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.

Centripetal acceleration

When an object moves in a circle at a constant speed its velocity is constantly changing. Its velocity is changing not because the magnitude of the velocity is changing but because its direction is. this constantly changing velocity means that the object is accelerating, in other words centripetal acceleration. For this acceleration to happen there must be resultant force, this force is called the centripetal force.

 Centripetal acceleration : a= v²/r

To make an object move round on a circular path, it is acted on by a resultant force which changes its direction of motion. This resultant force is called as the centripetal force because it acts in the same direction as the centripetal accelerations, which is towards the centre of the circle.

Centripetal force in real life:

• For a planet moving round the sun, the force of gravity between the planet and the sun is the centripetal force.
• For an object whirling round on the end of a string, the tension in the string is the centripetal force.

Gravitational fields

What do we already know about gravitational fields?

We know that the earth's gravitational field strength is 9.81Nkg ¯¹ .
When you throw a ball into the air and it returns to you because of the earth's gravitational force. The force of gravity on the ball pulls it back to earth.

The force of attraction between the ball and the earth is an example of gravitational attraction. This attraction exists between any two masses but we don't notice it since it's force is too weak. The time we notice it, it's when one of the masses is very large.

The mass of an object creates a force field around itself. Any other mass placed in the field is attracted towards the object. The second mass also has a force field around itself and this pulls on the first object with an equal force in the opposite direction. The force field round a mass is called a gravitational field.

The strength of a gravitational field(g) is the force(F) per unit mass(m)
= g=F/m : Nkg ¯¹

Gravitational potential

Gravitational potential energy is usually calculated using the formula ∆Ep=mg∆h.this can also be thought of as the work done moving the mass. The force being pushed against is given by mg and the distance moved is ∆h. This only works when the field is uniform like on the earth's surface.

The gravitational potential, V, at a point is the work done per unit mass to move a small object from infinity to that point.

The potential:  V=W/m
The work done to move a mass is:  ∆W=m∆V

Imagine you are in a space rocket, about to blast off from the surface of a planet. The planet's gravitational field extends far into space although it becomes weaker with increased distance from the distance from the planet. To escape from the planet's pull due to gravity, the rocket must do work against the force of gravity on it due to the planet. If the rocket fuel doesn't give enough energy to escape, the rocket will return.

Satellite motion

Satellite motion is not confined to artificial satellites orbiting the earth. Any small mass which orbits a larger mass is a satellite. The moon is the earth's only natural satellite.

Going back to my blog...

In this blog I want to talk about geostationary satellites.

Geostationary satellites

Geostationary satellites are in orbit above the equator. The height of their orbit, 36,000 km, is just the right distance so that it takes them 24 hours to make each orbit. This means that they stay in a fixed position over the Earth’s surface. 

The radius of orbit of a geostationary satellite can be calculated as follows using the equation:

r³/T² = GM/4π²

Uses of geostationary satellites:

• Communications - including satellite TV
• Global positioning or GPS - which is used for sat navs (satellite navigation systems

My first lesson with SHM

SHM- Simple Harmonic Motion

So what is simple harmonic motion??

It's the motion of an object if its acceleration is proportional to the displacement of the object from the equilibrium and is always directed towards the equilibrium position.

The acceleration,a,of an object oscillating in simple harmonic motion is given by the equations:

a= -(2пf)²x = -ω²x

It's helpful to think of a pendulum swinging backwards and forwards to help remember how displacement, velocity and acceleration vary. 

As the pendulum swings upwards from the equilibrium position its velocity is decreasing in other words it's accelerating. The force that makes this happen is a component of gravity that acts towards the equilibrium position.