# Oscillations

## Oscillating Systems

An oscillating object moves back and forth repeatedly, with each motion cycle (back and forth) called an oscillation. The object is undergoing periodic motion, where it returns to a given position after a fixed time interval every cycle. Examples of oscillations include atomic vibrations, pendulums, electromagnetic waves, etc.

An oscillating system has several important features:

• The equilibrium position is the point at which the oscillating object can remain at rest, with zero net force acting on it.
• The restoring force acts on the object and is always directed towards the equilibrium position. If the object is disturbed from this point, the restoring force causes it to oscillate back and forth past the equilibrium position.
• If no other forces act (such as dampening forces, like friction) then each oscillation will be identical to the previous one.

## Simple Harmonic Motion

Simple harmonic motion, or SHM, is a particular type of periodic motion where the restoring force is directly proportional to the object's position. The further the object is displaced from equilibrium, the greater the restoring force. This causes the object to oscillate back and forth repeatedly about the equilibrium point.

For example, imagine a block on a frictionless, flat surface, attached to a spring. If the block is pushed to x and released, the spring force accelerates it back towards the centre. When the block reaches the centre, no force is acting on it, so it moves past the centre due to its velocity. This stretches the spring, decelerating the block. When the block reaches -x, it has decelerated completely and begins to accelerate in the other direction. This cycle repeats indefinitely, since there is no friction to slow the block down.

• At x, displacement is maximum and the stretched spring means that the force is also at a minimum (acting opposite to displacement). Velocity is zero, as the particle is changing direction.
• At the origin, there is zero displacement or force, but velocity is at a maximum (or minimum, depending on the direction).
• At -x, displacement is minimum and force is maximum. Velocity is once again zero.
• The system's energy is constantly being transferred between kinetic energy in the block and potential energy in the spring.

The restoring force here is the spring force. Remember that spring force is given by the equation F = -kx , where k is the spring constant and x is the displacement. The negative sign indicates that the force is always in the opposite direction to the displacement - back towards equilibrium. Combining F=-kx and F=ma gives a relationship for the particle's acceleration:

ax = -(k/m)x

Acceleration is proportional to force (F=ma), and so is a maximum/minimum whenever force is maximum/minimum.

### Displacement, Acceleration and Velocity for SHM

Acceleration is not constant during simple harmonic motion, so the kinematics equations are no use. However, the repetitive oscillations of SHM may be easily represented by a sine or cosine function. The following equations describe a particle oscillating back and forth on the x axis.

The position (x) is proportional to cosine function of time (t) and constant factors:

x(t) = A * cos(ωt + φ)

A is the amplitude, or maximum displacement. ω is the angular frequency. ω = sqrt(k/m). φ is the phase constant, and depends on the particle's location when t=0. φ is zero if the particle is initially at x=A.

Together, the quantity (ωt + φ) is known as the phase.

Differentiating gives equations for velocity and acceleration in terms of time:

v(t) = -ωAsin(ωt + φ) a(t) = -ω2Acos(ωt + φ)

From these equations, several relationships are evident. These mirror the example of the block and spring, above.

• The particle oscillates between x=A and x=-A, as cos(ωt + φ) varies between 1 and -1.
• The acceleration is proportional to the displacement, but in the opposite direction.
• Velocity is a maximum/minimum when acceleration and displacement are 0, and vice versa.

These relationships are displayed on the graph below. The horizontal axes display time and the vertical axes display displacement, velocity, and acceleration respectively.

### Period and Frequency for SHM

• The period is the amount of time it takes for one full oscillation to occur. The symbol for period is T. On the graphs above, period is the difference in time between two peaks.
• Frequency is the number of cycles per second. The symbol for frequency is f.

For any system, frequency is inversely proportional to period.

f = 1/T

Frequency and period may also be expressed in terms of angular frequency (ω).

T = (2π/ω) f = (ω/2π)

Thus ω = 2πf = 2π/T

### Oscillation Energy

 In an oscillating system, energy is constantly transferred between kinetic and potential energy. No energy is lost, so the total energy of the system remains constant.

In the spring example, the energy is all stored as elastic potential energy when the spring is stretched. As the spring contracts, the energy is converted to kinetic energy as the block begins to move. At the equilibrium point, all the energy has been converted to kinetic. The energy is then converted back to potential energy as the spring compresses and the block slows. This can be represented on the displacement graph as shown below.

The total energy of a system in simple harmonic motion is constant, and is given by the equation:

E = ½kA2

The total energy at any time is also the sum of the kinetic and potential energies at that time:

E = K + U

The graph below shows the energy over time for the spring example.

### Uniform Circular Motion

 An object undergoing uniform circular motion may be treated as though undergoing SHM in two directions at once. There is an x component to the motion and a y component, which are offset by T/2. The vector sum of these motions traces out a circle.

### Pendulums

A pendulum exhibits SHM when swinging. Gravity provides the restoring force in this case.

A simple pendulum is one where an object with mass (m) hangs on the end of a light string (negligible mass) of length L. For a simple pendulum,

ω = sqrt(g/L)

Therefore, the period for a simple pendulum is given by:

T = 2π * sqrt(L/g)

Note that the period is independent of mass.

A physical pendulum is used when the object cannot be approximated as a simple pendulum. The physical pendulum can be used regardless of mass distribution. For a physical pendulum:

ω = sqrt((mgd)/I)

m = mass of pendulum g = acceleration due to gravity d = distance between pivot and centre of mass I = moment of inertia of object about the pivot

T = 2π * sqrt(I/(mgd))

## Dampened Oscillations

Damped oscillations occur when non-conservative forces act on the system, removing mechanical energy. For example, friction in the spring example would gradually remove energy from the system, damping the oscillations. The force which causes damping is the retarding force.

Unlike restoring force, which acts against displacement, the retarding force acts against velocity. The retarding force (R) is proportional to velocity and acts in the opposite direction. The constant of proportionality is the damping coefficient (b).

R = -bv

There are three types of damping that may occur, depending on the damping coefficient.

1. If the damping coefficient is equal to a critical value bc, the system is critically damped. The system will not oscillate, and motion will stop before the it reaches the equilibrium position.
2. If b > bc, overdamping occurs. The system will not oscillate, and motion stops when it reaches the equilibrium position.
3. If b < bc, underdamping occurs. The system oscillates with continually decreasing amplitude.

You only need a qualitative understanding of dampened oscillations for this course.

## Forced Oscillations

Forced oscillations occur when energy is fed into the system from an external driving force (often to combat retarding forces like friction). If the energy input (from forced oscillations) equals the energy lost (due to retarding forces), the oscillation amplitude will remain constant. If these two energies are not equal, the oscillations will gradually change until they are. For instance, a driving force on an object at rest will gradually increase the object's oscillation amplitude until energy losses through friction balance out the energy input.

### Resonance

If the driving force operates at the oscillating system's natural frequency, an increase in amplitude occurs. This is called resonance. The driving force is in phase with the oscillating system, so maximum power is transferred to the system.

You only need a qualitative understanding of forced oscillations and resonance for this course.

## End

This is the end of this topic. Click here to go back to the main subject page for Higher Physics 1A.