Simple Harmonic Motion Study Pack
Kibin's free study pack on Simple Harmonic Motion includes a 5-section study guide, 25 quiz questions, 30 flashcards, and 5 open-ended Explain review questions. 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 27, 2026
Simple Harmonic Motion Study Guide
Master the mechanics of oscillation by working through Hooke's Law, sinusoidal motion, and period formulas for spring-mass systems and simple pendulums. See how kinetic and potential energy trade off continuously while total mechanical energy stays constant.
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 perfectly sinusoidal, meaning position, velocity, and acceleration all vary as sine or cosine functions of time.
- •Period and frequency are independent of amplitude in SHM — doubling the amplitude of a spring-mass system does not change how long one oscillation takes.
- •For a mass on a spring, the period is T = 2π√(m/k), depending only on the mass and the spring constant; for a simple pendulum under small-angle conditions, T = 2π√(L/g), depending only on length and gravitational acceleration.
- •Energy in SHM is continuously exchanged between kinetic energy (maximum at equilibrium) and potential energy (maximum at maximum displacement), with total mechanical energy remaining constant in the absence of friction.
- •Velocity is greatest at the equilibrium position and zero at maximum displacement; acceleration is greatest at maximum displacement and zero at the equilibrium position.
Defining Simple Harmonic Motion
Simple harmonic motion is a specific category of periodic motion defined by the mathematical relationship between restoring force and displacement — not just any back-and-forth movement qualifies.
The Restoring Force Requirement
- •A system undergoes SHM only when the net force acting on it points back toward a fixed equilibrium position and its magnitude scales linearly with how far the object has moved from that position.
- •This relationship is expressed by Hooke's Law: F = −kx, where F is the restoring force, k is the spring constant (a measure of stiffness, in N/m), and x is the displacement from equilibrium.
- •The negative sign is essential — it means the force opposes the displacement, always pulling or pushing the object back rather than pushing it further away.
Equilibrium Position and Displacement
- •The equilibrium position is the location where the net force on the object is zero; left undisturbed, the object remains there indefinitely.
- •Displacement (x) is the signed distance from equilibrium at any instant — positive in one direction, negative in the other — and it is the quantity that drives the restoring force.
- •Amplitude (A) is the maximum magnitude of displacement reached during oscillation; it sets the scale of the motion but does not affect period or frequency.
Mathematical Description of SHM: Position, Velocity, and Acceleration
Because the restoring force in SHM is proportional to displacement, the resulting motion is sinusoidal — position traces out a perfect sine or cosine wave over time, and velocity and acceleration follow directly from that.
Position as a Function of Time
- •Position is described by x(t) = A cos(2πt/T), where A is the amplitude and T is the period — the time required to complete one full oscillation.
- •The cosine form assumes the object starts at maximum positive displacement (x = A) at t = 0; a sine form is used when the object starts at equilibrium moving in the positive direction.
- •Angular frequency ω = 2π/T = 2πf connects the period and frequency to the sinusoidal functions; it carries units of radians per second.
Velocity in SHM
- •Velocity is the time derivative of position: v(t) = −Aω sin(ωt), meaning velocity is 90° out of phase with position.
- •Maximum speed v_max = Aω occurs at the equilibrium position (x = 0), where all energy is kinetic.
- •Velocity drops to zero at the turning points (x = ±A), where the object momentarily stops before reversing direction.
Acceleration in SHM
- •Acceleration is the time derivative of velocity: a(t) = −Aω² cos(ωt), which means a = −ω²x — acceleration is always proportional to displacement and opposite in sign.
- •Maximum acceleration a_max = Aω² occurs at maximum displacement, where the restoring force is largest.
- •At the equilibrium position, acceleration is zero because the restoring force is zero there.
Period and Frequency for Common SHM Systems
Two physical systems — a mass attached to a spring and a pendulum swinging through small angles — are the canonical examples of SHM, and each has a distinct formula governing how fast it oscillates.
Mass-Spring System Period
- •For a mass m attached to a spring with spring constant k, the period is T = 2π√(m/k).
- •A heavier mass oscillates more slowly (larger T) because it has more inertia; a stiffer spring (larger k) produces faster oscillations because it exerts a stronger restoring force at any given displacement.
- •Critically, amplitude does not appear in this formula — stretching a spring twice as far does not change how long each oscillation takes, only how large it is.
Simple Pendulum Period
- •A simple pendulum — a point mass on a massless, inextensible string of length L — behaves as a SHM system only for small angles (roughly less than 15°), where sin θ ≈ θ in radians.
- •Under that small-angle approximation, the period is T = 2π√(L/g), where g is the local gravitational acceleration.
- •The mass of the bob does not appear in this formula; a heavier pendulum of the same length oscillates at exactly the same frequency as a lighter one.
Frequency and Angular Frequency
- •Frequency f = 1/T, measured in hertz (Hz), tells how many complete oscillations occur per second.
- •Angular frequency ω = 2πf = √(k/m) for a spring system, or ω = √(g/L) for a pendulum, and it appears directly in the sinusoidal equations for position, velocity, and acceleration.
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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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Question 1 of 25
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What is the mathematical expression for the restoring force in simple harmonic motion, as given by Hooke's Law?
Card 1 of 30
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Concept 1 of 5
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Restoring Force and Hooke's Law
Explain what a restoring force is and how Hooke's Law (F = −kx) describes it. Why is the negative sign important, and what role does this force play in making simple harmonic motion possible?
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