec

Digital Display x

·x ω = 0.002 r = 0.0314 · x rad/s

α=

∆ω ∆t

 rad  s2

0 2 4 . . .

Table 1: 25 gram mass - Small pulley

• Use the average of the values of α to calculate Iα. Then calculate the torque Γ = m(g−a)r ≃ mgr = F r where m is the suspended mass, g is the gravitational acceleration and r is the radius of the small pulley. Assume that the acceleration, a, is negligible with respect to g, the acceleration due to gravity. Record your calculations as shown in table 2

1. 2. 3. 4.

SYSTEM Fr (see note below) N·m 25 g. mass - small pulley 50 g. mass - small pulley 25 g. mass - large pulley 25 g. mass - small pulley & both steel discs

Iα N· m

% DIFF

Table 2: Comparison Table

Part 1b): Verify energy conservation • Determine the decrease in potential energy of the mass, the kinetic energy acquired by the falling mass and the kinetic energy acquired by the rotating disc. • For each measured height, tabulate mg ∆h, 21 Iω 2 , 12 mv 2 , the sum E = 12 Iω 2 + 21 mv 2 and the % difference between E and mg∆h. Plot mg∆h vs E and find the slope. Part 2: Verify conservation of angular momentum • Calculate the moments of inertia of the two discs considering them to be cylinders. Page 5

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• Calculate the angular velocities and angular momentum of the top and bottom discs from the frequency measurement before the collision (Note that momentum has a polarity depending upon the direction of rotation). • Calculate the total angular momentum of the system from the frequency measurement after the collision. Hand-in instructions Part 1: Angular Acceleration, 2nd Law of rotation, Conservation of Energy 1. Required Tabulated Data: 4 data tables. (a) Raw Data: Time/x/ω/α for all 3 systems (b) Newtons 2nd law for rotation: Torque/Iα (c) Conservation of Energy, Raw data: ∆h/t/α/v/ω (d) Conservation of Energy Results, ∆h/pot. E/Trans. KE/Rot. KE 2. Graphs/Figures: Lost Potential Energy vs. Total Kinetic Energy Gained (a) Plot this figure using Python (b) Use the least-squares method (see Text sections 5.2 5.6 and Thursdays lecture) to determine the best-fit line to your data and include this line on your figure along with your data (with error bars!). Also include the max/min lines in this figure to demonstrate your uncertainty in the parameters you determine (i.e. slope and intercept) using the least-squares method. (c) Plot your residuals; i.e. the discrepancy between your data and the best fit line. See section 5.6 of your text and the example is Fig. 5.11. 3. Discussion: (a) Are your results in agreement with conservation of energy? Why or why not(the difference between potential energy lost and the total kinetic energy gained should be part of this discussion). You must include your least-squares calculations. These can be done by hand or using Python/Jupyter. Excel spreadsheets will not be accepted. Part 2: Conservation of Angular Momentum 1. Required Tabulated Data. (a) Raw Data for angular momentum before collision, angular momentum after collision (with bottom disc stationary). Both Steel/Steel and Aluminum/Steel Page 6

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Lab 4. Rotational Dynamics

(b) Raw Data for angular momentum before collision, angular momentum after collision (with bottom disc non-stationary). Both Steel/Steel and Aluminum/Steel 2. Discussion: (a) Are your results in agreement with the principle of conservation of momentum? Why or why not? Explain. Note: You must include sample calculations for EVERY computed quantity that appears in a data table or figure; i.e. show sample calculations for each calculated quantity AND its associated uncertainty. It is really impossible to mark your work without these sample calculations. The relevant sample calculations for the data tables and figures below should follow the data table or figure in your report in a sensible and well-organized way Tables and Figures must be properly formatted. this is the last time I will mention this. It will be assumed from now on.

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