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Oscillations practice problems AP Physics C: Mechanics, Unit 7

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

Simple harmonic motion from its differential equation, springs, simple and physical pendulums, and energy in oscillations. 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 oscillations

d2x/dt2 = −ω2xthe SHM equation
x = A cos(ωt + φ)its solution
T = 2π√(m/k)mass on a spring
T = 2π√(L/g)simple pendulum, small angles
T = 2π√(I/(mgd))physical pendulum
E = ½kA2total energy

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
Find the period of a 0.50 kg mass on a spring with k = 50 N/m.
s

Hint

T = 2π√(m/k).

Worked solution
  1. T = 2π√(0.01) = 2π × 0.1.
  2. T = 0.628 s.
Q2Warm-up
Find the period of a simple pendulum 2.0 m long, for small swings.
s

Hint

T = 2π√(L/g).

Worked solution
  1. T = 2π√(2.0/9.8) = 2π × 0.452.
  2. T = 2.84 s.
Q3Warm-up
In simple harmonic motion, the magnitude of the acceleration is greatest:

Hint

a = −ω2x.

Worked solution
  1. The acceleration is proportional to the displacement.
  2. So it is greatest at maximum displacement: B.

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
An oscillator has position x(t) = 0.050 cos(10t), in metres. Find its maximum speed.
m/s

Hint

Differentiate, or use vmax = Aω.

Worked solution
  1. v = −0.50 sin(10t), so vmax = Aω = 0.050 × 10.
  2. vmax = 0.5 m/s.
Q5Exam-style
For the same oscillator, find the maximum acceleration.
m/s²

Hint

amax = Aω2.

Worked solution
  1. amax = 0.050 × 102 = 5 m/s².
Q6Exam-style
A mass on a spring with k = 200 N/m oscillates with amplitude 0.10 m. Find the total mechanical energy.
J

Hint

At maximum displacement all the energy is in the spring.

Worked solution
  1. E = ½kA2 = ½ × 200 × 0.010.
  2. E = 1 J.
Q7Exam-style
The amplitude of a mass-spring oscillator is doubled. What happens to its period?

Hint

Look at T = 2π√(m/k). Does A appear?

Worked solution
  1. The period depends only on m and k.
  2. So it stays the same: C. The maximum speed doubles instead.
Q8Challenge
A uniform rod 1.0 m long swings with small oscillations about a pivot at one end. Find its period.
s

Hint

Physical pendulum: T = 2π√(I/(mgd)), with I = mL2/3 and d = L/2.

Worked solution
  1. T = 2π√((mL2/3)/(mgL/2)) = 2π√(2L/(3g)).
  2. T = 2π√(2.0/29.4) = 2π × 0.261.
  3. T = 1.64 s.

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 · 6 points · about 15 minutes

A block of mass m on a frictionless horizontal surface is attached to a spring of spring constant k. It is pulled a distance A from equilibrium and released from rest at t = 0.

  1. (a) Write the differential equation for the block's position x(t).
  2. (b) Show that x(t) = A cos(ωt) is a solution, and find ω.
  3. (c) For m = 0.20 kg, k = 80 N/m and A = 0.050 m, find the maximum speed.

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)

m d2x/dt2 = −kx.

Part (b)

d2x/dt2 = −Aω2 cos(ωt) = −ω2x. Matching gives ω = √(k/m). x(0) = A, as required.

Part (c)

ω = √(80/0.20) = 20 rad/s. vmax = Aω = 1.0 m/s.

Your mark: 0 of 6

Common mistakes in oscillations

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