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4 ALL INVOLVE SIMPLE HARMONIC MOTION

©JParkinson 5 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 Time t + A - A

©JParkinson 6 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 x A A x or

©JParkinson 7 Displacement x time Amplitude ( A ): The maximum distance that an object moves from its rest position. x = A and x = - A. + A - A Period ( T ): The time that it takes to execute one complete cycle of its motion. Units seconds, T 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, 2 π per cycle.

©JParkinson 8 θ r Arc length s IN RADIANS FOR A FULL CIRCLE RADIANS

©JParkinson 9 EQUATION OF SHM x a Acceleration – Displacement graph Gradient = - ω 2 + A - A MAXIMUM ACCELERATION = ± ω 2 A = ( 2 πf ) 2 A

©JParkinson 10 EQUATION FOR VARIATION OF VELOCITY WITH DISPLACEMENT +x -x x v Maximum velocity, v = ± 2 π f A Maximum Kinetic Energy, E K = ½ mv 2 = ½ m ( 2 π f A ) 2

©JParkinson 11 Displacement x Velocity v Acceleration a t t t Velocity = gradient of displacement- time graph Maximum velocity in the middle of the motion ZERO velocity at the end of the motion Acceleration = gradient 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!

©JParkinson 12 THE PENDULUM The period, T, is the time for one complete cycle. l

©JParkinson 13 MASS ON A SPRING M F = Mg = ke e Stretch & Release A k = the spring constant in N m - 1

©JParkinson 14 =cResource.dspView&ResourceID=44 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

©JParkinson 15 ENERGY IN SHM potential E P Kinetic E K Potential E P PENDULUMSPRING M M M potential kinetic potential If damping is negligible, the total energy will be constant E TOTAL = E p + E K

©JParkinson 16 Maximum velocity, v = ± 2 π f A Energy in SHM Maximum Kinetic Energy, E K = ½ m ( 2 π f A ) 2 = 2π 2 m f 2 A 2 Hence TOTAL ENERGY = 2π 2 m f 2 A 2 m x = 0 F For a spring, m x = A energy stored = ½ Fx = ½ kx 2,[as F=kx] = MAXIMUM POTENTIAL ENERGY! MAXIMUM POTENTIAL ENERGY = TOTAL ENERGY = ½ kA 2

©JParkinson 17 Energy in SHM Energy Change with POSITION = kinetic = potential= TOTAL ENERGY, E Energy Change with TIME x -A+A 0 energy E time TT/2 N.B. Both the kinetic and the potential energies reach a maximum TWICE in on cycle. E

©JParkinson 18 time DAMPING DISPLACEMENT INITIAL AMPLITUDE THE AMPLITUDE DECAYS EXPONENTIALLY WITH TIME