 # Oscillations

An oscillation is a system in which a mass moves towards an equilibrium point. Examples of oscillations are a spring pushing and pulling a mass, a ball bobbing on water, a coin rolling up and down the inside of a curved bowl, and a skateboarder on a half-pipe.

Oscillation occurs when the force is zero at the mid-point (equilibrium), and if the mass moves away from the equilibrium point, a force increase proportionally to the distance away it is. As the spring stretches, the force pulling the object back to the mid-point increases.

All oscillations have cycles, which have a period and frequency.

## Pendulums

A pendulum is an example of an oscillation with a period and frequency. An oscillation is a motion which is self-replicating. It may be, as in the case of a spring, due to elastic potential converting to kinetic energy, and back again. Or it may be, as in the case of a pendulum, due to kinetic energy converting to gravitational potential, which then reconverts to kinetic.

Pendulum: can you time the period and calculate the length of the string?

The period of an oscillation is the time interval between two positions that are identical in space. This means that a period of a pendulum consists of two swings.

The frequency, λ, of a pendulum is the number of double swings (back and forth) per second, and is expressed in Hertz (Hz).

The mass and amplitude of swing has no effect on the period of a pendulum. Instead, only the length of the string will determine the period of oscillation for a any mass.

### The Maths

The period T of a pendulum of length L is:

\$\$T = 2π √{L/g}\$\$

Note: the value of π/√g is 0.987, or very close to 1. Therefore, an approximation of the time it takes a pendulum to make a single swing (half a period) is simply √L.

A pendulum 1.0m long has a period of 2 secs, or 1 second a swing. This is how Galileo 'swung' the pendulum clock industry into being!

# Hooke's Law

When a spring is compressed or stretched, a force is created which tries to return the spring to the length it has when it is neither compressed nor stretched. This point is called the 'equilibrium point'.

## Spring Force Hooke's Law states that the force is linearly proportional to the distance of compression or extension of a helical spring

Hooke's Law states that the force exerted by a spring opposes the force extending or compressing the spring. Since a spring always resists the force trying to stretch it or compress it, there is a negative sign in the equation:

F = - kx

where F is the force exerted by the spring, k is the spring (or force) constant (unit N⋅m-1 or kg⋅s-2), and x is the distance to which the spring changes length from its natural length.

Be careful! - the F in Hooke's Law refers to the spring force, not the force causing the spring to stretch or compress.

When a load (mass) is hung from a spring, it will extend the spring till the spring force equals the weight (mg) of the object. Similarly, a mass placed on a standing spring will be compressed till the spring force equals the weight (mg).

## Period of Oscillation:

Oscillation of a helical spring

Consider a spring hanging from a stand. A load (weight) is hung from it and allowed to fall. As it extends the spring beyond its natural length (the length it adopts when there is no load), the spring generates a force (F) opposing the extension. At a certain equilibrium point, this force (F) is equal to the weight of the load (mg). However, the system has some kinetic energy due to the acceleration of the load under gravity. This energy means the spring will continue to extend. As it does so, the spring force (F) continues to increase. The spring converts the kinetic energy of the load to elastic potential.

When all of the kinetic energy has been converted to elastic potential, the spring force F is greater than the weight of the load (mg), and causes the spring to reverse its direction of movement and accelerate the load back to the equilibrium point (where the forces are equal). In so doing, the spring converts its excess (due to the extension beyond the equilibrium point) elastic potential to kinetic again.

The spring is now travelling upwards. At the equilibirum point, the load does not stop, since its kinetic energy compresses the spring beyond the equilibrium point, converting the kinetic energy to elastic potential again (this time as compression). This new elastic potential now in turn reverts to kinetic energy as the system attempts to return to the equilibrium point, and the load falls. As before, at the equilibrium point there is kinetic energy, extending the spring again, and the cycle repeats.

This kind of cycle, where a constantly changing force is generated and oriented towards a central equilibrium, is called Simple Harmonic Motion (SHM).

The period T of a helical spring with spring constant k, and mass m, is:

\$\$T = 2π √{m/k}\$\$

## Work done by a spring

Recall that the work W done by a force F over a distance d is W = F.d

However, the force in this case is the average force the spring applies, which is half the maximum force (at full extension or compression).

\$\$F_{avg} = {F_{max}}/2 = - {kx}/2\$\$

where x is the distance from the end of the spring to the equilibrium point, and k is the spring constant. The spring constant is dependent on the characteristics of the spring. A strong spring has a larger spring constant than a weak spring.

The negative sign indicates that the spring always resists any force stretching or compressing it, so generates a force in the opposite direction to x.

### Limits of compression and extension

Every spring has physical limits to its compression and extension, depending on how much the coils can be compacted together or stretched. Graph of Hooke's Law

### Consequences

Hooke's Law provides a basis for the understanding of Simple Harmonic Motion (SHM).

The manometer works on the Hooke's Law principle. The manometer is an instrument that measures pressure. The compression of a spring inside the instrument will generate a force resisting the movement of the dial mechanism equal to the pressure being measured.

Clock mechanisms include a balance wheel, which works according to Hooke's Law.

Robert Hooke was an English scientist who lived 1635 - 1703. He was a polymath, and worked in a very large number of fields, including physics, microscopy, and architecture. Hooke was a contemporary of Isaac Newton (1643 - 1727), and they were antagonists, meaning they were always fighting and criticising each other in public. It was mainly for his dislike of Hooke, that Newton did not become involved with the Royal Society till he took over the presidency in 1703, after Hooke died! Robert Hooke was an early user of microscopes to study the world of the very small Robert Hooke, 1635 - 1703, scientist and experimentalist, who led the way in the use of the microscope in scientific investigation

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