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©JParkinson 1 ©JParkinson 2 ©JParkinson 3 ALL INVOLVE SIMPLE HARMONIC MOTION ©JParkinson 4 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 Time t -A x A cos 2ft ©JParkinson 5 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 F kx is Hooke' s Law and as ©JParkinson F ma then a x 6 Displacement x T +A time -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. Unit: seconds (s) Frequency ( f ): The number of oscillations the object 1 completes per unit time. f -1 ©JParkinson Units:Hertz (Hz) = s . T 7 Displacement x x A cos 2ft t Velocity v Velocity = slope of displacementtime graph Maximum velocity vo in the middle t of the motion ZERO velocity at the end of the motion Acceleration a t ©JParkinson Acceleration = slope of velocity time graph Maximum acceleration at the end of the motion – where the restoring force is greatest! ZERO acceleration in the middle of the motion! 8 THE PENDULUM The period, T, is the time for one complete cycle. l T 2 ©JParkinson l g 9 MASS ON A SPRING x M A F = Mg = kx Stretch & Release k = the spring constant in N m-1 ©JParkinson m T 2 k T 2 x g 10 The link below enables you to look 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 11 DAMPING DISPLACEMENT INITIAL AMPLITUDE time THE AMPLITUDE DECAYS EXPONENTIALLY WITH TIME If damping is negligible, the total energy will be constant ©JParkinson 12 ENERGY IN SHM SPRING PENDULUM potential M M potential Eg Kinetic EK Potential Eg M kinetic potential ETOTAL = EP + EK ©JParkinson 13 Energy in SHM = kinetic = potential = TOTAL ENERGY, E energy Energy Change with POSITION -A Energy Change with TIME E +A 0 energy x E N.B. Both the kinetic and the potential energies reach a maximum TWICE in one cycle. time T/2 ©JParkinson T 14 Energy in SHM Maximum Kinetic Energy, EK (max) = ½ m (vo)2 where vo is the velocity at the rest position (origin) = max velocity. TOTAL ENERGY = Eel + Ek For a spring, energy stored = ½ Fx = ½ kx2, [as F=kx] m F m x=A x=0 MAXIMUM POTENTIAL ENERGY = TOTAL ENERGY = ½ kA2 ©JParkinson 15 Formula Summary and Consolidation: ET = (½)mv2 + (½)kx2 ET=Eel+Ek ET=(½)mvo2 ET=(½)kA2 16