Simple Harmonic Motion Examples: 43 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Simple Harmonic Motion.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Oscillatory motion where the restoring force is proportional to displacement from equilibrium, producing sinusoidal position over time.

A spring or pendulum that bounces back and forth in a smooth, repeating pattern.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Simple Harmonic Motion asks what energy enters, leaves, stays stored, or changes form in the chosen system.

Common stuck point: Students often know a formula related to simple harmonic motion but skip the recognition step: Can I define the system and track energy before and after the interaction or process? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Can I define the system and track energy before and after the interaction or process?

Worked Examples

Example 1

easy

A mass-spring system has a spring constant

k = 50 \text{ N/m}

and mass

m = 2 \text{ kg}

. What is the period of oscillation?

Answer

T \approx 1.26 s

First step

The period of a mass-spring system is:

T = 2\pi\sqrt{\frac{m}{k}}

Full solution

2
$T = 2 π \sqrt{\frac{2}{50}} = 2 π \sqrt{0.04} = 2 π \times 0.2 = 0.4 π$
3
$T \approx 1.26 s$

Simple harmonic motion is periodic oscillation where the restoring force is proportional to displacement. The period depends on mass and spring constant but not on amplitude.

Example 2

medium

A simple pendulum has a length of

1 \text{ m}

. What is its period on Earth (

g = 9.8 \text{ m/s}^2

) and on the Moon (

g = 1.6 \text{ m/s}^2

Example 3

medium

0.5 \text{ kg}

mass on a spring oscillates at frequency

2 \text{ Hz}

. Find the spring constant.

Example 4

medium

0.2 \text{ kg}

mass on a spring (

k = 80 \text{ N/m}

) oscillates with amplitude

0.05 \text{ m}

. Find the total mechanical energy.

Example 5

medium

A mass of

0.4 \text{ kg}

oscillates on a spring with

k = 100 \text{ N/m}

and amplitude

A = 0.05 \text{ m}

. Find the maximum speed.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

0.3 \text{ kg}

mass on a spring oscillates with amplitude

0.1 \text{ m}

and period

0.8 \text{ s}

. What is the maximum speed of the mass?

Example 2

hard

0.5 \text{ kg}

mass on a spring (

k = 200 \text{ N/m}

) oscillates with amplitude

0.04 \text{ m}

. What is the total energy of the system, and what is the speed when the displacement is

0.02 \text{ m}

Example 3

easy

A mass-spring system has

m = 2 \text{ kg}

and

k = 8 \text{ N/m}

. Find the period. (

T = 2\pi\sqrt{m/k}

)

Example 4

easy

A pendulum has length

L = 2.45 \text{ m}

(use

g = 9.8

). Find its period. (

T = 2\pi\sqrt{L/g}

)

Example 5

easy

A SHM oscillation has frequency

f = 5 \text{ Hz}

. Find the angular frequency. (

\omega = 2\pi f

)

Example 6

easy

A pendulum's period is 4 s. What is its frequency? (

f = 1/T

)

Example 7

easy

In SHM

x = A\cos(\omega t)

with

A = 0.1 \text{ m}

, what is the maximum displacement from equilibrium?

Example 8

easy

Does increasing the amplitude of a mass-spring SHM change its period?

Example 9

easy

Where in its motion is a SHM oscillator moving fastest?

Example 10

easy

A spring has

k = 50 \text{ N/m}

and a 0.5 kg mass. Find

\omega

. (

\omega = \sqrt{k/m}

)

Example 11

medium

A mass-spring system has

T = \pi \text{ s}

and

k = 16 \text{ N/m}

. Find the mass. (

T = 2\pi\sqrt{m/k}

)

Example 12

medium

A pendulum has period 2 s on Earth (

g = 9.8

). To double its period to 4 s, by what factor must its length change?

Example 13

medium

A 0.2 kg mass on a spring (

k = 80 \text{ N/m}

) oscillates. How many full cycles occur in 10 s?

Example 14

medium

A pendulum on the Moon (

g = 1.6

) has length 1 m. Find its period. (

T = 2\pi\sqrt{L/g}

)

Example 15

medium

A spring oscillator has maximum speed 3 m/s and

\omega = 6 \text{ rad/s}

. Find the amplitude. (

v_{max} = \omega A

)

Example 16

medium

A spring with

k = 200 \text{ N/m}

is compressed 0.1 m. Find the elastic PE stored. (

PE = \frac{1}{2}kx^2

)

Example 17

medium

A 1 kg mass on a spring (

k = 100 \text{ N/m}

) has amplitude 0.2 m. Find its maximum speed. (

\omega = \sqrt{k/m}

v_{max} = \omega A

)

Example 18

challenge

A 0.5 kg mass on a spring (

k = 50 \text{ N/m}

) has amplitude 0.4 m. Find its maximum kinetic energy.

Example 19

challenge

Two springs

k_1 = 100

and

k_2 = 300 \text{ N/m}

connect in parallel to a 2 kg mass (effective

k = k_1 + k_2

). Find the period.

Example 20

challenge

A pendulum clock keeps perfect time on Earth (

g = 9.8

). Taken to a planet where it runs slow (period longer), is that planet's gravity stronger or weaker than Earth's?

Example 21

medium

A mass-spring system has

m = 0.5 \text{ kg}

and

k = 32 \text{ N/m}

. Find its frequency. (

f = \frac{1}{2\pi}\sqrt{k/m}

)

Example 22

medium

A pendulum has length 0.9 m (use

g = 10

). Find its frequency. (

T = 2\pi\sqrt{L/g}

f = 1/T

)

Example 23

easy

A mass-spring system has

m = 0.25 \text{ kg}

and

k = 100 \text{ N/m}

. Find the period of oscillation.

Example 24

easy

A simple pendulum of length

0.625 \text{ m}

is on Earth (

g = 9.8 \text{ m/s}^2

). Find its period.

Example 25

medium

A pendulum has period

T = 2 \text{ s}

. Find its length on Earth. Use

g = 9.8 \text{ m/s}^2

Example 26

medium

A mass-spring SHM has amplitude

A = 0.05 \text{ m}

and angular frequency

\omega = 20 \text{ rad/s}

. Find the maximum speed.

Example 27

medium

A SHM oscillator has

A = 0.1 \text{ m}

and

\omega = 10 \text{ rad/s}

. Find the maximum acceleration.

Example 28

medium

A pendulum has period

1.2 \text{ s}

on Earth. What is its period on the Moon (

g_M = 1.6 \text{ m/s}^2

Example 29

medium

A SHM oscillator's position is

x(t) = 0.04 \cos(5t)

m. Find the period.

Example 30

hard

1 \text{ kg}

mass on a spring (

k = 100 \text{ N/m}

) has amplitude

0.10 \text{ m}

. Find its speed when

x = 0.06 \text{ m}

Example 31

medium

A pendulum has

L = 0.8 \text{ m}

on a planet where

g = 5 \text{ m/s}^2

. Find its frequency.

Example 32

easy

A spring-mass SHM has

T = 0.5 \text{ s}

. Find

\omega

Example 33

medium

0.5 \text{ kg}

mass on a spring has total energy

0.2 \text{ J}

and amplitude

0.1 \text{ m}

. Find the spring constant.

Example 34

challenge

0.2 \text{ kg}

mass on a spring with

k = 50 \text{ N/m}

is set into SHM with amplitude

0.06 \text{ m}

. At what displacement is half the energy kinetic and half potential?

Example 35

medium

1 \text{ kg}

block on a spring (

k = 16 \text{ N/m}

) oscillates. Find its frequency.

Example 36

medium

A SHM has

x(t) = 0.2 \sin(4\pi t)

m. Find the maximum speed.

Example 37

hard

0.3 \text{ kg}

mass on a spring (

k = 75 \text{ N/m}

) is set in motion. The displacement equation is

x(t) = 0.04 \cos(\omega t)

. After how long does the mass first reach

x = 0

Example 38

medium

A pendulum has length

L

and period

T

on Earth. On a planet with twice Earth's gravity, what is the new period in terms of

T

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Related Concepts

Spring Force Kinetic Energy Potential Energy Waves Frequency

Background Knowledge

These ideas may be useful before you work through the harder examples.

spring forcekinetic energypotential energy