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Angular lab:

Simulation of rotational , rotational and conservation of

Purpose 1. To apply rotational and dynamics to a falling while suspended from a pulley;

2. To study the conservation of angular momentum and apply it to the situation of two colliding disks

Introduction This online lab exercise will allow you to simulate several experiments in rotational . In the first two parts, hanging will be allowed to fall, and you will calculate their final and compare them with the values presented by the simulators. The two different simulators used in Part 1 and Part 2 allows you to change the type of pulley, and therefore its of inertia. In Part 3, you will explore the concept of angular momentum through two videos including a video experiment. You will apply the conservation of angular moment to a system comprising two disks which undergo an inelastic , in order to find the final angular of the system.

Angular momentum is a vector denoted by �. For a point object of mass �, the of the angular momentum, �, is given by

� = � � sin � = �� = �� , (1) where �is the perpendicular from the point about which angular momentum is computed to the direction of p, the linear momentum of the particle object. Similarly � is the component of the linear momentum that is perpendicular to �. In this description, � is the radial vector from the axis about which angular momentum is computed to the particle.

For an extended object, � = ��, where � is the of the extended object or system, and � is the . Note that for a point object you can still use � = ��, and since I for a point object � = �� , and � = ��, we recover L = rꓕp or rpꓕ.

Software This lab runs in any web browser. The simulation tools used are: Part 1: https://ophysics.com/r5.html Part 2: http://physics.bu.edu/~duffy/HTML5/block_and_pulley_energy.html Part 3: 3 separate videos, links given below in the text.

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Part 1: Rotational dynamics For the diagram shown, the hanging mass is allowed to fall, thus pulling on the string and rotating the pulley.

1) Draw the free body diagram. Use Newton’s second law and also the rotational version of Newton’s second law: Net = � � , where � is the moment of inertia, and α is the angular . Using the two equations, derive a formula for the of the pulley in terms of �, � and � (and �). To get you started, note that:

• The torque acting on the pulley is caused by the weight of the pulling mass. That weight force, �� generates a torque of ��� on the pulley, about an axis running through its middle. • The magnitude of the acceleration of the hanging mass is equal to the magnitude of the acceleration of the rim of the pulley. • The tangential acceleration, �, of a rotating object at � from the axis of can be written in terms of the angular acceleration as follows: � = � �. • In this experiment you will use two different pulley types and therefore you will use two different expressions for the moment of inertia, � of the pulley. Those expression for � are given below.

2) Open the simulator: https://ophysics.com/r5.html

In the simulation type, select falling mass, and in mass distribution select solid cylinder. The moment of inertia of a solid cylinder of mass M, radius r about an axis that M, mass runs perpendicular to the cylindrical face, though its of pulley center, is ��.

� = 1.5 3) Adjust the mass of the falling mass to kg, r, radius of and the mass of the pulley, � to 5 kg, and the radius of pulley the pulley, � to 0.5 m. Notice that we are using the m, pulley as a solid cylinder. Click start and wait till � and � pulling become displayed then click pause. mass

4) Using step 1 above, calculate the angular acceleration � of the pulley and also the linear acceleration a of the rim of the pulley and of pulling mass �. Compare to the measured values given by the simulator.

5) Calculate the net torque on the pulley. Compare with the measured value by the simulator.

6) Repeat the above steps for the pulley as a solid sphere. The moment of inertia of a solid cylinder of mass M, radius r about an axis through its center is: ��.

After you complete the above for both a solid cylinder pulley and a solid sphere pulley, go to the next part of the lab: Part 2, considerations.

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Part 2: Energy considerations Recall, for a mass m at height y above some nominal zero of gravitational :

Gravitational potential energy = mgy, translational = ��2

For a rotating object:

Rotational kinetic energy = ��2, where � is the moment of inertia, and � is the angular .

Calculating the final velocity

1) Using the same diagram shown above, write the expression (using symbols: m, yi, v, I, q, g,…) for the initial energy of the system (Earth+ Block + pulley):

Gravitational potential energy of the block + translational KE of the block + rotational KE of the pulley.

2) Write an expression for the final energy of the system after the block has fallen through a distance ∆y.

3) Use the principle of conservation of energy to write an equation linking each term in the energy of the system at the start and end of the fall.

4) Open the Simulator to be used in this part of the lab: http://physics.bu.edu/~duffy/HTML5/block_and_pulley_energy.html

5) Under the energy graph, select “show energy for” Earth + Block + pulley. Adjust mass of the block to 1.2 Kg (default), mass of the pulley to 1 kg, choose the type of the pulley as uniform solid disk (default). Note that this simulator approximates the acceleration due to , � to 10 m/s2 instead of 9.81 m/s2.

6) The initial height of the block is y1 = 2.4 m and the final height of the block is y2 = – 3.16 cm. (These heights are coded into the simulator, but not shown to the user.)

From your equation in step 2 above, use the values of initial and final heights to calculate the final velocity of the block (which is equal to the final velocity of the rim of the pulley. Do you know why they are equal?)

Notice that to calculate a velocity here, you do not need the radius of the pulley. Do you know why?

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7) Run the simulator (hit Play) until its stops automatically. Compare the final velocity that you calculated in the previous step. with the final value of “v” measured by the simulator, shown in the column of data to the left of the screen. Note the elapsed at the end of the simulation.

Calculating the distance fallen

8) Let’s next calculate ∆� using

�|��| = |∆�| (2) where ∆� = �� − ��. Why is equation (2) correct?

Notice that ∆y in the change of gravitational potential energy should be negative because y2 is less than y1. We need r∆q to find Dy using this method. Recall the following from rotational kinematics:

� ∆� = � � + �� , � � where � is time of rotation and is given by the simulator, and α is the angular acceleration. If we multiply the equation for �� by r the radius of the pulley, we get:

��� = � � � + � � �. �

But �� = � ��� � � = � . Here, � = 0 �/�, (since the system starts from rest), and � is the linear acceleration of the rim of the pulley and also of the block, and is given by the simulator.

Calculate ∆� using �|��| = |∆�|. Notice that both a and Dy as vectors should have negative signs, if we take the downward vertical as negative.

Compare your computed value of ∆y with the values of initial and final height given to you in step 6.

9) Repeat the above calculations and simulation with the pulley type as a ring, given that y1= 2.4 m and y2= -1 cm in that instance. You will:

• First, use the method of conservation of energy in step 6 to calculate the final velocity from the new value of Dy. Run the simulation, compare the value of final velocity with that given at the end of the simulation. Note the (new value of the) final time elapsed. • Second, use the method of step 8 (rotational kinematics) to calculate the value of ∆y from the new total time elapsed and the new value of linear acceleration given by the simulator (i.e.

without using the given y1 and y2). Compare your computed value of Dy using this second

method with the value of Dy = y2- y1 using the values of initial and final height given in step 9.

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Part 3: Conservation of angular momentum You can provide your answers for part 3 in the data sheet on page 7

1) Watch this video, then answer the questions in step 2 below. : https://www.khanacademy.org/science/ap-physics-1/ap-torque-angular-momentum/conservation-of- angular-momentum-ap/v/conservation-of-angular-momentum

2a) What condition is required for the angular momentum of a system to be conserved? b) For a system undergoing rotational motion that has a conserved angular momentum, what is the effect of reducing the average distance of the mass particles of the system from the axis of rotation? c) What happens to a system that has conserved angular momentum, when the mass of the system is increased (in particular what happens to �, the angular speed of the system)?

3) Note that the magnitude of � = ��, where � is the perpendicular distance from the axis of rotation, and �is the angular speed. Now watch this video, then answer the questions in step 4 below: https://www.khanacademy.org/science/physics/torque-angular-momentum/torque- tutorial/v/constant-angular-momentum-when-no-net-torque

4) Derive the relation between net external torque applied to a particle of mass � and the angular momentum � of the mass.

5) When dealing with an extended object, we use � = ��, where � is the angular momentum of the system and � is the moment of inertia. Note that � and � are vector .

Watch this video https://www.youtube.com/watch?v=hgcudPr73LU. In the last part of this video, a problem is introduced: an initially stationary disk is dropped on an initially rotating disk. Using the values given in the video, calculate the final angular speed, of the system in the data sheet on page 7. Show your working.

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DATA SHEETS

Part 1: Rotational dynamics a) Pulley is a uniform solid cylinder

Show your here for calculating I, a, α, and net torque:

Calculated I (Kg m2) Calculated a (m/s2) Calculated α (rad/s2) Calculated net torque (N.m)

a measured by simulator α measured by simulator net torque measured by simulator

b) Pulley is a solid sphere

Show your work here for calculating I, a, α, and net torque:

Calculated I (Kg m2) Calculated a (m/s2) Calculated α (rad/s2) Calculated net torque (N.m)

a measured by simulator α measured by simulator net torque measured by simulator

Part 2: Energy considerations

Copy from pages 2 &3, step 1) Initial total energy expression step 2) Final total energy expression step 3) Conservation of energy equation

Brooklyn College 6 a) Pulley is a uniform solid disk calculated v final v final measured

Show your work here for calculating vfinal for step 6, using (m/s) by the simulator given initial and final heights: (m/s)

Dy= y2-y1 (y2 and Dy computed using step 8 time of rotation t measured by simulator= ...... s, y1 are given in magnitude of linear acceleration a, measured by the step 6, just subtract) simulator= ……….m/s2 = m = m Show your work here for calculating Dy for step 8, using the method discussed in step 8:

Why is the magnitude of rDq equal to the magnitude of Dy?

b) Pulley is a ring

Show your work here for calculating vfinal for step 9 (first bullet point), using given initial and final heights: calculated v final v final measured (m/s) by the simulator (m/s)

time of rotation t measured by simulator=...... s, magnitude of linear acceleration a, measured by the simulator= ……….m/s2

Show your work here for calculating Dy for step 9, Dy= y2-y1 (y2 and y1 Dy computed using step 9 using step 9, second point: are given in step 9, (second method) just subtract) = m = m

Part 3: Conservation of angular momentum

Write your answers to the questions listed in part 3 here.

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