Mass on a spring
Simple harmonic motion: stiffness and mass set the period, damping drains the amplitude.
Just need the number? Physics Formulas
Displacement vs time
Readouts
What's happening
The spring pulls back with force −kx, so acceleration is proportional to displacement and points toward equilibrium, which is the definition of simple harmonic motion. The period 2π√(m/k) does not depend on the amplitude. Damping removes energy each cycle, shrinking the oscillation exponentially without changing its period much.
- Period of a mass on a springT = 2π √(m / k)
- Hooke's lawF = −kx
On the syllabus
- GCSE Physics AQA 4.5.3Hooke's law F = kx
- A-Level Physics AQA 3.6.2simple harmonic motion, damping and resonance
Challenge
Predict first: if you quadruple the mass, what happens to the period? Run it at m and 4m and compare the graphs.
Quadrupling the mass doubles the period, because T ∝ √m.
Learn this properly
Lessons from GCSE Combined and Separate Science on this topic: worked steps, then exam-style questions with new numbers every time.
- Springs, elasticity and the force-extension practicalForces, work, elasticity, moments and pressurePlus
- Potential energy and energy conservationEnergy stores, transfers and efficiencyFree
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FAQ
What is the period of a mass on a spring?
T = 2π√(m/k). A heavier mass or a softer spring gives slower oscillations; the amplitude does not change the period.
What is Hooke's law?
The force from a spring is proportional to its extension, F = kx, as long as it is not stretched past its limit of proportionality. k is the spring constant in N/m.
What does damping do?
Damping removes energy every cycle, so the amplitude shrinks exponentially. Light damping barely changes the period; heavy damping can stop the oscillation altogether.
How do I find the spring constant from the period?
Rearrange T = 2π√(m/k) to k = 4π²m/T². A 0.50 kg mass bouncing with a period of 0.80 s gives k = 4π² × 0.50 / 0.80² = 30.8 N/m. Pick Spring constant under Solve for to see each step, or k from a stretch to use Hooke's law instead.