Revision Note 10: Simple Harmonic Motion

Conditions of Simple Harmonic Motion (SHM): Force is (a) restoring and (b) proportional to the displacement from the equilibrium position.

 

This solution applies to all SHM. The different quantities (ω, φ) can be found from initial conditions and A = maximum displacement.

Other formulas common to all SHM:

PE and KE goes back and forth, with PE + KE = constant.

Defining PE at equilibrium = 0, PE at other points:

Different Physical Systems:

Only k varies.

Spring:

k = Spring constant.

Pendulum: (for small θ)

Restoring force:

Notice mass = m cancels out and the period depends on square root of length.

Tip 10.1:

A vertical spring has the same time period as the equivalent horizontal spring – but its equilibrium position shifts due to mg, by an amount such that:

kδ = mg → δ = mg /k

Tip 10.2: Up-down intuition:

For springs:

k  ↑   T ↓

In words, stiffer spring leads to higher frequency and lower time period.

For pendulums:

L  ↑ T ↑

In words, longer pendulums have more stately swings.