Mechanical Oscillations

Basic Quantities

An oscillation is a motion that repeats in time.

• Period T (s), frequency ν = 1/T (Hz), angular frequency ω = 2πν.
• Harmonic oscillation: x = A·cos(ωt).
• During oscillation, energy is exchanged between kinetic and potential energy.

Pendulums

Pendulum	Period	Dependence
Simple	T = 2π√(l/g)	depends on length l, not on mass
Spring	T = 2π√(m/k)	on mass m and stiffness k

• The period of a simple pendulum does not depend on mass. The formula is valid for small angles of displacement.
• If the length increases 4 times, the period increases 2 times.
• For a spring pendulum, if the mass increases 4 times, the period increases 2 times.
• The period depends on the square root: if the quantity changes 4 times, the period changes 2 times.

Resonance

Resonance: when the frequency of the forced oscillation equals the natural frequency, the amplitude increases sharply. Example: when soldiers march across a bridge in step, the bridge can start to oscillate violently.

Calculation example: if T = 0,25 s, then ν = 1/0,25 = 4 Hz.