A2 Physics


Further Mechanics


Contents

    Newton's Laws of Motion
    Momentum and Impulse
    Radians
    Circular Motion
    Frequency, Linear and Angular Velocity
    Centripetal Force and Acceleration
    Simple Harmonic Motion (SHM)
    Mass on a Spring
    The Simple Pendulum
    Energy in an Oscillator
    Damped SHM

   

Newton's Laws of Motion 

1. An object remains at a constant velocity unless acted on by a force.

Constant velocity can include being at rest or moving at a constant speed in one direction.

2. The rate of change of momentum is proportional to the resultant force on it.

3. When two objects interact, they exert an equal and opposite force on each other.

If all forces are equal and opposite how does anything accelerate? Because this action-reaction pair are on different objects! You must determine the net force on an individual object to see whether it will accelerate or not e.g. you are an astronaut 'floating' in space and you try to pull a spaceship towards you via a rope. You pull the spaceship, but the spaceship pulls you with an equal and opposite force. Due to Newton's second law (F=ma) you will accelerate much faster towards the spaceship compared with the spaceship accelerating towards you because you have a much smaller mass! What about a horse and a cart...

Momentum

Momentum , p, is given by the following equation:
It is measured in kgms^-1. The following rule of thumb is helpful: an object with a lot of momentum is 'difficult to slow down when it is moving;.

Principal of conservation of momentum = for a system of interacting objects, the total momentum remains constant, provided no external resultant force acts on the system i.e. momentum is constant before and after a collision/explosion.

Collisions

Elastic = no loss in total momentum and kinetic energy of the system i.e. objects bounce apart
Inelastic = total loss in momentum and kinetic energy of the system.
Partially inelastic = colliding objects move apart, and have less momentum and kinetic energy after the collision.

Explosions

In an explosion the total momentum of all things flying apart equal zero. This is because the initial momentum before the explosion is zero. Therefore the total momentum after the collision will be zero.

Impulse

Impulse is the force multiplied by the time for which the force acts (measured in Ns), it is otherwise just the change in the momentum of an object:
The area of a force-time graph is the change of momentum, or 'impulse':






Things get more complicated when the force varies over time. You may need to calculate a sum of areas for simple scenarios. However, if the force does not change uniformly you would need to integrate the function of the force curve (something that you don't require at A level!).

Radians

Angles can be measured in degrees or radians.

One radian = the angle subtended at the centre of a circle by an arc of length equal to the radius. 

Uniform Circular Motion




T = period (s) = time taken for a single rotation
f = frequency (Hz)
v = velocity (ms-1)
r = radius (m)
ω = angular velocity (rad s-1)

Frequency, Linear and Angular Velocity

Rotational speeds are often given as rotations/revolutions per second, otherwise known as the frequency, f. It is perhaps more useful to think of a new quantity known as the angular velocity, ω, which is the angle that the radius arm moves through in 1 second. Below are some useful relations that are essential for understanding circular motion:
v is the linear velocity. It is the velocity at the tangent to the circular path. If the centripetal force keeping the object in circular motion were to disappear suddenly then the object will follow the tangential path of the velocity at that moment (as in the above diagram). The linear velocity is therefore the instantaneous velocity (m/s) at that moment in time.

Centripetal Force and Acceleration

The acceleration is the rate of change in the velocity. In uniform circular motion the speed remains constant. Acceleration is the change in speed or direction, or both.

In uniform circular motion the direction of the velocity is constantly changing. The direction of the acceleration is always towards the centre along with the Force that is creating this acceleration.

In the equations opposite r is the radius of curvature.

Simple Harmonic Motion (SHM)

Simple harmonic motion describes a regular periodic motion e.g. a pendulum, swing, object on a spring, object rocking from side to side.
The equilibrium position is the point of zero displacement i.e. where the pendulum/box on spring would come to rest if the motion was 'damped' (or where a person on a swing would come to rest when they stop moving). As the object reaches maximum displacement (i.e. amplitude) the velocity becomes zero. This is because, as the object changes direction it must momentarily stop. At the point where the object has stopped to change direction the acceleration is maximum, this is because it has the greatest change of speed to undergo at this point. When the object undergoing SHM is moving through its equilibrium position its acceleration is zero because it is travelling at maximum speed.

The acceleration in SHM is therefore:
Where x is the displacement from equilibrium. This equation shows that at zero displacement (i.e. maximum velocity) the acceleration is zero and in the opposite direction (i.e. the negative sign). The variation of the displacement is sinusoidal and is given by the following expression:
Where A is the amplitude i.e. when t = T, x = Acos(2π) = A, because cos(2π) = 1. Remember A is the amplitude (i.e. maximum displacement) whereas x is just the displacement from the equilibrium position (this solution can be derived in the knowledge that a=d2x/dt2).

Mass on a Spring

The force on the mass on a spring is given by Hooke's law F=-kx. By making this equal to F=ma, we can derive a very useful equation for the frequency of a mass on a spring undergoing SHM.
This reduces to the following (written in 2 ways):


or
This equation shows you that the tougher the spring (high value for k) the quicker it will oscillate. The more massive the spring the slower it will oscillate.

The Simple Pendulum

The force on a pendulum is the component of gravity acting in the direction that the pendulum is moving in.
This equation only works for small values of θ. Sinθ can be written as follows:
Using this and the equation for acceleration that relates to frequency and displacement the following can be derived:
This can then be rearranged to get the following:


or



The image opposite demonstrates how the length of a pendulum affects the frequency/period. The shorter length pendulum at the top swings fastest and makes all the pendulums eventually move apart. Absolute mayhem!

Energy in an Oscillator

In an ideal simple harmonic oscillator energy is never lost therefore:
The potential energy in the spring is:
And at maximum displacement, the total energy of the system is equal to:
Where A is the amplitude of the oscillations. Since kinetic energy is given by:
Then:
The equation for the angular velocity of a the system, is given by the following (as derived above): 
Combining these equations we get an expression for the velocity of a mass on a spring at various positions, x:
The energy of the system can be represented as follows, in the ideal case where no energy is lost. Notice that the total energy remains constant throughout and that the potential energy and kinetic energy are continuously 'replacing' each other as the mass moves up and down.

Damped SHM

The graph above shows 3 scenarios whereby the damping effects vary. Often damping leads to an exponential decay of displacement. It is important to notice that the frequency of a heavily damped and lightly damped system are equal given than the mass and spring constant are equal for that system.

Resonance

When a system is oscillating without an applied force it is oscillating at its natural frequency.

When an applied force matches the frequency of the natural frequency resonance occurs and the amplitude of the oscillations is maximum.

Either side of the resonance frequency the amplitude of the oscillations are lower.

The graph opposite shows a number of different scenarios of forces oscillations. The red curve shows the system undergoing resonance at its resonance frequency.