AP Physics 1 Unit 5: Torque and Rotational Dynamics

Model rotation with angular kinematics, torque, rotational equilibrium, rotational inertia, and Newton’s laws in rotational form.

[Maximum number: 7]

(7 points, suggested time 13 minutes)

Two pulleys with different radii are attached to each other so that they rotate together about a horizontal axle through their common center. There is negligible friction in the axle. Object 1 hangs from a light string wrapped around the larger pulley, while object 2 hangs from another light string wrapped around the smaller pulley, as shown in the figure above.
$m_{0}$ is the mass of object 1.
$1.5 m_{0}$ is the mass of object 2.
$r_{0}$ is the radius of the smaller pulley.
$2 r_{0}$ is the radius of the larger pulley.

(a)

At time t=0, the pulleys are released from rest and the objects begin to accelerate.

Derive an expression for the magnitude of the net torque exerted on the objects-pulleys system about the axle after the pulleys are released. Express your answer in terms of $m_{0}, r_{0}$ , and physical constants, as appropriate.

Object 1 accelerates downward after the pulleys are released. Briefly explain why.

[ 4 ]

i. For correct expressions for the torques from the weight of each object

The torques are $m_{0} g\left(2 r_{0}\right)$ for object 1 and $\left(1.5 m_{0} g\right) r_{0}$ for object 2

1 point

For indicating that the two torques are exerted in opposite directions

$\tau_{\text {net }}=m_{0} g\left(2 r_{0}\right)-\left(1.5 m_{0} g\right) r_{0}$

1 point

For the derivation of a correct answer of $0.5 m_{0} g r_{0}$

1 point

Example response for part (a)(i)

$\tau_{\text {net }}=\tau_{1}-\tau_{2}$ $\tau_{\text {net }}=m_{0} g\left(2 r_{0}\right)-\left(1.5 m_{0} g\right) r_{0}$ $\tau_{\text {net }}=0.5 m_{0} g r_{0}$

ii. For an explanation that object 1 exerts a larger torque than object 2

1 point

Total for part (a)

4 points

(b)

At a later time $t=t_{C}$ , the string of object 1 is cut while the objects are still moving and the pulley is still rotating. Immediately after the string is cut, how do the directions of the angular velocity and angular acceleration of the pulley compare to each other? Same direction Opposite directions
Briefly explain your reasoning.

[ 1 ]

(c)

On the axes below, sketch a graph of the angular velocity $\omega$ of the system consisting of the two pulleys as a function of time t. Include the entire time interval shown. The pulleys are released at t=0, and the string is cut at $t=t_{C}$ .

[ 2 ]

For a linear graph between 0 and $t_{C}$ , with an initial angular velocity of zero and nonzero slope 1 point
Scoring note: The slope can be positive or negative.
For a change in the sign of the slope at $t=t_{C}$ 1 point
AND
no discontinuity.
Example response for part (c)

Figure for Answer to Question (c) — AP Physics 1: Algebra-Based

Total for part (c)

for question 57 points

Figure for Question 5 Torque and Rotational Dynamics question 2 — AP Physics 1: Algebra-Based

(7 points, suggested time 13 minutes) Two ladybugs are standing on a rotating disk that is spinning counterclockwise, as shown in the figure above. Assume that friction in the bearings of the axle is negligible.

(a)

Figure for Question (a) — AP Physics 1: Algebra-Based

(a)

[ 2 ]

(i)

Figure for Question (i) — AP Physics 1: Algebra-Based

i. Is the angular speed of ladybug A greater than, less than, or the same as the angular speed of ladybug B ? Greater Less The same Briefly justify your answer.

(ii)

Figure for Question (ii) — AP Physics 1: Algebra-Based

ii. Is the linear speed of ladybug A greater than, less than, or the same as the linear speed of ladybug B ? Greater Less The same Briefly justify your answer.

ii. 1 point

Correct answer: "Greater"

For selecting "Greater" and explaining that linear speed is proportional to (or increases

with) distance from the center $(v=\omega R)$

Notes: No credit is earned if the wrong answer is selected.

No credit is earned for a correct selection with a wrong explanation or no explanation given.

(b)

Figure for Question (b) — AP Physics 1: Algebra-Based

(c) In a different scenario, a single ladybug is standing near the edge of the disk at a distance of 0.9 R from the center, where R is the radius of the disk, as shown in Figure 1 below. The rotational inertia of the ladybug- disk system is $I_{1}$ , and the disk completes one rotation in 2.5 s. The ladybug then walks toward the center of the disk to a distance of 0.1 R from the center and comes to a Now the rotational inertia of the system is $I_{2}$ , and the disk completes one rotation every 2.0 s.

[ 3 ]

(i)

Figure 1

Figure 2

ii. While the ladybug is walking toward the center of the disk, does it exert a torque on the disk? Yes No Briefly explain your reasoning. Wave 1 Tuning Fork Wave 2 Piano String Wave 3

[Maximum number: 2]

Figure for Question 5 Torque and Rotational Dynamics question 3 — AP Physics 1: Algebra-Based

8 points; suggested time: 15-20 minutes

A mass m is initially resting at the center of a rotating disk with

rotational inertia $(1 / 2) M R^{2}$ . The entire system is freely rotating with an

initial angular speed of $w_{0}$ . As time goes by, the mass is carefully

moved at a constant speed outward from the center. This is done in

such a way that no torque on the system is introduced.

(a)

(a) If the mass m is about half the mass of the spinning disk, estimate

the approximate angular speed of the two-mass system after the

mass m has been pushed all the way to the edge in terms of $w_{0}$ .

Justify your estimate using qualitative reasoning beyond

referencing equations.

[ 1 ]

4. (a) 1 point. Pushing the mass out to the edge will result in an increase

in the rotational inertia of the system of $m r^{2}$ or $(M / 2) r^{2}$ , which is

same as the original rotational inertia of the disk alone.

1 point. Since this is effectively doubling the rotational inertia, the

angular speed should be reduced to 1/2 of its original value.

(b)

(b) How can one push the small mass m such that no torque is

induced?

[ 1 ]

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5 Torque and Rotational Dynamics question 1

Question (a)

Question (b)

Question (c)

5 Torque and Rotational Dynamics question 2

Question (a)

Question (i)

Question (ii)

Question (b)

Question (i)

5 Torque and Rotational Dynamics question 3

Question (a)

Question (b)