Oscillations

Repetitive motion of an object around its equilibrium position

Object starts in an equilibrium position

Force applied to the object

Object is displaced

Begins to oscillate

Displaced from its equilibrium position and then released

Travels towards the equilibrium position at increasing speed

Slows down once it has gone past the equilibrium position

Reaches maximum positive displacement (amplitude)

Distance from equilibrium position

Maximum distance from equilibrium position

Time taken to complete one full oscillation

Number of complete oscillations per unit time

Difference in displacement between two oscillating objects

Difference in displacement of an oscillating object at different times

In phase

Oscillating in step

2 x maximum positive displacements

0 rad

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In antiphase

Oscillating out of step

1 x maximum positive displacement + 1 x maximum negative displacement

π rad

ω = 2π / T

ω = 2πf

a = -ω²x

• ω² is a constant for the object
• a ∝ x
• - means that a acts in the direction opposite to x (it returns the object to the equilibrium position)

• Straight line of constant, negative gradient
• Through the origin
• a ∝ x
• Gradient = -ω²

Constant gradient (which is equal to - angular frequency squared)

...Constant frequency and period of oscillation

......Period of oscilation independent of amplitude

.........Increase of amplitude = increase in average speed ∴ constant period of oscillation

• Zero displacement = pendulum is at/ moving through equilibrium position

• Maximum diplacement = pendulum is at the top of its swing

• t = 0 = maximum positive displacement

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• Gradient = velocity

• Zero displacement = maximum velocity

• Maximum displacement = zero gradient = zero velocity

• Looks as though the x-t graph has been shifted to the left by 1/4 oscillation

• v can be determined from the gradient of the x-t graph

• a can be determined from the gradient of the v-t graph

• Similar to the x-t graph, except 'inverted', ∴ a ∝ -x

Object begins oscillating from its amplitude: x = Acosωt

Object begins oscillating from its equilibrium position: x = Asinωt

v = +- ω√(A² - x²)

Zero velocity when x = A

Maximum velocity when x = 0 and oscillator at its equilibrium position

v_max = ωA

An oscillation is damped when an external force that acts on the oscillator has the effect of reducing the amplitude of its oscillations

• Small damping force
• Period of oscillations almost unchanged

• Large damping force
• Slight increase in period of oscillations
• Rapid decrease in amplitude

• Very large damping force
• No oscillatory motion
• Oscillator slowly moves towards its equilibrium

Transferred to other forms (usually heat)

Displaced from its equilibrium position and then allowed to oscillate without any external forces

The frequency of a free oscillation

An oscillation in which a periodic driver force is apllied to the oscillator

The frequency with which the periodic driver force is applied to the oscillator in a forced oscillation

A number of paper cone pendulums of varying lengths

......are suspended from a string

.........along with a heavy brass bob pendulum

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Bob = driver for the cones

...Oscillates at its natural frequency

......Forces the cones to oscillate at the same frequency

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The cone that has the same length as the bob also has the same natural frequency

...Resonates

......Its amplitude is greater than that of the other cones

For a forced oscillator with negligible damping, at resonance

driving frequency of the forced oscillation = natural frequency of the oscillating object

which causes a considerable increase in amplitude of the oscillation to the point at which the object fails

• The greatest possible transfer of energy from the driver to the forced oscillator occurs

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• The amplitude of the forced oscillator is maximum

• Many clocks keep time using the resonance of a pendulum/ quartz crystal

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• Many musical instruments have bodies that resonate to produce louder notes

For light damping, the maximum amplitude occurs at the natural frequency of the forced oscillator

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As the amount of damping increases:

• the amplitude of vibration at any frequency decreases
• the maximum amplitude occurs at a lower frequency than the natural frequency
• the peak on the graph become flatters and broader

Amplitude

• Briefly stationary
• Zero kinetic energy
• All its energy in the form of gravitational potential energy

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As it falls

• Loses gravitational potential energy
• Gains kinetic energy

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Equilibrium position

• Maximum velocity
• Maximum kinetic energy
• Zero gravitational potential energy

Mass oscillating vertically

Potential energy in the form of

• gravitational potential energy (due to the position of the mass in the Earth's gravitational field)
• elastic potential energy (stored in the spring)

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Mass oscillating horizontally

Potential energy in the form of

• elastic potential energy (stored in the spring)

• Total energy of an oscillating system remains unchanged

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• Continuous interchange between potential energy and kinetic energy

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• Sum at each displacement is always constant and equal to the total energy

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• Zero potential energy at the equilibrium position

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• Zero kinetic energy at the amplitude