Periodic and oscillatory motion
A motion that repeats itself at regular intervals of time is periodic: the Earth around the Sun, the hands of a clock. If the body moves to and fro about a fixed mean (equilibrium) position, the motion is also oscillatory: a pendulum, a guitar string, a mass on a spring.
The period T is the time for one complete oscillation.
The frequency ν = 1/T is the number of oscillations per second, measured in hertz (Hz).
The angular frequency ω = 2π/T = 2πν, in rad/s.
Every oscillation is periodic, but not every periodic motion is an oscillation (the Earth's orbit isn't to and fro).
Simple harmonic motion
Simple harmonic motion (SHM) is the oscillation in which the restoring force is proportional to the displacement from the mean position and directed towards it:
F = −kx
Its displacement varies sinusoidally with time:
x(t) = A cos(ωt + φ)
A is the amplitude, the maximum displacement.
(ωt + φ) is the phase, which tells the state of motion at time t.
φ is the phase constant, set by the starting position and velocity.
SHM is the projection, on a diameter, of uniform circular motion. A point moving around a circle of radius A with angular speed ω has a shadow on the x-axis that moves in SHM.
Velocity and acceleration in SHM
Quantity | Expression | At the mean position | At the extremes |
|---|---|---|---|
Displacement | x = A cos(ωt + φ) | 0 | ±A |
Velocity | v = −ωA sin(ωt + φ) = ±ω√(A² − x²) | maximum, ωA | 0 |
Acceleration | a = −ω²x | 0 | maximum, ω²A |
The key result a = −ω²x shows that the acceleration is proportional to the displacement and always points towards the mean position. Any motion satisfying this is SHM, with T = 2π/ω.
Energy in SHM
Kinetic energy: K = ½mv² = ½mω²(A² − x²)
Potential energy: U = ½kx² = ½mω²x²
Total energy: E = K + U = ½kA², constant
Energy continually changes between kinetic (all kinetic at the mean position) and potential (all potential at the extremes). K and U are each equal to half the total when x = A/√2.
The spring-mass system
A block of mass m attached to a spring of constant k on a smooth surface obeys F = −kx, so ω² = k/m and
T = 2π √(m/k)
A heavier mass or a softer spring gives a longer period. The period does not depend on the amplitude.
Example. A 0.5 kg block on a spring of k = 50 N/m: ω = √(50/0.5) = 10 rad/s, T = 2π/10 ≈ 0.63 s.
The simple pendulum
A small bob of mass m hanging from a light, inextensible string of length L, displaced by a small angle θ, feels a restoring torque proportional to sin θ. For small angles (sin θ ≈ θ), the motion is SHM with
T = 2π √(L/g)
T depends on L and g only: not on the mass of the bob, and (for small swings) not on the amplitude.
A seconds pendulum has T = 2 s; on Earth its length is about 1 m.
On the Moon, where g is about g/6, the same pendulum would have a period √6 ≈ 2.45 times as long.
Example. A pendulum 2.45 m long, with g = 9.8 m/s², has T = 2π √(0.25) = 2π × 0.5 ≈ 3.14 s.
Damped oscillations
Real oscillations slowly die out, because friction and air resistance remove energy. These are damped oscillations: the amplitude decreases with time while the period stays nearly the same. Shock absorbers in vehicles are designed to damp oscillations quickly.
Forced oscillations and resonance
When an oscillator is pushed by a periodic external force, it eventually oscillates at the driving frequency, not its own natural frequency. If the driving frequency is close to the natural frequency, the amplitude becomes very large. This is resonance.
Pushing a child on a swing at just the right moment makes the swing go higher.
Soldiers are ordered to break step when crossing a bridge, so that their marching doesn't drive the bridge at its natural frequency.
Tuning a radio means adjusting a circuit to resonate with one station's frequency.
The less the damping, the taller and sharper the resonance peak.
Summary
SHM: F = −kx, a = −ω²x, x = A cos(ωt + φ).
Speed is greatest at the mean position; acceleration is greatest at the extremes.
Total energy ½kA² is constant, shared between K and U.
Spring: T = 2π√(m/k); pendulum (small angles): T = 2π√(L/g).
Damping reduces amplitude; resonance occurs when the driving frequency matches the natural frequency.