S-Cool Revision Summary

S-Cool Revision Summary

Basic Oscillations

The time taken for an oscillating object to complete one full oscillation is called the time period, T. It is measured in seconds.

If a number of oscillations are involved we can work out the time period by dividing the total time taken by the number of oscillations completed:

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The frequency, f, of oscillations is the number of oscillations undergone in one second, and is measured in hertz (Hz).

The frequency and the period can therefore be related as:

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The displacement of an oscillating particle is the distance the particle has been moved from its equilibrium position.

The amplitude of an oscillation is the maximum displacement of the vibrating object from the equilibrium position (its usual position).

Simple Harmonic Motion

Simple Harmonic Motion is a vibration whose time period stays the same even when its amplitude changes.

Simple harmonic motion occurs when the displacement of the oscillator is proportional to the acceleration but in the opposite direction or

s ∝ -a

Simple Harmonic Motion

Finding Acceleration

The definition for simple harmonic motion tells us that:

a ∝ -s

We can get rid of the proportionality sign by putting in a constant. In this case, the constant is (2πf)2, so:

a = - (2πf)2 s

Finding Displacement and Velocity

As shm oscillations follow a sine or cosine wave, we can find the displacement at any point using:

s = A cos (2πf t) or s = A sin (2πf t)

Note: use cosine if your time starts when you are max displacement and sine if it starts when you are at the centre of the oscillation.

Where:

A = amplitude - not acceleration!

Velocity can be found using:

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Kinetic and Potential Energy

The total energy in S.H.M. is constant when no energy is lost to the environment. Energy changes from kinetic to potential.

Kinetic and Potential Energy

Energy and Amplitude

The amplitude of a wave gives an indication of the amount of energy the oscillator has. This makes sense if you think of the spring and mass. The greater the amplitude the larger the amount of energy stored in the spring when it is extended. However,

PE = 1/2 ks2 so energy must be proportional to the amplitude2.

Damping

If energy is being removed from the system the oscillations are damped and the amplitude will decrease with time.

The greater the damping the quicker the vibrations will diminish.

Critical damping occurs when the displacement returns to zero in the quickest time, without going past the equilibrium position.

Free vibrations:

Damping, Natural Frequency and Resonance

Forced vibrations:

Damping, Natural Frequency and Resonance

Natural Frequency

Hit anything and it will vibrate. The amazing thing is that every time you hit it, it will vibrate with exactly the same frequency, no matter how hard you hit it.

The frequency of un-damped oscillations in a system, which has been allowed to oscillate on its own, is called the natural frequency, f0.

In order to keep it vibrating after you've hit it, you need to keep re-hitting it periodically to make up for the energy being lost. We say that you need to apply a periodic force to it. (Although some people would just say that you are being unnecessarily violent.)

The frequency with which the periodic force is applied is called the forced frequency. If the forced frequency equals the natural frequency of a system (or a whole number multiple of it) then the amplitude of the oscillations will grow and grow. This effect is known as resonance.

Symbols

Simple Harmonic Motion

a = acceleration of the vibrating object, ms-2

s = displacement of the vibrating object from its equilibrium condition. (Note - it's a vector)

ω = the angular frequency of the oscillation, Hz.

f = frequency of the oscillations, Hz

T = the period of oscillation, s

t = the time since the start of the oscillations, s