Chapter 19: Oscillations

Oscillations
Many system mechanical and otherwise will oscillate freely when disturbed from equilibrium. In each case given below there is something that is oscillating.

Simple Harmonic Motion

Simple Harmonic Motion is defined as the motion of a particle about a fixed point such that its acceleration is proportional to its displacement from the fixed point, and is directed towards the point. The equilibrium is where the object comes to rest, in simple pendulum it at its lowest point. If we displace to object by displacement of x there will be force that brings the object back to the equilibrium point.

We can represent this as:

Since F= ma we can write:

F α –x

a α –x

Foe an object to be moving with simple harmonic motion, its acceleration must satisfy two conditions:

  • Acceleration is proportional to the displacement
  • Acceleration is in the opposite direction to the displacement.

a = -ω2x

Equations

ω is known as angular frequency of the oscillation

ω = 2πf

Solution of equation for s.h.m.:

x = x0 sin ωt

Or,

x = x0 cos ωt

Where, x0 is amplitude of oscillation and V is the gradient of displacement time graph.

v = x0ω cos ωt when x = x0 sin ωt

For the case where x is at zero at time t = 0, displacement and velocity are given by:

x = x0 sin ωt

v = x0ω cos ωt

Applying sin2Ø + cos2 Ø = 1

 X2/x02 + v2/x02 ω2 = 1

v2 = x02 ω2 – x2 ω2

So,

 v =±   ω √(x02– x2)

a = – x02 ω2 sin ωt when x = x0 sin ωt

a = – ω2x

The K.E. of the particle oscillating with s.h.m. is ½ mv2:    

Ek = ½ m ω2(x02-x2)

The restoring force is F=ma;

Fres= – m ω2x

The potential energy;

Ep = ½ m ω2x2                                                                                                             

The total energy of the oscillating particle is:

Etot = Ek + Ep

        = ½ m ω2(x02-x2) + ½ m ω2x2

Etot = ½ m ω2 x02

Damping
Damping forces oppose the motion of the oscillating body, they slow or stop simple harmonic motion from occurring. Damping force act in the opposite direction to the velocity.

Light Damping: It slowly reduces the amplitude of the oscillations, but keeps the time period almost constant.

Heavy Damping: It allows the body to oscillate but brings it quickly to rest.

Critical Damping: It brings the body back to the equilibrium point very quickly without oscillation.

Over Damping: It also prevents oscillation but makes the body take a longer time to reach equilibrium.

Resonance
Resonance is an important physical phenomenon that can appear in a great many different situations. Resonance occurs when a natural frequency of vibration of an object is equal to the driving frequency, giving a maximum amplitude of vibration. The frequency at which resonance occurs is called the resonant frequency.