Oscillation

Continuously repeated movements

Period (T)

The time taken for an object to complete one full oscillation

Simple harmonic motion

The motion of an oscillating system.

There’s usually restoring force during the SHM, which tries to return the object to its Centre position

Restoring force

F = -kx

Where the force is proportional to the distance from the center position and k is a constant that depends on the particular oscillating system

Amplitude

The maximum displacement from the equilibrium position

Period of a pendulum

T = 2π√l/g

Where L is the length of the pendulum string

Angular velocity

• ω = θ/t

Where t is the time in seconds

• ω = 2πf

• ω = 2π/T

Where T is the period

Horizontal distance

x = r cosθ

Moving in a circle

x = r cos(ωt)

R can also be amplitude

ω of a pendulum is √g/L

L is the length of the string of the pendulum

ω of a string is √k/m

K is the restoring force constant

Acceleration acts in the opposite direction to the displacement

When displacement is zero so is the acceleration

And when X is max so is acc

The displacement graph is a cosine graph

The acceleration graph is a negative cosine graph

The velocity is a negative sine graph

During a swinging motion of a pendulum, at each end of the swing, it’s velocity is zero

So it has zero kinetic energy and a maximum potential energy at this point

When the bob passes through the central position, it’s velocity is at a maximum

And the kinetic energy is at a maximum while the potential energy is at a minimum

Free oscillation

A continuous exchange of PE and KE, caused by a restoring force which is proportional to the displacement.

Natural frequency

The frequency at which the oscillating system naturally oscillates without external forces

Forced oscillation

Forcing the the oscillating system to oscillate at another frequency than its natural one.

The frequency is called driving frequency

Damped oscillations

These suffer a loss of energy in each oscillation reducing the amplitude over time

Types of damping:

Under damping

Over damping

Critically damped (overshooting)

Under damping

It’s when the oscillator completed several oscillations and the amplitude decreases exponentially

E.G. A normal pendulum

Overdamping

The amplitude of the oscillation may drop so rapidly that the oscillator does not even complete one cycle

E.g. under water

Critical damping or overshooting

When the oscillator returns to its equilibrium position in the quickest possible time without going past that position

Undamped oscillation

Is when no energy is lost during oscillations

Resonance

Is when a system is forced to vibrate at its natural frequency, absorbing more and more energy and thus increasing the amplitude.

A simple setup of several pendulums can demonstrate resonance. They heavy swinging ball acts through the suspension wire to drive all the other pendulums at its natural frequency.

Only the pendulum with the same natural frequency will vibrate with a large amplitude. The pendulum has the same length of wire as the driving pendulum. The other pendulums show little movement because the driving frequency doesn’t match their natural frequency

As the driving frequency applied to an oscillating system changes, it will pass through Natural frequencies of the system which causes large amplitude vibrations

The size of the vibrations at resonant frequencies can be so great that they damage the system