Oscillations Adapted by Rob Dickens from a presentation by John Spivey To help with learning and revision of the ‘Waves and Our Universe’ section of the AS/A level physics course.

Oscillations 1.Going round in circlesGoing round in circles 2.Circular Motion CalculationsCircular Motion Calculations 3.Circular Motion under gravityCircular Motion under gravity 4.Periodic MotionPeriodic Motion 5.SHMSHM 6.Oscillations and Circular MotionOscillations and Circular Motion

Going round in circles Speed may be constant But direction is continually changing Therefore velocity is continually changing Hence acceleration takes place

Centripetal Acceleration Change in velocity is towards the centre Therefore the acceleration is towards the centre This is called centripetal acceleration

Centripetal Force Acceleration is caused by Force (F=ma) Force must be in the same direction as acceleration Centripetal Force acts towards the centre of the circle CPforce is provided by some external force – eg friction

Examples of Centripetal Force Friction Tension in string Gravitational pull

Centripetal Force 2 What provides the cpforce in each case ?

Centripetal force 3

Circular Motion Calculations Centripetal acceleration Centripetal force

Period and Frequency The Period (T) of a body travelling in a circle at constant speed is time taken to complete one revolution - measured in seconds Frequency (f) is the number of revolutions per second – measured in Hz T = 1 / f f = 1 / T

Angles in circular motion Radians are units of angle An angle in radians = arc length / radius 1 radian is just over 57º There are 2π = 6.28 radians in a whole circle

Angular speed Angular speed ω is the angle turned through per second ω = θ/t = 2π / T 2π = whole circle angle T = time to complete one revolution T = 2π/ω = 1/f f = ω/2π

Force and Acceleration v = 2π r / T and T = 2π / ω v = r ω a = v² / r = centripetal acceleration a = (r ω)² / r = r ω² is the alternative equation for centripetal acceleration F = m r ω² is centripetal force

Circular Motion under gravity Loop the loop is possible if the track provides part of the cpforce at the top of the loop ( S T ) The rest of the cpforce is provided by the weight of the rider

Weightlessness True lack of weight can only occur at huge distances from any other mass Apparent weightlessness occurs during freefall where all parts of you body are accelerating at the same rate

Weightlessness This rollercoaster produces accelerations up to 4g (40m/s²) These astronauts are in freefall Red Arrows pilots experience up to 9g (90m/s²)

The conical pendulum The vertical component of the tension (Tcosθ) supports the weight (mg) The horizontal component of tension (Tsinθ) provides the centripetal force

Periodic Motion Regular vibrations or oscillations repeat the same movement on either side of the equilibrium position f times per second (f is the frequency) Displacement is the distance from the equilibrium position Amplitude is the maximum displacement Period (T) is the time for one cycle or or 1 complete oscillation

Producing time traces 2 ways of producing a voltage analogue of the motion of an oscillating system

Time traces

Simple Harmonic Motion1 Period is independent of amplitude Same time for a large swing and a small swing For a pendulum this only works for angles of deflection up to about 20º

SHM2 Gradient of displacement v. time graph gives a velocity v. time graph Max veloc at x = 0 Zero veloc at x = max

SHM3 Acceleration v. time graph is produced from the gradient of a velocity v. time graph Max a at V = zero Zero a at v = max

SHM4 Displacement and acceleration are out of phase a is proportional to - x Hence the minus

SHM5 a = -ω²x equation defines SHM T = 2π/ω F = -kx eg a trolley tethered between two springs

Circular Motion and SHM The peg following a circular path casts a shadow which follows SHM This gives a mathematical connection between the period T and the angular velocity of the rotating peg T = 2π/ω