---
title: "Mass-Spring System — AP Physics C Mechanics Definition"
description: "A mass-spring system is a mass on a spring that oscillates in simple harmonic motion with T = 2π√(m/k). The go-to SHM model on the AP Physics C exam."
canonical: "https://fiveable.me/ap-physics-c-mechanics/key-terms/mass-spring-system"
type: "key-term"
subject: "AP Physics C: Mechanics"
unit: "Unit 6"
---

# Mass-Spring System — AP Physics C Mechanics Definition

## Definition

A mass-spring system is a mass attached to a spring that, when displaced from equilibrium, experiences a linear restoring force F = -kx and oscillates in simple harmonic motion with angular frequency ω = √(k/m) and period T = 2π√(m/k), independent of amplitude.

## What It Is

A mass-spring system is exactly what it sounds like, a block of [mass](/ap-physics-c-mechanics/key-terms/mass "fv-autolink") *m* attached to a spring with spring constant *k*. Pull the mass away from its equilibrium position and the spring pushes or pulls it back with a [force](/ap-physics-c-mechanics/unit-2/2-forces-and-free-body-diagrams/study-guide/2LH73zRqxtRXtAKH "fv-autolink") given by Hooke's Law, F = -kx. That negative sign is the whole story. The force always points back toward equilibrium, which is the defining feature of a restoring force and the reason the system oscillates instead of flying off.

Because the restoring force is directly proportional to [displacement](/ap-physics-c-mechanics/unit-1/2-displacement-velocity-and-acceleration/study-guide/robnlCwaanT6NImP "fv-autolink"), Newton's second law gives the differential equation a = -(k/m)x, which is the signature of simple harmonic motion (SHM). Solving it gives sinusoidal motion x(t) = A cos(ωt + φ) with angular frequency ω = √(k/m) and period T = 2π√(m/k). Two things about that period surprise people. It does not depend on amplitude (a big swing and a small swing take the same time), and it does not depend on gravity. A vertical mass-spring system oscillates with the same period as a horizontal one; gravity just shifts where equilibrium sits.

## Why It Matters

The mass-spring system lives in Topic 6.1 (Simple Harmonic Motion, Springs, and Pendulums) and is the default example [AP Physics C](/ap-physics-c-mechanics "fv-autolink") uses to define SHM itself. When the exam asks you to show that a system exhibits simple harmonic motion, the expected move is the mass-spring playbook. Write the net force, show it has the form F = -(constant)·x, and read off ω from the constant. It is also a workhorse for energy analysis, since the system trades kinetic energy ½mv² and spring potential energy ½kx² back and forth with total mechanical energy ½kA² conserved. That makes it a natural crossover problem connecting [oscillations](/ap-physics-c-mechanics/unit-7 "fv-autolink") back to the dynamics and energy units, which is exactly the kind of synthesis Physics C FRQs love.

## Connections

### Hooke's Law and the Restoring Force (Units 2 & 6)

[Hooke's Law](/ap-physics-c-mechanics/key-terms/hookes-law "fv-autolink"), F = -kx, is the engine of the mass-spring system. Any system whose net force looks like negative-constant-times-displacement is mathematically a mass-spring system in disguise, which is why the exam keeps handing you weird setups and asking you to prove they're SHM.

### Energy Conservation and Spring Potential Energy (Unit 3)

The spring stores potential energy U = ½kx², and during [oscillation](/ap-physics-c-mechanics/key-terms/oscillation "fv-autolink") the system constantly converts it to kinetic energy and back. Setting ½kA² equal to ½mv² at equilibrium is the fastest way to find maximum speed, no calculus required.

### [Simple Pendulum (Unit 6)](/ap-physics-c-mechanics/key-terms/simple-pendulum)

The [pendulum](/ap-physics-c-mechanics/key-terms/pendulum "fv-autolink") is the mass-spring system's sibling in Topic 6.1, but its restoring force comes from gravity instead of a spring. Its period T = 2π√(L/g) depends on gravity and not on mass, the exact opposite of the spring's T = 2π√(m/k).

### [Angular Frequency (ω) (Unit 6)](/ap-physics-c-mechanics/key-terms/angular-frequency-w)

For a mass-spring system, ω = √(k/m) falls straight out of the differential equation a = -ω²x. Once you have ω, you get period, frequency, max velocity (ωA), and max acceleration (ω²A) almost for free.

## On the AP Exam

Mass-spring systems show up in both multiple-choice and free-response. MCQs test the period formula (what happens to T if you double the mass or the spring constant?), amplitude independence, and where velocity and acceleration are maximized (max speed at equilibrium, max acceleration at the endpoints). FRQs go further. A classic Physics C task is deriving SHM from scratch, meaning you write Newton's second law, show a = -(k/m)x, identify ω, and write x(t). Energy versions ask you to use ½kA² = ½mv² + ½kx² to find speeds at arbitrary positions. Watch for hybrid problems too, like a collision (Unit 4 momentum) that launches a block into a spring, then asks for the amplitude and period of the resulting oscillation.

## Mass-Spring System vs Simple Pendulum

Both oscillate in SHM and live in Topic 6.1, but the periods depend on different things. A mass-spring system has T = 2π√(m/k), so heavier mass means longer period and gravity is irrelevant. A pendulum has T = 2π√(L/g), so mass is irrelevant and gravity matters (a pendulum on the Moon slows down, a mass-spring system doesn't). Also, the pendulum is only approximately SHM (small angles), while an ideal spring is exactly SHM at any amplitude.

## Key Takeaways

- A mass-spring system oscillates in simple harmonic motion because Hooke's Law gives a restoring force proportional to displacement, F = -kx.
- The period is T = 2π√(m/k), which depends only on mass and spring constant, not on amplitude and not on gravity.
- To prove SHM on an FRQ, write Newton's second law, show the acceleration has the form a = -ω²x, and identify ω = √(k/m).
- Total mechanical energy ½kA² is conserved, so speed is maximum at the equilibrium position and zero at the endpoints, while acceleration does the opposite.
- A vertical mass-spring system has the same period as a horizontal one; gravity only shifts the equilibrium position downward by mg/k.
- Unlike a pendulum, doubling the mass on a spring increases the period (by √2), and the motion is exactly sinusoidal at any amplitude for an ideal spring.

## FAQs

### What is a mass-spring system in AP Physics C?

It's a mass attached to a spring that oscillates in simple harmonic motion when displaced from equilibrium. The restoring force F = -kx produces sinusoidal motion with period T = 2π√(m/k), and it's the standard SHM example in Topic 6.1.

### Does the period of a mass-spring system depend on amplitude?

No. For an ideal spring, T = 2π√(m/k) contains no amplitude term, so stretching the spring twice as far doesn't change the period. The block travels farther but also moves faster, and the two effects cancel exactly.

### How is a mass-spring system different from a pendulum?

The spring's period T = 2π√(m/k) depends on mass but not gravity, while the pendulum's period T = 2π√(L/g) depends on gravity but not mass. Also, a spring is exactly SHM at any displacement, while a pendulum is only approximately SHM for small angles.

### Does gravity change a vertical mass-spring system's period?

No. Gravity stretches the spring to a new equilibrium position (lower by mg/k), but the oscillation about that new equilibrium has the exact same period T = 2π√(m/k) as a horizontal system. The same setup would oscillate at the same rate on the Moon.

### Where is the speed of a mass-spring system the greatest?

At the equilibrium position, where all the energy is kinetic and v_max = ωA = A√(k/m). At the endpoints (x = ±A) the speed is zero but the acceleration is at its maximum, ω²A.

## Related Study Guides

- [6.1 Simple Harmonic Motion, Springs, and Pendulums ](/ap-physics-c-mechanics/unit-6/simple-harmonic-motion-springs-pendulums/study-guide/V3L3GpIEmeGgpS239Xdd)
- [Unit 6 Overview: Oscillations](/ap-physics-c-mechanics/unit-Ro4fvf7uNKWRj5Zi/review/study-guide/u9jCpEdPbPJalemc8fvS)

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