AP Physics 1: Algebra-Based 5 Torque and Rotational Dynamics Questions

Model rotational motion with angular graphs, torque, inertia and equilibrium, then connect Newton’s laws to rolling, pulley and experimental systems.

Exam points

calculate angular displacement and acceleration from rates, elapsed time and rotational graphs
connect linear and angular speeds for points on one rotating rigid system using v = ωr
use force lines, perpendicular components and lever arms to identify and calculate torques
draw force diagrams and combine signed torques for beams, disks and pulley systems
rank torque effects or select force combinations that produce zero net torque
explain and calculate rotational inertia as mass distribution or rotation axis changes
apply force and torque equilibrium to solve support forces, cable tensions, masses or positions
distinguish translational motion from rotation in balanced systems and after release
apply τnet = Iα with angular kinematics to predict acceleration, speed, slowing or reversal
couple translational and rotational equations for rolling objects and massive pulleys
design and linearise experiments to find unknown mass, rotational inertia or frictional torque

Covered syllabus points

5.1 Rotational Kinematics 5.2 Connecting Linear and Rotational Motion 5.3 Torque 5.4 Rotational Inertia 5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form 5.6 Newton’s Second Law 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

[Maximum number: 3]

Figure for 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)

[ 2 ]

(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.

[ 1 ]

(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.

[ 1 ]

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)

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.

[ 1 ]

(i)

Figure 1

Figure 2

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

[ 1 ]

[Maximum number: 2]

Figure for 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)

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)

How can one push the small mass m such that no torque is induced?

[ 1 ]

[Maximum number: 10]

Students are investigating balancing systems using the following setup. The students have a spring scale of negligible mass that is fixed to one end of a uniform meterstick. The center of the meterstick is attached to a stand on which the meterstick can pivot. There is a hook of negligible mass fixed to the top of a block of mass $m_{0}$ . The hook can be attached to the meterstick through one of the small holes in the meterstick, as shown in Figure 1. The students do not have a direct way to measure the mass of the block. The block cannot be attached to the spring scale.

Figure 1

The students are asked to take measurements that will allow the students to create a linear graph whose slope could be used to determine the mass $m_{0}$ of the block.

(a)

Describe an experimental procedure to collect data that would allow the students to determine $m_{0}$ . Include any steps necessary to reduce experimental uncertainty.

[ 2 ]

A For describing a procedure that includes both of the following:

Point A1

- Attaching the block to the meterstick

- Measuring the force exerted on the left end of the meterstick while the block is

attached

For a procedure that indicates a reasonable method of reducing experimental

Point A2

uncertainty

Examples of acceptable responses may include the following:

- Repeating the force measurement for one location of the block

- Collecting force measurements for multiple locations of the block

Example Response

Attach the block of unknown mass $m_{0}$ through one of the small holes in the meterstick.

Measure the distance from the stand to the block. Record the force reading from the

spring scale when the meterstick is horizontal. Repeat data collection by attaching the

block to the meterstick through a different small hole in the meterstick. Repeat the

measurements by attaching the block at each of the small hole locations numerous

times.

(b)

Describe how the data collected in part A could be graphed and how that graph would be analyzed to determine $m_{0}$ .

[ 2 ]

B For indicating appropriate quantities that can be plotted so that $m_{0}$ can be determined

Scoring Notes:

- Any response that correctly identifies the functional dependence between force

and distance earns this point, regardless of any coefficients that contain

numbers or physical/fundamental constants, or which axis is chosen to graph

each of those quantities.

- This point may be earned independently of the response in part A.

For describing how the slope is related to the unknown mass $m_{0}$

Examples of acceptable responses may include the following:

Graph	Analysis
Force as a function of the distance from the stand to the block	The slope is equal to $m_{0}$ times g divided by the distance from the stand to the spring scale.<br>OR<br> $m_{0}$ is equal to the slope times the distance from the stand to the spring scale divided by g.
Force as a function of the ratio of the distance from the stand to the block to the distance from the stand to the spring scale	The slope is equal to $m_{0}$ times g.<br>OR<br> $m_{0}$ is equal to the slope divided by g.
Force as a function of g times the distance from the stand to the block	The slope is equal to $m_{0}$ divided by the distance from the stand to the spring scale.<br>OR<br> $m_{0}$ is equal to the slope times the distance from the stand to the spring scale.
Force as a function of g times the ratio of the distance from the stand to the block to the distance from the stand to the spring scale	The slope is equal to $m_{0}$ .<br>OR<br> $m_{0}$ is equal to the slope.
Force times the distance from the stand to the spring scale as a function of the distance from the stand to the block	The slope is equal to $m_{0}$ times g.<br>OR<br> $m_{0}$ is equal to the slope divided by g.
Force times the distance from the stand to the spring scale as a function of g times the distance from the stand to the block	The slope is equal to $m_{0}$ .<br>OR<br> $m_{0}$ is equal to the slope.

Example Response

A graph of the force as a function of the distance from the stand to the block could be

used to determine the unknown mass. The slope of the graph times the distance from the

stand to the spring scale divided by g will result in the value of the unknown mass $m_{0}$ .

(c)

The students have an identical meterstick of mass M that is now attached to an axle that is fixed to a wall. The meterstick is free to rotate with negligible friction about the axle. The meterstick is suspended horizontally by a string that is connected to a spring scale of negligible mass, as shown in Figure 2.

Figure 2

The angle $\theta$ that the string makes with the meterstick can be varied by attaching the string to one of the pegs located along the wall. The students use the spring scale to measure the tension $F_{T}$ required to hold the meterstick horizontal. Table 1 shows the measured values of $\theta$ and $F_{T}$ .

Table 1

The students correctly determine that the relationship between $F_{T}$ and $\theta$ is given by

F_{T}=\frac{5 M g}{6 \sin \theta} .

The students create a graph with $\frac{1}{\sin \theta}$ plotted on the horizontal axis.

[ 4 ]

(i)

Indicate what measured or calculated quantity could be plotted on the vertical axis to yield a linear graph whose slope can be used to calculate an experimental value for the mass M of the meterstick.

Vertical axis:

Horizontal axis: $\frac{1}{\sin \theta}$

[ 1 ]

C (i) For listing a quantity that could be plotted on the vertical axis to produce a linear graph

Point C1

whose slope can be used to determine the mass of the meterstick ${ }^{1}$

Examples of acceptable responses may include the following:

- $F_{T}$

- $\frac{F_{T}}{g}$

- $\frac{6}{5} \frac{F_{T}}{g}$

- $\frac{6}{5} F_{T}$

Scoring Note: Any response that correctly identifies the functional dependence

between tension and angle earns this point, regardless of any coefficients that contain

numbers or physical/fundamental constants.

Example Response

Vertical axis: $F_{T}$

(ii)

On the blank grid provided, create a graph of the quantities indicated in part C (i) that can be used to determine M.

- Use Table 2 to record the data points or calculated quantities that you will plot.

- Clearly label the vertical axis, including units as appropriate.

- Plot the points you recorded in Table 2.

Figure 3

[ 2 ]

(iii)

Draw a straight best-fit line for the data graphed in part C (ii).

[ 1 ]

(iii) For drawing an appropriate best-fit line that approximates the trend of the plotted data

Point C4

Scoring Note: If the graph produced is nonlinear, then a line or curve that approximates

the trend of that data can earn this point.

${ }^{1}$ Update made post-scoring to correct typo that did not impact scoring.

Example Response

Figure 3

(d)

Using the best-fit line that you drew in part C (iii), calculate an experimental value for the mass M of the meterstick.

[ 2 ]

D For correctly relating the slope of the best-fit line to the value of M

Examples of acceptable responses may include the following:

Graph	Analysis
$F_{T}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=\frac{5}{6} M g \text { OR } M=\frac{6}{5} \frac{\text { slope }}{g} \end{gathered}$
$\frac{F_{T}}{g}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=\frac{5}{6} M \text { OR } M=\frac{6}{5} \text { slope } \end{gathered}$
$\frac{6}{5} \frac{F_{T}}{g}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=M \text { OR } M=\text { slope } \end{gathered}$
$\frac{6}{5} F_{T}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=M g \text { OR } M=\frac{\text { slope }}{g} \end{gathered}$

For a value of M within a range of 0.90 kg to 1.15 kg

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (i)

Question (ii)

Question (b)

Question (i)

Question 3

Question (a)

Question (b)

Question 4

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (iii)

Question (d)