Simple Harmonic Motion Study Pack

Kibin's free study pack on Simple Harmonic Motion includes a 3-section study guide, 8 quiz questions, 10 flashcards, and 1 open-ended Explain review question. Sign up free to track your progress toward mastery, plus upload your own notes and recordings to create personalized study packs organized by course.

Last updated May 21, 2026

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Simple Harmonic Motion Study Guide

Master the mechanics of oscillation by working through the core principles of simple harmonic motion, from Hooke's Law (F = −kx) to sinusoidal position, velocity, and acceleration functions. This pack covers period formulas for spring-mass systems and simple pendulums, the energy exchange between kinetic and potential forms, and why amplitude never affects period or frequency — exactly what you need for SHM exam questions.

Key Takeaways

  • •Simple harmonic motion (SHM) occurs when a restoring force is directly proportional to displacement from equilibrium and always directed back toward that equilibrium point, described mathematically by Hooke's Law: F = −kx.
  • •The motion is sinusoidal in time, meaning position, velocity, and acceleration all vary as sine or cosine functions with the same angular frequency ω.
  • •The period of a spring-mass system depends only on mass and spring constant (T = 2π√(m/k)), while the period of a simple pendulum depends only on length and gravitational acceleration (T = 2π√(L/g)) — neither depends on amplitude for small oscillations.
  • •At equilibrium, a SHM oscillator reaches maximum speed; at maximum displacement (amplitude), speed drops to zero and restoring force reaches its peak magnitude.
  • •Energy in SHM continuously converts between elastic or gravitational potential energy and kinetic energy, with total mechanical energy remaining constant in the absence of friction.
  • •Amplitude, period, and frequency are independent parameters: changing the amplitude of an ideal SHM system does not alter its period or frequency.

Defining Simple Harmonic Motion

Simple harmonic motion is a specific category of periodic motion defined by a precise relationship between force and displacement, making it one of the most mathematically tractable and physically important types of oscillation in nature.

The Restoring Force Condition

  • •SHM requires that the net force on an object always points toward a fixed equilibrium position and has a magnitude that scales linearly with how far the object is from that position.
  • •This relationship is expressed by Hooke's Law: F = −kx, where x is displacement from equilibrium and k is the spring constant (a measure of stiffness in units of N/m).
  • •The negative sign is critical — it means force and displacement are always in opposite directions, ensuring the object is always pushed or pulled back toward equilibrium rather than away from it.

What Qualifies as a SHM System

  • •A mass attached to an ideal, massless spring on a frictionless surface is the canonical example of SHM because the spring force obeys Hooke's Law exactly.
  • •A simple pendulum — a point mass on a massless string — approximates SHM only for small angular displacements (typically less than about 15°), where the restoring force component is approximately proportional to displacement.
  • •Many real-world systems, from vibrating molecules to oscillating bridges, behave approximately as SHM systems near their equilibrium configurations.

Kinematic Description: Position, Velocity, and Acceleration

Because the restoring force in SHM varies linearly with displacement, the resulting motion follows a sinusoidal pattern, allowing position, velocity, and acceleration to be written as exact mathematical functions of time.

Position as a Function of Time

  • •The displacement of an SHM oscillator is given by x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant that sets the initial position.
  • •The amplitude A is the maximum displacement from equilibrium, representing the 'reach' of the oscillation; it is always a positive quantity.
  • •The phase constant φ is determined by initial conditions — for example, if the mass starts at maximum displacement and is released from rest, φ = 0.

Velocity and Acceleration in SHM

  • •Velocity is the time derivative of position: v(t) = −Aω sin(ωt + φ), so maximum speed v_max = Aω occurs at the equilibrium position where x = 0.
  • •Acceleration is the time derivative of velocity: a(t) = −Aω² cos(ωt + φ), which can also be written as a = −ω²x, directly linking acceleration back to displacement.
  • •Acceleration reaches its maximum magnitude at the turning points (x = ±A) where displacement is greatest, and equals zero at the equilibrium position.

Angular Frequency, Period, and Frequency

  • •Angular frequency ω (in radians per second) is related to the ordinary frequency f (in hertz) by ω = 2πf, and to the period T (in seconds) by ω = 2π/T.
  • •Period and frequency are reciprocals: T = 1/f, so a system with a period of 0.5 s oscillates at 2 Hz.

About this Study Pack

Created by Kibin to help students review key concepts, prepare for exams, and study more effectively. This Study Pack was checked for accuracy and curriculum alignment using authoritative educational sources. See sources below.

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