Simple Harmonic Motion & Springs

You’ll be able to

Define simple harmonic motion and the restoring force
Apply Hooke’s law, F = −kx
Locate where speed and acceleration are greatest in an oscillation

A restoring force that grows with displacement

Simple harmonic motion (SHM) occurs whenever the force pulling an object back toward equilibrium is proportional to how far it has been displaced and points back toward equilibrium. The farther you pull the object, the harder it is pulled back. This single rule produces the smooth back-and-forth of springs, pendulums, and countless vibrating systems.

Hooke’s law and the turning points

For a spring, the restoring force follows Hooke’s law, F = −kx, where k is the spring constant and x is the displacement from equilibrium (the minus sign means "back toward equilibrium"). At the extremes (x = ±A, the amplitude) the force and acceleration are greatest but the object is momentarily at rest. At equilibrium (x = 0) the force is zero but the speed is greatest — energy has fully become kinetic.

Hooke’s law and spring period

F = −kx · T = 2π√(m/k)

k is in N/m. A stiffer spring (larger k) gives a shorter period; a heavier mass gives a longer one.

Worked example

A spring with k = 50 N/m is stretched 0.4 m from equilibrium. Find the restoring force.

1.Use Hooke’s law magnitude: F = kx.
2.Substitute: F = 50 × 0.4.
3.Compute: F = 20 N.
4.The force points back toward equilibrium, opposite the stretch (hence the minus sign in F = −kx).

Answer: F = 20 N, directed back toward equilibrium

Tip

Track the two extremes: at maximum displacement the force and acceleration peak while the speed is zero; at equilibrium the speed peaks while the force is zero. They are exactly out of step.

Checkpoint

A spring with spring constant 50 N/m is stretched 0.4 m from equilibrium. What is the magnitude of the restoring force?

Checkpoint

A mass oscillates on a spring. Where is its speed the greatest?

On the exam

A frequent SHM question asks "where is acceleration maximum / speed maximum?" Remember they are opposite: acceleration peaks at the extremes (max force), speed peaks at equilibrium (max KE).

Answer the 2 checkpoints as you read.