Oscillations

A levels

Characteristics of Simple Harmonic Motion

When the body’s acceleration is

1. Directly proportional to the displacement from its fixed equilibrium position

2. and is always in the opposite direction to this displacement

What is the net force on the object

Is directly proportional to the displacement and is always directed towards the equilibrium position

Formula for acceleration

a = -ω^{² x}

Formula for velocity

v = ±ω√(x₀² - x²)

Where does maximum velocity occur at

x=0 V_max = ωX₀

Formula for displacement (at eqm position)

x = ± X₀ sin(ωt)

Formula for displacement (at amplitude position)

x = ± X₀ cos(ωt)

Angular Frequency

Measured in radians per second,
ω = 2π/T or 2πf

ω

√(k/m)

Kinetic energy

E_k = ½ mω² (x₀² - x²)

Potential energy

E_p = ½ mω² x²

Total energy

E = ½ mω² x₀²

Damping

Process in which the amplitude of oscillations diminishes with time as a result of dissipative forces that reduce the total energy of the oscillations

Critical damping

When the system (when displaced and released) returns to the equilibrium position in the shortest possible time without any oscillations occuring

Forced oscillations

Oscillations where a periodic driving force is applied to the system to cause it to oscillate at the frequency at which the periodic force is applied

Resonance

The phenomenon which occurs when the frequency at which the object is made to vibrate is equal to the natural frequency of the vibration
Amplitude is at a maximum