Unit 4: Section 1 - Further Mechanics

Momentum(Kg msֿ¹) = Mass(Kg) x Velocity(msֿ¹)

p = mv

Assuming that there is no external force acting on an object, momentum will always be conserved.

Momentum Before = Momentum After

In an elastic collision Kinetic energy is conserved as well as momentum, So no energy is lost in heat, sound, etc. However if the collision is inelastic then Kinetic energy is not conserved.

"The rate of change of momentum of an object is directly proportional to the resultant force which acts on the object"

F = mv / t

If the mass is constant this can be written as:

F = ma

Impulse is the change in momentum.

Impulse = mv - mu

Impulse is the area under a force-time graph.

The force caused by an impact can be reduced by increasing the time of the impact.

Linear speed, v = Distance / Time

Angular speed, ω = θ / t

θ is measured in radians.

These two formula can be linked by:

v = rω

Frequency, f is the number of complete revolutions per second. f = 1 / T

Time period, T is the time taken for one complete revolution, T = 1 / f

For a complete circle, an object turns through 2π radians.

ω = 2πf

ω = 2π / f

An object in circular motion has a constantly changing velocity since its direction is constantly changing. Since acceleration is the rate of change of velocity, the object is constantly accelerating. However the object can still be travelling at a constant speed.

Centripetal acceleration, a = v² / r

and

a = ω²r

Based on Newton's laws. If there if a centripetal acceleration, then there has o be a centripetal force.

Using the two formula above and F = ma

F = mv² / r

and

F = mw²r

SHM: An oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint, and is directed towards the mid point.

In SHM the type of potential energy(PE) depends on what restoring force is required. EG. gravitational PE for pendulums and Elastic PE for masses on springs.

As an object oscillates its energy changes from PE to KE.

At the midpoint, PE is zero and KE is maximum. At maximum displacement, PE is maximum and KE is zero.

The sum of KE and PE is the mechanical energy. If the system isn't damped then the mechanical energy is constant.

Graphs: Displacement,x: A some wave. Max value is A.

Velocity: gradient of displacement-time graph. Max value (2πf)A. 1/4 cycle in front of displacement.

Acceleration: Gradient of velocity-time graph. Max value (2πf)²A. Anti phase of displacement.

In simple harmonic motion, the frequency and period are independent to the amplitude. EG A pendulum clock will still tick in regular time intervals even if its swing becomes very small, or very large.

Acceleration:

a = -(2πf)²x

a max = - (2πf)²A

Velocity:

v = ± 2πf √ A² - V²

v max = 2πfA

Displacement:

x = A cos(2πft)

An example of a SHO is a mass on a spring.

When the mass is pushed or pulled from its equilibrium position, a force is exerted on it.

F = -kx, k is the spring constant of the spring in Nmֿ¹

T = 2π √ m / k

A simple pendulum: If you set up a simple pendulum and change the lenght, l, the mass of the bob, m, and the amplitude, A. You would get the following graphs:

T2 - l: T α √ l, so T² α l.

T - m: T is not affected by m

T - A: T is not affected by A

T = 2π √ l / g

If you stretch and release a mass on a spring, it will oscillate at its natural frequency.

If no energy is transferred to or from the surroundings then it will keep oscillating with the same amplitude forever. In practise there never happens. However a spring vibrating in air is called a free vibration.

A system can be forced to vibrate by a periodic external force. The frequency of this force is called the driving frequency.

If the driving frequency is much less than the natural frequency then the two are in phase.

If the driving frequency is much greater than the natural frequency then the oscillator won't be able to keep up.

As the driving frequency gets closer and closer to the natural frequency the system gains more and more energy from the driving force and so vibrates with a rapidly increasing amplitude. This is called resonating.

Example of resonance: A glass resonating will smash.

There are four types of damping: Light, heavy, Critical and overdamping.

Light: Displacement slowly decreases

Heavy: Displacement decreases faster than Light damping

Critical: Reduces the displacement to zero in the quickest possible time

Over-damping: Takes longer to reach zero than critical damped systems.

For objects in resonance:

Light: The system has a very sharp resonance peak. Amplitude only increases dramatically when the driving and natural frequency are very close.

Heavy: Has a flatter resonance, amplitude doesn't increase much during resonance.

Structures are damped to avoid being damaged by resonance. Loud speakers are heavily damped so that they have a flatter resonance, to avoid "colouring" the sound.