1Week 14: Chapter 15Oscillatory MotionPeriodic Motion Periodic motion is motion of an object that regularly returns to a given position after a fixed time interval A special kind of periodic motion occurs in mechanical systems when the force acting on the object is proportional to the position of the object relative to some equilibrium position If the force is always directed toward the equilibrium position, the motion is called simple harmonic motionMotion of a Spring-Mass System A block of mass m is attached to a spring, the block is free to move on a frictionless horizontal surface Use the active figure to vary the initial conditions and observe the resultant motion When the spring is neither stretched nor compressed, the block is at the equilibrium position x = 0Hooke’s Law Hooke’s Law states Fs= - kx Fsis the restoring force It is always directed toward the equilibrium position Therefore, it is always opposite the displacement from equilibrium k is the force (spring) constant x is the displacementMore About Restoring Force The block is displaced to the right of x = 0 The position is positive The restoring force is directed to the leftMore About Restoring Force, 2 The block is at the equilibrium position x = 0 The spring is neither stretched nor compressed The force is 02More About Restoring Force, 3 The block is displaced to the left of x = 0 The position is negative The restoring force is directed to the rightAcceleration The force described by Hooke’s Law is the net force in Newton’s Second LawHooke NewtonxxFFkx makaxmMotion of the Block The block continues to oscillate between –A and +A These are turning points of the motion The force is conservative In the absence of friction, the motion will continue forever Real systems are generally subject to friction, so they do not actually oscillate foreverSimple Harmonic Motion –Mathematical Representation Model the block as a particle The representation will be particle in simple harmonic motion model Choose x as the axis along which the oscillation occurs Acceleration We let  Then a = -2x22dx kaxdt m2kmSimple Harmonic Motion –Mathematical Representation, 2 A function that satisfies the equation is needed Need a function x(t) whose second derivative is the same as the original function with a negative sign and multiplied by 2 The sine and cosine functions meet these requirements Simple Harmonic Motion –Graphical Representation A solution is x(t) = Acos (t +  A, are all constants A cosine curve can be used to give physical significance to these constants3Simple Harmonic Motion –Definitions A is the amplitude of the motion This is the maximum position of the particle in either the positive or negative directionis called the angular frequency Units are rad/sis the phase constant or the initial phase angleSimple Harmonic Motion, cont A and  are determined uniquely by the position and velocity of the particle at t = 0 If the particle is at x = A at t = 0, then  = 0 The phase of the motion is the quantity (t + ) x (t) is periodic and its value is the same each time t increases by 2 radiansPeriod The period, T, is the time interval required for the particle to go through one full cycle of its motion The values of x and v for the particle at time tequal the values of x and v at t + T2TFrequency The inverse of the period is called the frequency The frequency represents the number of oscillations that the particle undergoes per unit time interval Units are cycles per second = hertz (Hz)1ƒ2TSummary Equations – Period and Frequency The frequency and period equations can be rewritten to solve for  The period and frequency can also be expressed as:22ƒT12ƒ2mkTkmAn object of mass m is hung from a spring and set into oscillation. The period of the oscillation is measured and recorded as T. The object of mass m is removed and replaced with an object of mass 2m. When this object is set into oscillation, what is the period of the motion?A. 2TB. 1.4T C. TD. 0.7T E. T/2T22/T4Period and Frequency, cont The frequency and the period depend only on the mass of the particle and the force constant of the spring  They do not depend on the parameters of motion The frequency is larger for a stiffer spring (large values of k) and decreases with increasing mass of the particleMotion Equations for Simple Harmonic Motion Simple harmonic motion is one-dimensional and so directions can be denoted by + or - sign Remember, simple harmonic motion is not uniformly accelerated motion222() cos ( )sin( t )cos( t )xt A tdxvAdtdxaAdt  Maximum Values of v and a Because the sine and cosine functions oscillate between ±1, we can easily find the maximum values of velocity and acceleration for an object in SHMmax2maxkvAAmkaAAmGraphs The graphs show: (a) displacement as a function of time (b) velocity as a function of time (c ) acceleration as a function of time The velocity is 90oout of phase with the displacement and the acceleration is 180oout of phase with the displacement SHM Example 1 Initial conditions at t = 0 are x (0)= A v (0) = 0 This means = 0 The acceleration reaches extremes of ±2A at A The velocity reaches extremes of ±A at x = 0SHM Example 2 Initial conditions att = 0 are x (0)=0 v (0) = vi This means = /2 The graph is shifted one-quarter cycle to the right compared to the graph of x (0) = A5Energy of the SHM Oscillator Assume a spring-mass system is moving on a frictionless surface This tells us the total energy is constant The kinetic energy can be found by K = ½ mv2= ½ m2A2sin2(t + ) The elastic potential energy can be found by U = ½ kx2= ½ kA2cos2(t + ) The total energy is E = K + U = ½ kA 2Energy of the SHM Oscillator, cont The total mechanical energy is constant  The total mechanical energy is proportional to the square of the amplitude Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the block Use the active figure to investigate the relationship between