Oscillations

 An oscillating object moves repeatedly one way then in the opposite direction through its equilibrium position. The displacement of the object from equilibrium continually changes during the motion. In one full cycle after being released from a non-equilibrium position, the displacement of the object:

• Decreases as it returns to equilibrium, then
• Reverse and increases as it moves away from equilibrium in the opposite direction, then
• Decreases as it returns to equilibrium
• Increases as it moves away from equilibrium towards its starting position

Examples of oscillating motion:

1. A pendulum moving to and fro repeatedly
   
2. A ball bearing rolling from side to side
3. A small boat rocking from side to side

Forced oscillations and resonance

Forced oscillations:

• When the system oscillates without a periodic force being applied to it,its frequency is referred to as its natural frequency.
• Periodic force is force applied at regular intervals. 
• When a periodic force is applied to an oscillating system, the response depends on the frequency of the periodic force. The system undergoes forced oscillations when a periodic force is applied to it.
• example:

Resonance:

• Applied frequency = natural frequency of system  
• the amplitude of oscillations becomes very large. the lighter the damping in the system, the larger the amplitude becomes. the system is in resonance when the applied frequency equals the natural frequency,
• the phase difference between the displacement and the periodic force is 1/2π at resonance. the periodic force is then exactly in phase with the velocity of the oscillating object. 
• Applied frequency > natural frequency of system
• the amplitude of oscillations decreases more and more,
• the phase difference between the displacement and the periodic force increases from 1/2π until the displacement is π radians out of phase with the periodic force.

 = Resonance curves

The principles of simple harmonic motion

Simple harmonic motion is defined as oscillating motion in which the acceleration is

Proportional to the displacement

• Always in the opposite direction to the displacement
• Acceleration,a = - constant x displacement

 The minus sign tells that the acceleration is in the opposite direction to the displacement. The constant of proportionality depends on the time period T of the oscillations. The shorter the time period, the faster the oscillations, which means the larger the acceleration at any given displacement. So the constant is greater the shorter the time period. The constant in this equation is (2πf)² , where f is the frequency.

Frequency= 1/T

Simple harmonic motion equation: a= -(2πf)²x 

Energy and simple harmonic motion

As a pendulum swings back and forth energy is changing from kinetic energy to potential energy constantly. If there were no energy losses it would continue forever. The total energy at any time would be constant.

If energy loss is taken into account then the amplitude of the oscillation decreases. This exponential decrease is called damping. Sometimes energy loss is not wanted so that there is minimum damping but also it can be desirable to limit oscillations such as in the suspension of cars.

Applications of SHM

Another simple device besides the spring-mass that can produce simple harmonic motion is the simple pendulum. A simple pendulum consists merely of a point-mass (m) suspended from a fixed point by a rod or string of length (L). The mass of the rod or string is assumed to be so much less than the suspended mass that it can be ignored. If the suspended mass is displaced to the left or right, while the rod or string is kept firm, and then released, the mass will swing freely back and forth under the gravity’s influence. For small horizontal displacements the restoring force on the suspended is given by:

F=-mgsinϴ

Therefore acceleration: a=F/m = -mgsinϴ/m = -gsinϴ

If ϴ does not exceed about 10°, then sinϴ = s/L,

Due to this, the acceleration a=-(g/L)s = -(2πf)²s, where (2πf)²=g/L

So the object oscillates with simple harmonic motion because its acceleration is proportional to the displacement from equilibrium and always acts towards equilibrium.

Newton's law of gravitation

Newton's law of gravitation assumes that the gravitational force between any two points objects is 

• always an attractive force,
• proportional to the mass of each object
• proportional to 1/ r², where r is their distance apart.

These last two requirements can be summarised as:

 

Gravitational force F= Gm₁m₂/r²

where m₁ and m₂ are masses of the two objects

The constant of proportionality,G, in the above equation, is called the universal constant of gravitation. 

G can be given units of Nm²kg¯². The value of G is 6.67 x 10¯¹¹  Nm²kg¯².

Planetary fields

The field lines of a spherical mass are always directed towards the centre, so the field pattern is just the same as for a point mass are always directed towards the centre, so the field pattern is just the same as for a point mass.

For a spherical mass M of radius R, the force of attraction on another mass (m) at distance r from the centre of M is the same as if mass M was concentrated at its centre. Therefore, the force of attraction between m and M,

F= GMm/r²

Therefore, the magnitude of the gravitational field strength at distance r is given by g=F/m= GMm/r², provided distance r is greater than or equal to the radius R of the sphere.