Circular Motion and Oscillations

• v=rw    where v=linear velocity  r=radius of circle  w=anglar speed rads^-1

• Circular motion also has frequency and period

• Period = time taken for one complete revolution

• Frequency = 1/T

• In a complete circle the object turns 2 pi radians in time T

• w=2piF  w=2pi/T

Centripetal Acceleration

• The velocity of an object moving in a circle is constantly changing

• Since the velocity is changing, the object must be accelerating

• Centripetal acceleration is always directed towards the centre of the circle

• a=v^2/r

Centripetal Force

• From Newton's laws, there must be a resultant force towards the centre of the circle, causing the acceleration

Since F=ma, the centripetal acceleration must equal F=mv^2/r

F=GMm/r^2

Any two objects in the universe attract each other with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

• This type of law is an inverse square law

• Therefore, if distance doubles, the force will be a quarter of its original value

• Any object with mass will experience an attractive force if placed in the gravitational field of another object

• Only objects with a large mass can be significantly seen. eg Sun and the planets

When drawing gravitational field lines remember:

• Gravitational forces are always attractive (towards an object)

• If a gravitational field is radial, the gravitational field lines should look as if they meet at the centre of an object

• NEVER draw gravitational field lines inside an object eg. the Earth

Gravitational Field Strength

Defined as the force per unit mass on a small test mass placed at a point in the field g=F/m

In a radial field, g is inversely proportional to r^2

• Again, as this is an inverse square law, as r, the distance from the centre of the point mass increases, the gravitational field strength, g, decreases

• For example if r doubles, g will be 1/4 of the orignial value

A satellite is just a smaller mass which orbits a larger mass

We can think of planets as being satellies, orbiting around the Sun

• Because planets have near circular orbits, we can use the equations of uniform circular motion
• For circular motion, the period = 2 x pi x r / v

Situation: Earth orbiting the Sun

The centripetal force on the Earth must be equal to the gravitational force due to the Sun.

Therefore, mv^2/r = GMm/r^2 ..... rearrange to get v = sqr(GM/r)

Substitute this equation into T = 2 pi r/v

This dervies Kepler's Third Law..

T^2 is proportional to r^3

The period of the orbit squared is proportional to the radius of the orbit cubed.

Geostationary Satellites

• Orbit directly over the equator
• Are always above the same point on Earth
• Travels at the same angular speed (w) as the Earth below it
• Orbit takes exactly 24 hours

What uses do geostationary satellites have?

• TV and telephone signals
• Satellite appears to stay still relative to the Earth - don't have to alter position of receivers

SHM: motion where the acceleration of an object is directly proportional to the displacement from the the midpoint, and the in opposite direction. Examples: mass on a spring/simple pendulum.

During SHM an object will exchange PE and KE. The type of PE the object gains/loses depends on the system. eg. grav PE for pendulums and elastic PE for masses on springs.

The force which causes the acceleration towards the midpoint does work. Energy is converted from PE (max at ends) to KE (max at midpoint). The sum of the PE and KE will be constant unless the system is damped (energy loss to surroundings).

• If displacement from the midpoint is maximum at t=0, then displacement is a cos graph
• v=dy/dx x displacement
• The differential of a cos graph is a -sin graph.
• Also when the displacement is maximum at the endpoints, the velocity is zero
• Acceleration is just the differential of a -cos graph --> -sin graph

• From max +ve displacement to max -ve displacement and back again is one full oscillation - this is called the period
• The frequency is the number of complete cycles (oscillations) per second
• Frequency is measured in Hertz
• The period is the time taken for one complete cycle

Relating to SHM

• The frequency and period DO NOT depend on the amplitude of the oscillation
• For example - a pendulum clock will continue to swing at regular time intervals even if the swing is very small

The definition of SHM: motion where the acceleration of an object is directly proportional to the displacement from the midpoint and in the opposite direction.

• a is proportional to -x  (the minus sign indicates that these two quantities are in opposite directions
• The constant of proportionality is (2 pi f)^2
• a=-(2 pi f)^2 x

• If velocity is positive, the object is moving away from the midpoint
• If the velocity is negative, the object is moving back towards the midpoint
• Maximum velocity is reached when the displacement from the midpoint = 0

• The equations for displacement vary depending on where the object is in its cycle
• Maximum displacement --> x=Acos(2 pi ft) (at endpoints of the oscillation)
• Minimum displacement --> x=Asin (2 pi ft) (from midpoint)

A free vibration is when no energy is transferred to or from the surroundings.

A forced vibration is when an external force provides energy into the system.

• If you stretch a mass on a spring and let it go, it osciallates at its natural frequency

• In practice, energy is lost to the surroundings (damping) and the amplitude decreases

Resonance

Resonance occurs when the driving frequency = natural frequency

In this situation, the system gains more and more energy and vibrates with a rapidly increasing amplitude.

Examples of resonance include:

• An organ pipe - the column of air resonates
• Glass smashing - When the driving frequency (sound waves) equals the natural frequency

Damping is when energy is lost to the surroundings in an oscillating system

• Damping usually happens due to frictional forces

• Sometimes sytems are deliberately damped to stop them oscillating/minimise the effects of resonance

Amounts of Damping

• Light damping - decreases the amplitude slowly over time
• Critical damping - reduces the amplitude to zero in the shortest time possible
• Overdamping - Take much longer to reach equilibrium than critically damped systems

Damping also affects resonance

• Lightly damped systems have a sharp resonance peak
• Heavily damped systems have a flatter response, the resonance peak is not as sharp