BitWithBite
BitWithBite / Mechanics / Simple Harmonic Motion
📘 Lesson 19 of 19 · Mechanics

📈 Simple Harmonic Motion

Simple harmonic motion (SHM) is the most fundamental type of oscillation, where the restoring force is directly proportional to displacement — producing smooth, predictable sine-wave motion.

Course progress: 100%

01 Key Concepts

Defining Condition for SHM

The restoring force (and acceleration) is directly proportional to displacement from equilibrium, and always points opposite to that displacement: F = -k*x.

Mass on a Spring

A classic SHM example, governed by Hooke's Law F=-kx, where k is the spring constant; the period is T = 2*pi*sqrt(m/k).

Simple Pendulum (Small Angles)

For small swing angles, a pendulum approximates SHM with period T = 2*pi*sqrt(L/g), where L is the pendulum's length.

Position, Velocity, and Acceleration in SHM

Position varies as a sine or cosine wave over time; velocity and acceleration are also sinusoidal, but shifted in phase relative to position.

Energy in SHM

Total mechanical energy stays constant, continuously converting between kinetic energy (maximum at equilibrium) and potential energy (maximum at the extremes of motion).

SHM: displacement x(t) = A cos(ωt)

02 Key Formulas

03 Solved Examples

Example 1 A 2 kg mass is attached to a spring with spring constant k=8 N/m. Find the period of oscillation.
  1. Apply T = 2*pi*sqrt(m/k) = 2*pi*sqrt(2/8) = 2*pi*sqrt(0.25).
  2. sqrt(0.25)=0.5.
  3. 2*pi*0.5.
Answer: T = pi ≈ 3.14 seconds
Example 2 Find the period of a simple pendulum with length 1m (g=9.8 m/s^2).
  1. Apply T = 2*pi*sqrt(L/g) = 2*pi*sqrt(1/9.8).
  2. sqrt(1/9.8) ≈ 0.319.
  3. 2*pi*0.319.
Answer: T ≈ 2.0 seconds
Example 3 In SHM, where is kinetic energy at its maximum, and where is potential energy at its maximum?
  1. At the equilibrium position, the object moves fastest, so KE is maximum there (and PE is minimum).
  2. At the extremes of motion (maximum displacement), the object momentarily stops, so KE is zero and PE is maximum.
Answer: KE is maximum at equilibrium; PE is maximum at the extremes of motion

04 Practice Questions

1A 1 kg mass on a spring with k=4 N/m. Find the period.
T=2*pi*sqrt(1/4)=2*pi*0.5=pi≈3.14s
2Find the period of a pendulum with length 4m (g=9.8).
T=2*pi*sqrt(4/9.8)≈4.01s
3What is the defining force equation for SHM?
F = -k*x
4What does the negative sign in F=-kx represent?
The restoring force always points opposite to the displacement
5At what point in SHM is speed at its maximum?
At the equilibrium position

📄 Simple Harmonic Motion — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet