G484- Circular Motion and Oscillations

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Radians to Degrees

x   

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Degrees to Radians

x   

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1 Radian

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

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angular displacement equation (circles)

(in radians)

                s

r

Ø    

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Conical pendulum mg=... (Ø at top)

mg=TcosØ    

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conical pendulum Fc=... (Ø at top)

Fc=TsinØ    

(also Fc=)

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Centripetal Force definition

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

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How does an object move in a circle?

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

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Centripetal force equation

Fc=

(also Fc=mac)

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Centripetal acceleration definition

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

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Centripetal acceleration equation

 

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Period definition

time taken for one complete cycle

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Period equation

(f= cyclic frequency)

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Cyclic frequency definition

number of cycles per unit time

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Cyclic frequency equation

also

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Linear (immediate) velocity equation

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Angular frequency definition

change in angle per unit time

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angular frequency equation

 =2f

also

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Circular Motion in vertical plane: Fc at top

Fc=T+mg

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Circular Motion in vertical plane: Fc at bottom

Fc=T-mg

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Newton's Law of Gravitation

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

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Gravitational force of attraction equation

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Gravitational field strength definition

the force per unit mass (like acceleration)

(on Earth's surface g=9.81)

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Gravitational field strength equation

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Geostationary orbit definition

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

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Geostationary satellites must...

  • 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
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Kepler's 3rd Law

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

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free oscillation definition

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

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displacement definition (SHM)

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

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amplitude definition (SHM)

( , A)

The maximum displacement from the equilibrium position

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Period definition (SHM)

(T,  )

Time taken for one complete oscillation
and
is independent from amplitude

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Frequency definition (SHM)

(f,  )

number of oscillations per unit time

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angular frequency (SHM)

(  , 2f, )


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

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Phase difference definition (SHM)

( Ø )

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

(in radians)

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Simple Harmonic Motion definition

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

(  )

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eqn. for displacement starting from max displaceme

x=A cos(2ft)

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eqn. for displacement starting from equilibrium

x=A sin(2ft)

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Kinetic + Potential energy at equilibrium

maximum Kinetic energy

0 potential energy

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Kinetic + Potential energy at max. displacement

0 Kinetic energy

maximum Potential energy

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maximum velocity equation

 

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Forced oscillations

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

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Damping

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

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Critical damping

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

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Resonance

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

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Resonance (useful and unuseful examples)

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

(  )

The frequency of the free oscillations of a system

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At resonance...

  • 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
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