Unit 5 / Torque and Rotational Dynamics
Unit 5: 36 practice problems
Work from memory first. Draw a diagram, choose a system, and say why your model applies. The 30 selected questions revisit every topic in Torque and Rotational Dynamics; 6 additional problems at the end ask you to explain and apply the models. The worked solutions live on their own page.
This is mixed topic review: short concept checks, calculations, and experiment tasks appear together. Choose a topic below to revisit its lesson, then return to the questions that challenge your model.
Question 1 · AP topic 5.1
1. Rotational Kinematics · 4
A wheel initially turns at +12 rad/s and comes to rest with constant α = −3.0 rad/s². Find the stopping time and angular displacement.
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Question 2 · AP topic 5.2
2. Connecting Linear and Rotational Motion · 4
A string unwinds without slipping from a 0.040 m-radius axle. It moves 0.60 m while the wheel turns. Find the angular displacement magnitude. If the string then has acceleration 0.20 m/s², find the angular acceleration magnitude.
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Question 3 · AP topic 5.3
3. Torque · 4
A light horizontal bar pivots at its center. A 6.0 N downward force acts 0.30 m to the left, and a 4.0 N downward force acts 0.20 m to the right. Determine the net torque using counterclockwise as positive.
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Question 4 · AP topic 5.4
4. Rotational Inertia · 4
Two 0.50 kg point masses are attached at the two endpoints of a massless 0.80 m rod. Find I about a perpendicular axis through its center and about a parallel axis through one end.
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Question 5 · AP topic 5.5
5. Rotational Equilibrium and Newton’s First Law in Rotational Form · 4
A uniform 2.0 m board weighs 60 N and is pivoted 0.80 m from its left end. A 100 N load hangs from the left end. What downward load at the right end balances it?
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Question 6 · AP topic 5.6
6. Newton’s Second Law in Rotational Form · 4
Compare two separate trials of a rotating frame. In trial A its masses remain fixed in a configuration with I = 0.050 kg·m². In trial B they remain fixed farther outward, with I = 0.080 kg·m². Apply constant net torque +0.40 N·m for 2.0 s in each trial. Compare α and the change in ω.
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Question 7 · AP topic 5.1
7. Unwrap the filmed turn
A mark on a wheel has camera orientations 350°, 30°, 90°, and 170° at t = 0, 1, 2, and 3 s. It turns continuously counterclockwise and completes less than one turn between frames. Construct its unwrapped angular displacement from the first frame in radians. Use the data to decide whether constant angular acceleration is a reasonable model, and state one limitation of drawing that conclusion from only these four frames.
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Question 8 · AP topic 5.2
8. Check two measurements against rigidity
Two marks are fixed to a rigid disk at radii 0.10 m and 0.30 m. During the same steady-rotation interval, a student reports tangential speeds 0.40 m/s and 0.90 m/s. Each speed has a possible error of at most 0.03 m/s; take the radii as exact for this task. Are the measurements consistent with one angular speed? Show a comparison that decides, then propose a specific measurement check before claiming that the outer mark really has a different angular speed.
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Question 9 · AP topic 5.3
9. Decide which pull turns harder
A meterstick is held about the same pivot in two trials. Trial A uses a 6.0 N pull applied 0.40 m from the pivot at 90° to the position vector. Trial B uses a 12 N pull applied 0.30 m from the pivot at 30° to that vector. Both pulls tend to turn the stick the same way. A student says B must make the larger torque because its force is larger. Evaluate the claim. Keeping B’s force and angle unchanged, choose a new application distance that would match A’s torque, and identify what distance must actually be measured.
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Question 10 · AP topic 5.4
10. Choose a graph that can test the inertia model
A light rotating frame has inertia 0.010 kg·m². Two 0.20 kg point masses are attached at equal adjustable radii r, and a known constant net torque 0.040 N·m drives each fixed configuration. You can determine angular acceleration α for several radii. Choose between plotting α against r² or plotting 1/α against r² if your goal is a straight-line test of the point-mass inertia model. Derive the predicted slope and intercept for your choice, and explain what mistake results from ignoring the frame’s inertia.
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Question 11 · AP topic 5.5
11. Use the off-center balance
A uniform meterstick is horizontal and balanced on a pivot at its 0.40 m mark. A 0.20 kg mass hangs at 0.10 m and a 0.10 kg mass hangs at 0.90 m. The meterstick’s mass is unknown; neglect attachment masses. Infer it without using a scale. Then predict which way the stick would initially turn if the pivot moved to 0.50 m while both hanging masses stayed in place. Explain why the meterstick’s weight matters in one configuration but has zero torque in the other.
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Question 12 · AP topic 5.6
12. Separate the bearing torque from the inertia
A fixed-axis frame has the same inertia in three trials. Measured applied driving torques are 0.20, 0.40, and 0.60 N·m, and the corresponding angular accelerations are 1.0, 3.0, and 5.0 rad/s². A constant unknown bearing torque opposes the drive. Choose axes for a linear graph that would let you infer both inertia and bearing torque, and find them from the data. A classmate labels the vertical axis “net torque” without changing the plotted values. Explain the error and where a graph of true net torque would meet the axis.
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Question 13 · AP topic 5.1
13. Rotational Kinematics · 1
A wheel has ω = −6 rad/s and α = +2 rad/s². What is happening at this instant?
- It turns clockwise and slows down.
- It turns counterclockwise and speeds up.
- It is stopped because the signs differ.
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Question 14 · AP topic 5.2
14. Connecting Linear and Rotational Motion · 1
Two points are fixed to the same rigid disk at radii r and 2r. Which quantity is the same for both?
- Tangential speed.
- Angular velocity.
- Radial acceleration magnitude.
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Question 15 · AP topic 5.3
15. Torque · 1
A force acts at distance r from a pivot but points directly toward the pivot. What is its torque about that pivot?
- rF
- F/r
- Zero.
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Question 16 · AP topic 5.4
16. Rotational Inertia · 1
A point mass is moved from radius r to 2r about the same axis. How does its rotational inertia change?
- It doubles.
- It becomes four times as large.
- It stays the same because its mass is unchanged.
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Question 17 · AP topic 5.5
17. Rotational Equilibrium and Newton’s First Law in Rotational Form · 1
A wheel is in rotational equilibrium. Which statement must be true?
- Its angular velocity is zero.
- Its angular velocity is constant.
- Its center of mass is at rest.
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Question 18 · AP topic 5.6
18. Newton’s Second Law in Rotational Form · 1
A rigid wheel’s inertia doubles while its net torque stays the same. How does α change?
- It doubles.
- It stays the same.
- It is halved.
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Question 19 · AP topic 5.1
19. Rotational Kinematics · 2
What does signed area under an angular-velocity versus time graph give?
- Angular acceleration.
- Angular displacement.
- Rotational inertia.
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Question 20 · AP topic 5.2
20. Connecting Linear and Rotational Motion · 2
A disk turns at constant nonzero angular velocity. Which statement is correct?
- All points have zero acceleration.
- Points away from the axis have tangential acceleration only.
- Points away from the axis have inward acceleration but zero tangential acceleration.
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Question 21 · AP topic 5.3
21. Torque · 2
A 20 N force acts on a 0.40 m wrench at 30° to its position vector. What torque magnitude results?
- 8.0 N·m
- 4.0 N·m
- 16 N·m
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Question 22 · AP topic 5.4
22. Rotational Inertia · 2
Which conditions justify I = I_cm + Md²?
- The two axes are parallel and one passes through the center of mass.
- Any two axes separated by d.
- Both axes pass through the center of mass.
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Question 23 · AP topic 5.5
23. Rotational Equilibrium and Newton’s First Law in Rotational Form · 2
Two equal opposite forces act along separated parallel lines on a bar. What can happen?
- Its center of mass accelerates but it cannot rotate.
- Both net force and net torque must vanish.
- Net force can be zero while net torque is nonzero.
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Question 24 · AP topic 5.6
24. Newton’s Second Law in Rotational Form · 2
A hanging mass is released from rest and accelerates downward while driving a pulley with rotational inertia. How does string tension compare with mg?
- T is smaller than mg.
- T equals mg.
- T is larger than mg.
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Question 25 · AP topic 5.1
25. Rotational Kinematics · 3
A wheel rotates through one complete counterclockwise turn. What is its unwrapped angular displacement?
- Zero.
- π rad.
- +2π rad.
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Question 26 · AP topic 5.2
26. Connecting Linear and Rotational Motion · 3
A point travels an arc 0.60 m at radius 0.20 m. What angular displacement magnitude produced it?
- 3.0 rad
- 0.12 rad
- 0.33 rad
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Question 27 · AP topic 5.3
27. Torque · 3
Two forces have equal magnitudes. Which makes the larger torque about a given axis?
- The one with the greater perpendicular lever arm.
- The one applied nearer the center of mass, always.
- They must make equal torques.
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Question 28 · AP topic 5.4
28. Rotational Inertia · 3
A hoop and a disk have equal mass and outer radius. Which has greater inertia about its center axis?
- The disk, because it looks more solid.
- They are equal.
- The hoop, because more mass is farther from the axis.
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Question 29 · AP topic 5.5
29. Rotational Equilibrium and Newton’s First Law in Rotational Form · 3
Why choose a support point as the axis when solving a beam in equilibrium?
- Its force has zero torque about that point.
- Gravity stops acting there.
- The support force must be zero.
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Question 30 · AP topic 5.6
30. Newton’s Second Law in Rotational Form · 3
A graph plots net torque vertically against angular acceleration horizontally. What is the slope for a rigid system?
- Angular speed.
- Rotational inertia.
- Angular displacement.
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Explain and apply the model
Try these problems before opening the solutions. State your assumptions, show the diagram or calculation, and explain what makes your answer follow.
Match models and the full equation set
A rigid wheel rotates about a fixed central axle. Its initial angle is θ₀ = 0.50 rad, its initial angular velocity is +3.0 rad/s, and its constant angular acceleration is +2.0 rad/s² for 4.0 s. Counterclockwise is positive.
(a) Calculate ω at 4.0 s using ω = ω₀ + αt, calculate θ using θ = θ₀ + ω₀t + ½αt², and independently check the result using ω² = ω₀² + 2α(θ − θ₀). Show all three calculations and state their shared domain condition.
(b) Write the linear one-dimensional counterpart of each of the three angular equations and pair angular displacement, angular velocity and angular acceleration with their linear analogues. Sketch ω versus t and use its slope and area to check α and Δθ.
(c) Explain why a spinning rigid wheel cannot be replaced by a point at its stationary center when describing the motion of opposite rim points. Contrast this with treating Earth as an object for its revolution about the Earth–Sun center of mass while ignoring daily axial spin. State what the two models preserve and omit.
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Compare both angular rates on a rigid disk
Two marks are fixed to the same rigid disk at radii 0.10 m and 0.30 m. During the same 2.0 s interval the disk’s angular velocity rises from +1.0 to +5.0 rad/s at constant rate.
(a) Which complete comparison correctly gives both marks’ final angular velocity, angular acceleration and tangential acceleration magnitudes?
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Construct an extended force diagram
A rigid horizontal 1.0 m bar has a marked reference axis perpendicular to the page through its left end. At this instant force A is 6.0 N downward at x = 0.20 m, force B is 9.0 N upward at x = 0.60 m, and force C is 3.0 N downward at x = 1.0 m. There are no other forces; take counterclockwise positive for the one-plane turning calculation.
(a) Construct an extended-object force diagram of the bar. Mark the stated axis and every force’s application point relative to it, label the source force A/B/C and each magnitude, and draw arrow lengths proportional to magnitude with correct directions. Explain how this diagram resembles a free-body diagram and what a single-dot free-body diagram would erase.
(b) Use your diagram to calculate each torque and net torque about the marked axis. Calculate net vertical force as a separate sum. Explain why equal net force does not determine net torque.
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Prove the center-axis minimum
A rigid planar assembly has total mass 3.0 kg and provided inertia I_cm = 0.12 kg·m² about a center-of-mass axis perpendicular to its plane. Every axis in this task is parallel to that axis. Let signed offset d measure axis position relative to the center-of-mass axis.
(a) Use I(d) = I_cm + Md² to prove that the center-of-mass axis has the minimum inertia among every axis in this parallel family. State the equality condition and explain why this proof does not compare differently oriented axes.
(b) Calculate I at d = +0.20 m, −0.20 m and +0.40 m. Sketch I versus d with labeled axes, units, those points and the minimum; explain the left/right symmetry.
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Separate translation and rotation
A freely moving bar has mass 2.0 kg and provided I_cm = 0.20 kg·m². Ignore all other forces in this ideal comparison. In arrangement A, two 5.0 N forces both point down, one 0.30 m left of the center and one 0.30 m right. In arrangement B, the left 5.0 N force points down and the right 5.0 N force points up. Upward and counterclockwise are positive.
(a) For arrangement A, draw both a dot free-body diagram and an extended force diagram about the center. Use the first to find net force/a_cm and the second to find net torque/α. Decide independently whether the bar has translational and rotational equilibrium. If initial ω is +1.0 rad/s, predict whether it changes.
(b) Repeat the separate force and torque sums for arrangement B. State the two equilibrium judgments and explain why the unbalanced torque requires angular velocity to change. Find the initial rate of change of ω; if an apparatus maintains this net torque for 0.10 s, find Δω.
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Couple a falling mass and pulley
A 0.80 kg mass falls while a light inextensible string unwinds without slipping from a fixed central axle pulley. Pulley radius is 0.10 m and provided inertia about that axle is 0.0040 kg·m². Bearing torque is negligible. Use g = 10 m/s²; the mass starts from rest. Downward mass motion and the corresponding pulley turn are positive.
(a) Draw the mass’s free-body diagram and an extended force diagram for the pulley, including tension at the rim and axle support. Independently write the mass’s linear balance and the pulley’s rotational balance, then write the no-slip constraint. Explain why the axle force has no axle torque and why tension is not mg.
(b) Solve the coupled equations for a, T and α. Then, after the mass falls 0.30 m, calculate its speed and the pulley angular speed and independently check that gravitational energy lost equals the sum of mass and pulley kinetic-energy gains.
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