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Torque and rotational dynamics practice problems AP Physics C: Mechanics, Unit 5

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Torque, rotational inertia by integration and the parallel axis theorem, and Newton's second law for rotation. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for torque and rotational dynamics

τ = rF sin θtorque
I = ∫ r2 dmrotational inertia
I = Icm + Md2parallel axis theorem
Στ = IαNewton's second law for rotation
a = rαrolling or string without slipping

Use g = 9.8 m/s², as on the AP reference sheet. You can type answers like 4.9, 1/3 or 3.6e2. The full sheet is in our AP Physics C equation sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
A 20 N force acts perpendicular to a wrench, 0.50 m from the bolt. Find the torque.
N·m

Hint

τ = rF when the force is perpendicular.

Worked solution
  1. τ = 0.50 × 20 = 10 N·m.
Q2Warm-up
Find the rotational inertia of a 2.0 kg point mass 0.30 m from an axis.
kg·m²

Hint

I = mr2.

Worked solution
  1. I = 2.0 × 0.302 = 0.18 kg·m².
Q3Warm-up
A hoop and a solid disk have the same mass and radius. Which has the greater rotational inertia about its central axis?

Hint

Rotational inertia grows with mass placed far from the axis.

Worked solution
  1. All of the hoop's mass is at radius R; much of the disk's is closer in.
  2. Ihoop = MR2 > Idisk = ½MR2: A.

Exam-style questions

Questions at the level of the AP Physics C: Mechanics exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
A uniform rod of mass 3.0 kg and length 2.0 m rotates about one end. Find its rotational inertia.
kg·m²

Hint

Integrate r2 dm with dm = (M/L) dr from 0 to L.

Worked solution
  1. I = ∫0L r2(M/L) dr = ML2/3.
  2. I = 3.0 × 4.0/3 = 4 kg·m².
Q5Exam-style
A net torque of 6.0 N·m acts on a uniform disk of mass 4.0 kg and radius 0.50 m, about its central axis. Find its angular acceleration.
rad/s²

Hint

I = ½MR2 for a disk.

Worked solution
  1. I = ½ × 4.0 × 0.502 = 0.50 kg·m².
  2. α = τ/I = 6.0/0.50 = 12 rad/s².
Q6Exam-style
Find the rotational inertia of a uniform disk of mass 2.0 kg and radius 0.40 m about an axis through its edge, perpendicular to the disk.
kg·m²

Hint

Parallel axis theorem, with d = R.

Worked solution
  1. I = ½MR2 + MR2 = 1.5MR2.
  2. I = 1.5 × 2.0 × 0.16 = 0.48 kg·m².
Q7Exam-style
A solid sphere and a hoop are released from rest at the top of the same incline and roll without slipping. Which reaches the bottom first?

Hint

The one with less of its energy going into rotation moves faster.

Worked solution
  1. The sphere has I = (2/5)MR2, the hoop MR2.
  2. The sphere puts a smaller share of its energy into rotation, so it moves faster and arrives first: A.
Q8Challenge
A 1.0 kg block hangs from a light string wrapped around a pulley, a uniform disk of mass 2.0 kg and radius 0.10 m on a frictionless axle. Find the block's acceleration.
m/s²

Hint

Write F = ma for the block and τ = Iα for the pulley, with a = Rα.

Worked solution
  1. Block: mg − T = ma. Pulley: TR = (½MR2)(a/R), so T = ½Ma.
  2. mg = (m + ½M)a, so a = 9.8/(1.0 + 1.0).
  3. a = 4.9 m/s².

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Mathematical routines style · 7 points · about 20 minutes

A uniform rod of mass M and length L is pivoted at one end on a frictionless axle. It is held horizontal and then released from rest.

  1. (a) Use integration to show that the rod's rotational inertia about the pivot is ML2/3.
  2. (b) Find the angular acceleration just after release, for L = 1.2 m.
  3. (c) Show that the free end initially accelerates faster than g, and explain how that is possible.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

I = ∫0L r2(M/L) dr = (M/L)(L3/3) = ML2/3.

Part (b)

τ = MgL/2, so α = (MgL/2)/(ML2/3) = 3g/(2L) = 12.25 rad/s2.

Part (c)

a = αL = 3g/2 > g. The rod is rigid, so the pivot and the rest of the rod push the end down faster than free fall.

Your mark: 0 of 7

Common mistakes in torque and rotational dynamics

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Physics C: Mechanics past papers, with official scoring guidelines. · All AP Physics C: Mechanics units