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Notes simple harmonic motion, damped and forced oscillation for Cambridge A level

Oscillation

Oscillation is one complete movement from the starting or rest position, up, then down and finally back up to the rest position.

Examples are pendulum, the beating of the heart, vibration of a guitar string, the motion of a child on a swing e.t.c.

Free Oscillation

A free oscillation means

  • No energy loss
  • No external force acting
  • Constant energy
  • Constant amplitude

Diagram of oscillation

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Understanding of terms
Period : it is the time taken for one complete oscillation
Frequency: it is the number of oscillations per unit time
Amplitude : it is the maximum displacement from the rest position.
Displacement : it is the distance from the equilibrium position
F= 1/T unit is Hertz (Hz)
F is frequency, T is period

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Simple harmonic motion
It is defined as the motion of a particle about a fixed point such that its acceleration a is proportional to its displacement x from the fixed point, and is directed towards the point.
Mathematically,

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The negative sign tells us that the acceleration is always in opposite direction to the displacement x

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The slope or gradient of this graph = ω^2

Conditions necessary for a body to execute S.H.M

  • when the body is displaced from equilibrium, there must exist a restoring force
  • this restoring force must be proportional to the displacement of the body

(it is always directed to the equilibrium position)

.a is acceleration

. x is the displacement

Changes in displacement, velocity and acceleration during simple harmonic motion

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This means when displacement is maximum, velocity is zero and acceleration maximum but in opposite direction.

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V is the instantaneous speed
Interchange between Kinetic and potential energy
K. E is maximum when displacemet is zero
P.E is maximum when the displacement is maximum

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Damping : This is an influence within or upon an oscillatory system that has the effect of reducing oscillations

It can now be properly define as reduction in energy of oscillations/ reduction in amplitude due to force opposing motion/ resistive force.

Note

During damping amplitude of oscillation does not decrease linearly also the frequency of the oscillations does not change as the amplitude decreases.

Types of Damping

Light damping: Amplitude decreases gradually as the oscillations continues for a long time

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Critical damping: displacement decreases to zero in the shortest time without oscillation

Over damping : displacement decreases to zero in a longer time than for critical damping without any oscillation

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Forced oscillation

This occur when an external force is applied to the original frequency causing a change in the frequency of the oscillation

For resonance to occur, there must be a system capable of oscillating freely and also have a way in which the system is forced to oscillate.

some terms

forced frequency: frequency at which object is made to vibrate

natural frequency of vibration: frequency at which object vibrates when free to do so

Resonance

resonance occurs when the natural frequency of vibration of an object is equal to the driving frequency giving a maximum amplitude of vibration

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Uses of resonance

oscillation of a child’s swing

microwave to cook food

tuning of radio receiver

Problems of resonance

high-pitched sound waves can shatter fragile object

metal panels on machinery vibrate

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