Mass-Spring Systems and Pendulums

The two most important physical systems that exhibit SHM at A-Level are the mass-spring system and the simple pendulum. Each has its own period equation, and understanding where these equations come from — and their limitations — is essential for exam success.

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The Mass-Spring System

Deriving the Period

For a mass m attached to a spring of spring constant k, Hooke's law gives the restoring force:

F = −kx

By Newton's second law, F = ma, so:

ma = −kx

a = −(k/m)x

Comparing this with the SHM equation a = −ω²x:

ω² = k/m

ω = √(k/m)

Since T = 2π/ω:

T = 2π√(m/k)

Key Features

• The period increases with mass (heavier objects oscillate more slowly)
• The period decreases with spring constant (stiffer springs give faster oscillations)
• The period does not depend on amplitude (isochronicity)
• The period does not depend on gravitational field strength — a mass-spring system would have the same period on the Moon as on Earth

Assumptions