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Unit 6 / Energy and Momentum of Rotating Systems

Unit 6: 35 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 Energy and Momentum of Rotating Systems; 5 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.

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Question 1 · AP topic 6.1

1. Rotational Kinetic Energy · 4

Wheel A has I = 0.020 kg·m² and wheel B has I = 0.080 kg·m². Both turn about fixed axles and each has 4.0 J of kinetic energy. Find their angular speeds.

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Question 2 · AP topic 6.2

2. Torque and Work · 4

A wheel with I = 0.050 kg·m² turns at 12 rad/s. A constant opposing torque of magnitude 0.30 N·m brings it to rest. Find the angle it turns through while stopping.

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Question 3 · AP topic 6.3

3. Angular Momentum and Angular Impulse · 4

A wheel has I = 0.20 kg·m² and initial angular velocity +5.0 rad/s. A constant braking torque of −0.40 N·m acts for 1.5 s. Find L_f and ω_f.

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Question 4 · AP topic 6.4

4. Conservation of Angular Momentum · 4

A rotating system initially has I_i = 2.0 kg·m² and ω_i = +3.0 rad/s. Its inertia falls to 1.0 kg·m² while an external braking angular impulse of −1.0 kg·m²/s acts. Find ω_f.

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Question 5 · AP topic 6.5

5. Rolling · 4

A hoop of mass 1.5 kg rolls without slipping at 2.0 m/s. Its provided inertia is I_cm = MR². Find translational, rotational, and total kinetic energy without needing its radius.

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Question 6 · AP topic 6.6

6. Motion of Orbiting Satellites · 4

For a model planet, GM = 3.20 × 10¹⁴ m³/s². A satellite is at center-to-center distance 8.0 × 10⁶ m. Find its circular speed and the threshold escape speed there.

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Question 7 · AP topic 6.1

7. Compare a different speed change

A 2.0 kg wheel has center-of-mass rotational inertia 0.040 kg·m². In trial A its center speed is 3.0 m/s and angular speed is 15 rad/s. In trial B the center speed is 6.0 m/s and angular speed is 7.5 rad/s. Draw scaled translation and rotation energy bars for each trial. A classmate predicts equal total kinetic energies because one speed doubled while the other halved. Evaluate the prediction without assuming a rolling constraint.

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Question 8 · AP topic 6.2

8. Choose the work coordinate

A sensor records driving torque 2.0 N·m at angle 0, 4.0 N·m at 1.0 rad, and 4.0 N·m at 2.0 rad. Assume linear interpolation between measurements. A constant opposing bearing torque has magnitude 0.50 N·m. The wheel begins with 1.0 J of rotational kinetic energy and has a fixed axle. Decide whether a torque–time or torque–angle graph can directly determine the work here, then use the supplied data to predict final kinetic energy. State the energy-transfer assumption needed for that prediction.

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Question 9 · AP topic 6.3

9. Follow a pulse and an opposing torque

A fixed-axis wheel starts with angular momentum +0.50 kg·m²/s. From t = 0 to 2.0 s its net torque forms a positive triangle, rising from zero to +2.0 N·m at 1.0 s and returning to zero at 2.0 s. From 2.0 to 3.5 s the net torque is −1.0 N·m. Sketch angular momentum against time with labeled values and explain its changing slopes. Does the wheel ever reverse its turning sense during this interval? Justify your answer from the data.

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Question 10 · AP topic 6.4

10. Judge a conservation claim with uncertainty

A rotating stool system has effective inertia 2.4 kg·m² with arms out and 0.80 kg·m² with arms in. Video gives angular speeds 2.0 ± 0.10 rad/s before the change and 5.5 ± 0.10 rad/s afterward; treat each quoted interval as possible measurement values and the inertias as exact for this task. Are these data consistent with zero external angular impulse? Determine the sign of any supported transfer and propose a control measurement that could test whether bearing drag explains it.

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Question 11 · AP topic 6.5

11. Diagnose the contact motion

A disk and hoop, each of radius 0.25 m, move right on a rough level stationary floor with clockwise spin. At one instant the disk has center speed 3.0 m/s and angular speed 8.0 rad/s; the hoop has 2.5 m/s and 10 rad/s. Identify which is instantaneously consistent with rolling without slipping. For the slipping object, predict the direction of kinetic friction and its qualitative effects on translation, spin, and mechanical energy. Can these data alone rank the two objects’ total kinetic energies? Explain what else is needed.

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Question 12 · AP topic 6.6

12. Check an elliptical orbit ledger

A 100 kg satellite has a gravity-only elliptical orbit around a model central object with GM = 1.20 × 10¹⁴ m³/s². At its nearest point, r = 2.0 × 10⁷ m and speed is 3.0 × 10³ m/s; at its farthest point, r = 6.0 × 10⁷ m. Predict the far-point speed and build labeled kinetic/potential energy bars at both points using zero gravitational potential energy at infinity. Decide whether the orbit is bound. Explain why the same satellite can change speed while keeping angular momentum constant.

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Question 13 · AP topic 6.1

13. Rotational Kinetic Energy · 1

A fixed-axis wheel reverses from +10 rad/s to −10 rad/s with the same inertia. How do the endpoint kinetic energies compare?

  1. The final energy is negative.
  2. They are equal.
  3. The final energy is zero.
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Question 14 · AP topic 6.2

14. Torque and Work · 1

A graph plots torque vertically against angular position horizontally. What does its signed area represent?

  1. Angular impulse.
  2. Angular acceleration.
  3. Work done by the torque.
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Question 15 · AP topic 6.3

15. Angular Momentum and Angular Impulse · 1

A puck travels along a straight line. Can its angular momentum about a reference point be nonzero?

  1. No; only curved paths have angular momentum.
  2. Yes, if the reference point is off the line of motion.
  3. Only if a force is acting.
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Question 16 · AP topic 6.4

16. Conservation of Angular Momentum · 1

A rotating system pulls its mass inward with negligible external torque. Which statement is correct?

  1. Angular speed must remain constant.
  2. Both kinetic energy and angular momentum must remain constant.
  3. Angular momentum stays constant while angular speed can increase.
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Question 17 · AP topic 6.5

17. Rolling · 1

For ideal rolling without slipping on a stationary surface, which statement is correct?

  1. The center of mass is at rest.
  2. The instantaneous contact point is at rest in the surface frame.
  3. Every point on the wheel moves at v_cm.
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Question 18 · AP topic 6.6

18. Motion of Orbiting Satellites · 1

For U = −GMm/r with zero at infinity, what happens to U as a satellite moves closer to the central object?

  1. It becomes more negative.
  2. It becomes positive.
  3. It stays constant in every orbit.
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Question 19 · AP topic 6.1

19. Rotational Kinetic Energy · 2

A point mass moves in a circle about a fixed axis. You calculate both ½mv² and ½Iω². How should they be used?

  1. They are two descriptions of the same energy.
  2. Add them to get total energy.
  3. Use only the smaller value.
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Question 20 · AP topic 6.2

20. Torque and Work · 2

A wheel stays at a fixed angle while a torque is exerted on it. What rotational work does that torque do on the wheel?

  1. Zero.
  2. Torque times the elapsed time.
  3. It cannot be determined without inertia.
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Question 21 · AP topic 6.3

21. Angular Momentum and Angular Impulse · 2

A net torque–time triangle has duration 0.50 s and peak 4.0 N·m above zero. What angular impulse does it supply?

  1. +1.0 kg·m²/s
  2. +2.0 J
  3. +8.0 rad/s²
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Question 22 · AP topic 6.4

22. Conservation of Angular Momentum · 2

Two coaxial disks exchange angular momentum through contact friction. Which system can have constant total L if bearing torque is negligible?

  1. Both disks together.
  2. Only the initially spinning disk.
  3. Only the initially stationary disk.
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Question 23 · AP topic 6.5

23. Rolling · 2

A wheel spins clockwise with its center initially at rest on a rough floor. Its bottom slips left. Which way does kinetic friction on the wheel act?

  1. Right.
  2. Left.
  3. There is no friction because the center is at rest.
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Question 24 · AP topic 6.6

24. Motion of Orbiting Satellites · 2

Which quantities stay constant in an unpowered elliptical orbit with gravity as the only interaction?

  1. Speed and kinetic energy.
  2. Total mechanical energy and angular momentum.
  3. Gravitational potential energy and radius.
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Question 25 · AP topic 6.1

25. Rotational Kinetic Energy · 3

Which inertia belongs in K_total = ½Mv_cm² + ½Iω²?

  1. Inertia about any convenient axis.
  2. Inertia about the contact point, always.
  3. Inertia about the center-of-mass rotation axis.
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Question 26 · AP topic 6.2

26. Torque and Work · 3

A fixed-axis wheel starts from rest. Its net work doubles with the same inertia. What happens to its final angular speed?

  1. It doubles.
  2. It becomes √2 times as large.
  3. It becomes four times as large.
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Question 27 · AP topic 6.3

27. Angular Momentum and Angular Impulse · 3

A graph of angular momentum against time is a horizontal line at a positive value. What is the net external torque?

  1. Positive.
  2. Equal to the angular momentum.
  3. Zero.
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Question 28 · AP topic 6.4

28. Conservation of Angular Momentum · 3

At fixed nonzero angular momentum, inertia is halved. How does rotational kinetic energy change?

  1. It is halved.
  2. It doubles.
  3. It stays constant.
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Question 29 · AP topic 6.5

29. Rolling · 3

A hoop and disk roll without slipping from rest through the same height. Which has greater center speed at the bottom?

  1. The hoop.
  2. They must match.
  3. The disk.
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Question 30 · AP topic 6.6

30. Motion of Orbiting Satellites · 3

How does threshold escape speed compare with circular-orbit speed at the same radius?

  1. It is twice as large.
  2. They are equal.
  3. It is √2 times as large.
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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.

Explain energy with a stationary center

Two 0.40 kg masses are fixed to opposite ends of a massless rigid arm, each 0.25 m from a fixed axle through their center of mass. The arm rotates at angular speed 8.0 rad/s.

(a) Calculate each material point’s linear speed and kinetic energy and add the energies. Explain physically why the rigid system has kinetic energy even though its center of mass is at rest, and why opposite velocities do not cancel that energy.

6.1.A6.1.A.2

(b) Find the rotational inertia and check the total using ½Iω². Identify the center-of-mass translation term and explain whether the two individual kinetic energies should be added again to ½Iω².

6.1.A.1

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Use four factors and derive angular impulse

For a straight-moving puck, m = 0.50 kg and v = 4.0 m/s. At one instant relative to a fixed point O, r = 0.80 m and the angle between the radial vector and velocity is 30°. Separately, a wheel with fixed I = 0.30 kg·m² receives constant net torque +0.60 N·m for 0.50 s; counterclockwise is positive.

(a) Calculate the puck’s angular-momentum magnitude about O. Holding all other factors fixed, predict the new magnitude when (i) mass doubles, (ii) speed doubles, (iii) r doubles, or (iv) the angle changes to 90°. State why these are independent arrangement comparisons rather than changes that can all occur independently on one unchanged straight path.

6.3.A6.3.A.2.ii

(b) Find the puck’s perpendicular path distance b. Predict L at the closest point on that same force-free path where r=b and the radial vector is perpendicular to velocity. Contrast with a reference point directly on the path.

6.3.A.2.ii

(c) Starting from τ_net = Iα, derive the rotational impulse–momentum theorem for constant I. Show the chain τ_net = IΔω/Δt = ΔL/Δt and explain how τ_netΔt = ΔL follows. Then calculate the wheel’s angular impulse and Δω.

6.3.C6.3.C.2.ii

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Account for both disks and the room

Two coaxial disks have I_A = 0.30 kg·m², ω_A = +4.0 rad/s and I_B = 0.20 kg·m², ω_B = −1.0 rad/s. They make contact and eventually turn together. Neglect external torque during contact. After contact, a brake connected to the room removes 0.20 kg·m²/s from the pair. All quantities refer to the same fixed axle; counterclockwise is positive.

(a) Calculate the initial component angular momenta and their signed sum. Find the common final angular velocity, then calculate the final component momenta and verify their sum.

6.4.A6.4.A.1

(b) Calculate the angular impulse on each disk during contact. Derive why these impulses must be equal and opposite from Newton’s third law for tangential contact forces at matching radii over the same interval. Explain why either disk alone does not have constant angular momentum.

6.4.A.2.i

(c) After braking, find the disk pair’s angular momentum and the compensating change in the brake plus Earth. Calculate the enlarged-system total and explain how angular momentum conservation remains valid while the pair’s own momentum changes.

6.4.B6.4.B.1

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Connect rolling acceleration and contact work

A rigid 3.0 kg solid disk of radius 0.20 m rolls from rest down a stationary rigid incline with sinβ = 0.50. Its provided inertia is I_cm = ½MR² = 0.060 kg·m². Use g = 10 m/s². There is enough static friction to prevent slipping, and deformation losses are negligible. Downhill translation and the corresponding clockwise rotation are positive.

(a) Write separate center-of-mass force and rotational torque balances, derive the no-slip displacement, speed and acceleration constraints, and use them to find a_cm, α and the static-friction magnitude/direction. State when the acceleration constraint is valid.

6.5.B6.5.B.1

(b) After 0.60 s, find center displacement, angular displacement, center speed and angular speed, and numerically verify all three constraints. Calculate the two kinetic-energy shares and check them against the gravitational potential-energy decrease.

6.5.A6.5.B.1

(c) Explain using the actual contact-point velocity why static friction does not dissipate energy in this ideal case. At t=0.60 s calculate its separate center-of-mass and rotational power contributions and their sum. State what changes if the contact slips, without quantitatively modeling the slipping evolution.

6.5.B.2

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Justify the satellite approximation and derive escape

A gravity-only, collision-free satellite–central-body model has GM = 3.0 × 10¹⁴ m³/s², G = 6.67 × 10⁻¹¹ N·m²/kg² and center separation r₀ = 6.0 × 10⁶ m. The satellite mass is 200 kg. Use an inertial center-of-mass frame, and U=0 at infinite separation. Treat central-body motion as negligible only after justifying it.

(a) Infer the central mass M from GM and G, then use the two-body center-of-mass condition to estimate the central body’s orbital radius relative to the satellite’s. Explain why neglecting central-body motion is valid here and how that approximation fails for comparable masses.

6.6.A6.6.A.1

(b) For a circular orbit at r₀, derive the circular speed from gravitational force balance and calculate K, U, E and angular-momentum magnitude L about the central body. Identify which of these four quantities are constant along that orbit and justify each constancy.

6.6.A.2.i

(c) Derive threshold escape speed at r₀ from conservation of energy, beginning with the zero-speed, zero-potential-energy limit at infinity. Show why threshold total energy is zero, solve for v_esc and calculate it. State the force and trajectory assumptions needed.

6.6.A.36.6.A.3.ii

(d) For a launch exactly at threshold, derive speed at a later radius r from the same E=0 condition. Find it at 4r₀ and explain whether the satellite reaches zero speed at any finite distance or only as an infinite-distance limit. Contrast with a launch above threshold.

6.6.A.3.i

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