Springs & pendulums: practice problems with step-by-step solutions

Free Springs & pendulums practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: mass on a spring, pendulum, speed, acceleration and energy.

Free practice problemEasy
4.8 kgk = ?

A 4.8 kg mass on a spring oscillates with a period of 2.5 s. Find the spring stiffness k.

m = 4.8 kgT = 2.5 s
N/m
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What you practise

  1. Easymass on a spring
  2. Mediumpendulum
  3. Hardspeed, acceleration and energy

Key formulas

Mass on a spring
T = 2π·√(m/k)
Pendulum
T = 2π·√(L/g)
Angular frequency
ω = √(k/m) = 2π/T
Maximum speed and acceleration
vmax = A·ω, amax = A·ω²
Energy
E = ½·k·A²

Learn Springs & pendulums from scratch

Before this: Newton’s laws

1What an oscillation is

Pull a mass on a spring to one side and let go. It swings back past the middle, out to the other side, and back again. A child on a swing and a guitar string do the same.

This repeating motion is an oscillation. The middle point, where the mass would rest, is the equilibrium position.

m0−A+Axk
  • The amplitude A is the largest distance from equilibrium, in m.
  • The period T is the time for one complete oscillation, there and back, in s.
  • The frequency f is the number of oscillations per second, in hertz (Hz).
f = 1 / T
Self-check A mass on a spring makes 20 complete oscillations in 10 s. What is the period?
  1. 0.5 s
  2. 10 s
  3. 2 s
  4. 200 s

Answer: 0.5 s. T = time / number of oscillations = 10/20 = 0.5 s. 2 is the frequency in Hz (20/10), the number of oscillations per second.

2Simple harmonic motion

A spring pulls back with a force proportional to the stretch: F = −k·x. The stiffness k is in N/m. The minus sign means the force always points back to equilibrium.

A restoring force like this gives simple harmonic motion (SHM). The position follows a cosine curve in time:

txA−AT
x = A · cos(ω·t)

The angular frequency ω (omega), in rad/s, sets how fast the motion repeats. One period is 2π rad.

ω = 2π / T = 2π · f

The amplitude does not change the period. A bigger swing covers more distance but also moves faster.

3Spring and pendulum formulas

For a mass on a spring, a heavier mass oscillates more slowly and a stiffer spring faster:

ω = √(k / m) T = 2π · √(m / k)

For a simple pendulum of length L, with small swings, the mass does not matter at all:

T = 2π · √(L / g)

If a spring hangs vertically, the weight stretches it to a new equilibrium: k·x = m·g. Use this to find k from a measured stretch.

The speed is largest at equilibrium and zero at the ends. The acceleration is largest at the ends, where the spring force is largest:

vmax = A · ω amax = A · ω²

The total energy stays the same: E = ½·k·A². At a position x it is shared as ½·k·x² + ½·m·v².

Self-check You replace the mass on a spring with one twice as heavy. What happens to the period?
  1. It becomes 1.41 times shorter
  2. It becomes 4 times longer
  3. It becomes √2 ≈ 1.41 times longer
  4. It doubles

Answer: It becomes √2 ≈ 1.41 times longer. T = 2π·√(m/k), so T grows with the square root of m. Doubling m multiplies T by √2. Saying it doubles forgets the square root.

4The method step by step

  1. Convert everything to SI units: grams to kg, centimetres to m.
  2. Decide which system it is: a mass on a spring (use k and m) or a pendulum (use L and g).
  3. If the period or the number of oscillations is given, find T first: T = time / number.
  4. Find ω from the system: ω = √(k/m) or ω = 2π/T.
  5. Use the formula for what is asked: T, f, vmax = A·ω, amax = A·ω² or the energy ½·k·A².
  6. Check that the answer is reasonable: a period of a fraction of a second for a stiff spring, a few seconds for a long pendulum.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

A 0.5 kg mass on a spring with k = 200 N/m is pulled 5 cm from equilibrium and released. Find the period T, the frequency f, the maximum speed vmax and the maximum acceleration amax.

1

Units

A = 5 cm = 0.05 m m = 0.5 kg k = 200 N/m
2

Angular frequency

ω = √(k / m) = √(200 / 0.5) = √400 = 20 rad/s
3

Period and frequency

T = 2π / ω = 2π / 20 = 0.314 s
f = 1 / T = 3.18 Hz
4

Maximum speed

The mass is fastest as it passes equilibrium.

vmax = A · ω = 0.05 · 20 = 1 m/s
5

Maximum acceleration

The acceleration is largest at the ends, where the mass turns round.

amax = A · ω² = 0.05 · 400 = 20 m/s²
6

Check the answer

Energy check: ½·k·A² = ½ · 200 · 0.05² = 0.25 J, and ½·m·vmax² = ½ · 0.5 · 1² = 0.25 J. All the spring energy becomes kinetic energy at equilibrium.

Answer

T ≈ 0.314 s, f ≈ 3.18 Hz, vmax = 1 m/s and amax = 20 m/s².

6Common mistakes

  • Using the amplitude in cm instead of m.
  • Turning the fraction upside down: T = 2π·√(m/k), not √(k/m). For a pendulum it is √(L/g).
  • Forgetting the 2π, or mixing up ω (rad/s) and f (Hz).
  • Thinking the mass changes the period of a pendulum, or the amplitude changes any period.
  • Mixing up vmax = A·ω and amax = A·ω².
Self-check A mass oscillates with amplitude 10 cm and ω = 10 rad/s. What is its maximum acceleration?
  1. 1 m/s²
  2. 10 m/s²
  3. 100 m/s²
  4. 1000 m/s²

Answer: 10 m/s². amax = A·ω² = 0.1 · 100 = 10 m/s². Using A = 10 (cm) gives 1000. 1 m/s is A·ω, the maximum speed, not the acceleration.

7How exam questions look

  • Easy: a mass on a spring. Find T, f or k, sometimes from a count of oscillations or from how far the mass stretches a hanging spring.
  • Medium: the pendulum. Find its period, the length for a given period, or g on another planet. Some questions ask the speed at a position or at a given time.
  • Hard: speed, acceleration and energy: vmax, amax, the kinetic energy at a position, or the amplitude after a bullet hits a block on a spring.

A quick check for any pendulum: a 1 m pendulum on Earth has a period of about 2 s.

Self-check A pendulum is 2.45 m long. With g = 9.81 m/s², what is its period for small swings?
  1. 0.50 s
  2. 1.57 s
  3. 12.6 s
  4. 3.14 s

Answer: 3.14 s. T = 2π·√(2.45/9.81) = 2π · 0.5 = 3.14 s. √(g/L) upside down gives 12.6 s, forgetting the root gives 1.57 s, and forgetting 2π gives 0.50 s.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
3.2 kgk = 210 N/m

A 3.2 kg mass on a spring with stiffness 210 N/m oscillates on a frictionless surface. Find the period T.

m = 3.2 kgk = 210 N/m

Find T (s)

Show hints
  1. The period of a mass–spring system does not depend on the amplitude.
  2. T = 2π·√(m/k).
  3. √(m/k) = 0.1234 s.
Show the answer and the step-by-step solution

Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².

Period

T = 2π·√(3.2/210) = 0.7756 s

Answer

T = 0.7756 s
Example 2Medium
1.1 kgk = 120 N/mA = 7 cmx = 6 cm

A 1.1 kg mass on a spring with k = 120 N/m oscillates on a frictionless surface with an amplitude of 7 cm. Find its speed v when it is 6 cm from equilibrium.

m = 1.1 kgk = 120 N/mA = 7 cmx = 6 cm

Find v (m/s)

Show hints
  1. Energy is conserved: ½·k·A² = ½·k·x² + ½·m·v².
  2. So v = ω·√(A² − x²) with ω = √(k/m).
  3. ω = 10.44 rad/s.
Show the answer and the step-by-step solution

Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².

Angular frequency

ω = √(120/1.1) = 10.44 rad/s

Speed at x

v = ω·√(A² − x²) = 10.44·√(0.07² − 0.06²) = 0.3766 m/s

Answer

v = 0.3766 m/s
Example 3Hard
1.7 kgk = 380 N/mA = 11 cm

A 1.7 kg mass on a spring with k = 380 N/m is pulled 11 cm from equilibrium and released. Find its maximum acceleration amax.

m = 1.7 kgk = 380 N/mA = 11 cm

Find amax (m/s²)

Show hints
  1. ω = √(k/m).
  2. The acceleration is largest at the ends: amax = A·ω².
  3. ω = 14.95 rad/s, A = 0.11 m.
Show the answer and the step-by-step solution

Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².

Angular frequency

ω = √(380/1.7) = 14.95 rad/s

Maximum acceleration

amax = A·ω² = 0.11·14.95² = 24.59 m/s²

Answer

amax = 24.59 m/s²

Questions and answers

What is Springs & pendulums?

Pull a mass on a spring to one side and let go. It swings back past the middle, out to the other side, and back again. A child on a swing and a guitar string do the same.

This repeating motion is an oscillation. The middle point, where the mass would rest, is the equilibrium position.

How do I solve Springs & pendulums problems step by step?

Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².

What are common mistakes in Springs & pendulums?
  • Include 2π: T = 2π·√(m/k) and f = 1/T.
  • Check the fraction under the root: T = 2π·√(m/k), mass on top.
  • That is the maximum speed, at equilibrium. At x the speed is lower: v = ω·√(A² − x²).
What do the Springs & pendulums exercises cover?
  • Easy: mass on a spring
  • Medium: pendulum
  • Hard: speed, acceleration and energy

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Topics before and after this one

Learn first

Newton’s laws

Continue with

Mechanical vibrationsWaves, sound & optics

Calculators

Pendulum and spring period calculatorSpring constant calculator (Hooke’s law)

Courses with this topic

DynamicsPhysics 1: MechanicsWaves and OpticsAP Physics 1 practice problems

More topics in Physics

KinematicsNewton’s lawsCircular motionMomentum & collisionsCoulomb’s law & electric fieldsMagnetic fields & inductionWaves, sound & optics

Printable worksheet

Springs & pendulums worksheet with answers
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