Simple Harmonic Motion 3

Simple Harmonic Motion 2 Learning Objectives Book Reference : Pages 36-37 To arrive at the relationship between displacement, velocity and acceleration for a system in SHM To be able calculate the magnitude & direction of the acceleration at any time for an object in SHM

Circular Motion & SHM y Consider an object in circular motion... At any given time the x and y coordinates are given by: x = r cos  y = r sin  r cos  r sin  r  x Graphically the x coordinate changes as shown x +r  / rad -r

Displacement 1 We have seen that the displacement of an object in SHM is described by a sinusoidal function. However, there are differences between where we start the motion at time t=0 x Maximum displacement A at time t=0 (cosine) +A -A x Zero displacement at time t=0 (sin) +A -A

Displacement 2 Providing that we operate in radians, the displacement x of an object in SHM is given by: x = A cos 2ft (for displacement A at time t=0) x = A sin 2ft (for zero displacement at time t=0) Where A is the maximum displacement Calculator must be in radians for this to work

Velocity The velocity of an object in SHM is given by: v =  2f (A2 – x2) Where v is the velocity, f is the frequency, A is the maximum displacement and x is the displacement The velocity is considered positive if moving away from the equilibrium point and negative if moving towards it Note this will be a maximum when x=0 vmax = 2fA

Maximum Acceleration In a similar way the maximum value for acceleration given by the acceleration equation yesterday will be.... Acceleration = - (2f)2 x displacement Accelerationmax = - (2f)2 A Where f is the frequency, and A is the maximum displacement .

Problems 1 A spring oscillates in SHM with a period of 3s and an amplitude of 58mm. Calculate: The frequency The maximum acceleration The displacement of an object oscillating in SHM changes with time and is described by X (mm) = 12 cos 10t where t is the time in seconds after the object’s displacement was at its maximum value. Determine: The amplitude The time period The displacement at t = 0.1s

Problems 2 An object on a spring oscillating in SHM has a time period of 0.48s and a maximum acceleration of 9.8m/s2. Calculate: Its frequency Its Amplitude An object oscillates in SHM with an amplitude of 12mm and a period of 0.27s. Calculate The frequency Its displacement and direction of motion 0.1s, and 0.2s after the displacement was +12mm