G484- Circular Motion and Oscillations

x   

x   

the angle subtended at the centre of a circle by an arc of length equal to the circle's radius

(in radians)

                s

r

Ø    

mg=TcosØ    

Fc=TsinØ    

(also Fc=)

Resultant force, acting towards the centre of a circle, causing an object to move in a circular path

If there is a force acting perpendicular to the velocity, the object moves in a circle

Fc=

(also Fc=mac)

the acceleration of a body moving in a circle with constant velocity, acting towards the circle's centre

time taken for one complete cycle

(f= cyclic frequency)

number of cycles per unit time

also

change in angle per unit time

 =2f

also

Fc=T+mg

Fc=T-mg

The gravitational force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres

the force per unit mass (like acceleration)

(on Earth's surface g=9.81)

an orbit of the earth by a satellite that has the same time period as the rotation of the earth and is in the equatorial plane

• have their orbit centred on the Earth's centre
• be over the equator
• travel from west to east
• have a period of 24hrs equal to the Earth's period of rotation

The square of the period (T) of a planet is directly proportional to the cube of its separation (r) from the sun

an oscillation with no driving mechanism, no variation in amplitude and no friction (e.g. a swinging clock pendulum)

(x) the distance an object has moved from its equilibrium position

( , A)

The maximum displacement from the equilibrium position

(T,  )

Time taken for one complete oscillation
and
is independent from amplitude

(f,  )

number of oscillations per unit time

(  , 2f, )

related to the frequency of the motion and inversely related to the period  

( Ø )

The fraction of an oscillation that one wave leads/lags behind another

(in radians)

when the acceleration (a) of an object is proportional and in the opposite direction to the displacement (x) from the equilibrium position

(  )

x=A cos(2ft)

x=A sin(2ft)

maximum Kinetic energy

0 potential energy

0 Kinetic energy

maximum Potential energy

occur when an object is forced to vibrate at the frequency of an external source

converts kinetic energy into other forms, so reduces the amplitude, and is the result of friction

the minimum amount of damping needed to cause a system to return to equilibrium without oscillating

the dramatic build up of a large amplitude oscillation when the frequencies of the vibrating objects match

• uses : (microwave cooking, a person pushing a child on a swing)
• nuisance : (unnecessary vibration of car parts, swinging of the Milennium bridge)

(  )

The frequency of the free oscillations of a system

• the natural frequency of the forced oscillator is equal to the driving frequency
• the forced oscillator has maximum amplitude
• the forced oscillator absorbs maximum energy from an external source
• the degree of damping affects both the resonant frequency and the amplitude of the forced oscillator