Simple Harmonic Motion Examples in Physics

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: The motion repeats with a specific period that depends on the system, not how far you pull.

Common stuck point: Period of a pendulum depends on length and g, NOT on mass or amplitude (for small angles).

Sense of Study hint: When solving an SHM problem, first identify whether it is a mass-spring system (T = 2\pi\sqrt{m/k}) or a pendulum (T = 2\pi\sqrt{L/g}). Then use x = A\cos(\omega t) for position, v = -A\omega\sin(\omega t) for velocity, and a = -A\omega^2\cos(\omega t) for acceleration. Remember: amplitude does not affect the period.

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?

Solution

  1. 1
    The period of a mass-spring system is: T = 2\pi\sqrt{\frac{m}{k}}.
  2. 2
    T = 2\pi\sqrt{\frac{2}{50}} = 2\pi\sqrt{0.04} = 2\pi \times 0.2 = 0.4\pi
  3. 3
    T \approx 1.26 \text{ s}

Answer

T \approx 1.26 \text{ 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)?

Practice Problems

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

Example 1

medium
A 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
A 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}?
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Background Knowledge

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

spring forcekinetic energypotential energy