PENDULUM ©JParkinson

SPRING ©JParkinson

SIMPLE HARMONIC MOTION SIMPLE HARMONIC MOTION ©JParkinson

SIMPLE HARMONIC MOTION ALL INVOLVE SIMPLE HARMONIC MOTION ©JParkinson

A body will undergo SIMPLE HARMONIC MOTION when the force that tries to restore the object to its REST POSITION is PROPORTIONAL TO the DISPLACEMENT of the object. A pendulum and a mass on a spring both undergo this type of motion which can be described by a SINE WAVE or a COSINE WAVE depending upon the start position. Displacement x + A - A Time t ©JParkinson

DEFINITION SHM is a particle motion with an acceleration (a) that is directly proportional to the particle’s displacement (x) from a fixed point (rest point), and this acceleration always points towards the fixed point. Rest point a a x x or ©JParkinson

MORE DEFINITIONS T + A - A Displacement x time T + A - A Amplitude ( A ): The maximum distance that an object moves from its rest position. x = A and x = - A . Period ( T ): The time that it takes to execute one complete cycle of its motion. Units seconds, Frequency ( f ): The number or oscillations the object completes per unit time. Units Hz = s-1 . Angular Frequency ( ω ): The frequency in radians per second, 2π per cycle. ©JParkinson

RADIANS IN RADIANS FOR A FULL CIRCLE RADIANS θ r Arc length s ©JParkinson

EQUATION OF SHM Acceleration – Displacement graph Gradient = - ω2 x a Gradient = - ω2 - A + A MAXIMUM ACCELERATION = ± ω2 A = ( 2πf )2 A ©JParkinson

EQUATION FOR VARIATION OF VELOCITY WITH DISPLACEMENT +x -x x v Maximum velocity, v = ± 2 π f A Maximum Kinetic Energy, EK = ½ mv2 = ½ m ( 2 π f A )2 ©JParkinson

Velocity = gradient of displacement- time graph Displacement x t Velocity v Velocity = gradient of displacement- time graph Maximum velocity in the middle of the motion t ZERO velocity at the end of the motion Acceleration = gradient of velocity - time graph Acceleration a Maximum acceleration at the end of the motion – where the restoring force is greatest! t ZERO acceleration in the middle of the motion! ©JParkinson

l THE PENDULUM The period, T, is the time for one complete cycle. ©JParkinson

MASS ON A SPRING k = the spring constant in N m-1 M F = Mg = ke e A Stretch & Release A k = the spring constant in N m-1 ©JParkinson

The link below enables you to look ( for 5mins only) at the factors that influence the period of a pendulum and the period of a mass on a spring http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=44 ©JParkinson

If damping is negligible, the total energy will be constant ENERGY IN SHM PENDULUM SPRING M potential kinetic potential EP Potential EP Kinetic EK potential If damping is negligible, the total energy will be constant ETOTAL = Ep + EK ©JParkinson

Energy in SHM Maximum velocity, v = ± 2 π f A Maximum Kinetic Energy, EK = ½ m ( 2 π f A )2 = 2π2 m f2 A2 Hence TOTAL ENERGY = 2π2 m f2 A2 = MAXIMUM POTENTIAL ENERGY! For a spring, energy stored = ½ Fx = ½ kx2, [as F=kx] m x = A m x = 0 F MAXIMUM POTENTIAL ENERGY = TOTAL ENERGY = ½ kA2 ©JParkinson

Energy Change with POSITION Energy in SHM = kinetic = potential = TOTAL ENERGY, E Energy Change with POSITION x -A +A energy E Energy Change with TIME energy time E N.B. Both the kinetic and the potential energies reach a maximum TWICE in on cycle. T/2 T ©JParkinson

DAMPING DISPLACEMENT INITIAL AMPLITUDE time THE AMPLITUDE DECAYS EXPONENTIALLY WITH TIME ©JParkinson