Periodic Motion Periodic motion is motion of an object that Week 14: Chapter 15 regularly returns to a given position after a fixed time interval A special kind of periodic motion occurs in Oscillatory Motion 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 motion Motion of a Spring-Mass System Hooke’s Law A block of mass m is Hooke’s Law states F = - kx attached to a spring, the s block is free to move on a Fs is the restoring force frictionless horizontal It is always directed toward the equilibrium position surface Therefore, it is always opposite the displacement from Use the active figure to vary equilibrium the initial conditions and observe the resultant k is the force (spring) constant motion x is the displacement When the spring is neither stretched nor compressed, the block is at the equilibrium position x = 0 More About Restoring Force More About Restoring Force, 2 The block is displaced The block is at the to the right of x = 0 equilibrium position The position is positive x = 0 The restoring force is The spring is neither directed to the left stretched nor compressed The force is 0 1 More About Restoring Force, 3 Acceleration The block is displaced The force described by Hooke’s Law is the to the left of x = 0 net force in Newton’s Second Law The position is negative The restoring force is FFHooke Newton directed to the right kx max k ax x m Simple Harmonic Motion – Motion of the Block Mathematical Representation The block continues to oscillate between –A Model the block as a particle and +A The representation will be particle in simple harmonic motion model These are turning points of the motion Choose x as the axis along which the oscillation The force is conservative occurs dx2 k Acceleration In the absence of friction, the motion will ax2 continue forever dt m 2 k Real systems are generally subject to friction, so We let they do not actually oscillate forever m Then a = -2x Simple Harmonic Motion – Simple Harmonic Motion – Mathematical Representation, 2 Graphical Representation A function that satisfies the equation is A solution is x(t) = A needed cos (t + Need a function x(t) whose second derivative is A, are all the same as the original function with a negative constants sign and multiplied by 2 A cosine curve can be The sine and cosine functions meet these used to give physical requirements significance to these constants 2 Simple Harmonic Motion – Definitions Simple Harmonic Motion, cont A is the amplitude of the motion A and are determined uniquely by the This is the maximum position of the particle in position and velocity of the particle at t = 0 either the positive or negative direction If the particle is at x = A at t = 0, then = 0 is called the angular frequency The phase of the motion is the quantity (t + Units are rad/s ) is the phase constant or the initial phase x (t) is periodic and its value is the same each angle time t increases by 2 radians Period Frequency The period, T, is the time interval required The inverse of the period is called the for the particle to go through one full cycle of frequency its motion The frequency represents the number of The values of x and v for the particle at time t oscillations that the particle undergoes per equal the values of x and v at t + T unit time interval 1 2 ƒ T T 2 Units are cycles per second = hertz (Hz) Summary Equations – Period An object of mass m is hung from a spring and set into oscillation. The period of the oscillation is measured and and Frequency 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? The frequency and period equations can be rewritten to solve for 2 A. 2T 2ƒ T B. 1.4T2T The period and frequency can also be C. T expressed as: D. 0.7TT / 2 mk1 T 2ƒ E. T/2 km2 3 Motion Equations for Simple Period and Frequency, cont Harmonic Motion xt() A cos ( t ) The frequency and the period depend only on dx the mass of the particle and the force vAsin( t ) constant of the spring dt 2 They do not depend on the parameters of dx 2 aA cos( t ) motion dt 2 The frequency is larger for a stiffer spring Simple harmonic motion is one-dimensional and so (large values of k) and decreases with directions can be denoted by + or - sign increasing mass of the particle Remember, simple harmonic motion is not uniformly accelerated motion Maximum Values of v and a Graphs The graphs show: Because the sine and cosine functions (a) displacement as a oscillate between ±1, we can easily find the function of time maximum values of velocity and acceleration (b) velocity as a function of time for an object in SHM (c ) acceleration as a function of time k The velocity is 90o out of phase with the vAAmax m displacement and the acceleration is 180o out k aAA 2 of phase with the max m displacement SHM Example 1 SHM Example 2 Initial conditions at t = 0 Initial conditions at are t = 0 are x (0)= A x (0)=0 v (0) = 0 v (0) = vi This means = 0 This means = /2 The acceleration The graph is shifted reaches extremes of ± one-quarter cycle to the 2A at A right compared to the The velocity reaches graph of x (0) = A extremes of ± A at x = 0 4 Energy of the SHM Oscillator, Energy of the SHM Oscillator cont Assume a spring-mass system is moving on a The total mechanical frictionless surface energy is constant The total mechanical This tells us the total energy is constant energy is proportional to the square of the amplitude The kinetic energy can be found by 2 2 2 2 Energy is continuously K = ½ mv = ½ m A sin (t + ) being transferred between The elastic potential energy can be found by potential energy stored in 2 2 2 the spring and the kinetic U = ½ kx = ½ kA cos (t + ) energy of the block 2 The total energy is E = K + U = ½ kA Use the active figure to investigate the relationship between the motion and the energy Energy of the SHM Oscillator, cont Energy in SHM, summary As the motion continues, the exchange of energy also continues Energy can be used to find the velocity k vAx22 m 22A x 2 Importance of Simple Harmonic Oscillators SHM and Circular Motion Simple harmonic oscillators This is an overhead view of are good models of a wide a device that shows the variety of physical relationship between SHM phenomena and circular motion Molecular example As the ball rotates with If the atoms in the molecule constant angular speed, its do not move too far, the shadow moves back and forces between them can forth in simple harmonic be modeled as if there were motion springs between the atoms The potential energy acts similar to that of the SHM oscillator 5 SHM and Circular Motion, 2 SHM and Circular Motion, 3 The circle is called a The particle moves reference circle along the circle with constant angular Line OP makes an velocity angle with the x axis OP makes an angle at t = 0 with the x axis Take P at t = 0 as the At some time, the angle reference position between OP and the x axis will be t + SHM and Circular Motion, 4 SHM and Circular Motion, 5 The points P and Q always have the same x The x component of the coordinate velocity of P equals the velocity of Q x (t) = A cos (t + ) These velocities are This shows that point Q moves with simple v = -A sin (t + ) harmonic motion along the x axis Point Q moves between the limits ±A SHM and Circular Motion, 6 Simple Pendulum The acceleration of point P A simple pendulum also exhibits periodic motion on the reference circle is directed radially inward The motion occurs in the vertical plane and is P ’s acceleration is a = 2A driven by gravitational force The x component is –2 A cos (t + ) The motion is very close to that of the SHM This is also the acceleration oscillator of point Q along the x axis If the angle is <10o 6 Simple Pendulum, 2 Simple Pendulum, 3 The forces acting on the In the tangential direction, bob are the tension and ds2 the weight Fmgmsin t 2 T is the force exerted on dt the bob by the string The length, L, of the pendulum is constant, and for mg is the gravitational small values of force dg2 g The tangential component 2 sin of the gravitational force is dt L L a restoring force This confirms the form of the motion is SHM Simple Pendulum, 4 Simple Pendulum, Summary The function can be written as The period and frequency of a simple = cos (t + ) pendulum depend only on the length of the max string and the acceleration due to gravity The angular frequency is The period is independent of the mass g All simple pendula that are of equal length L and are at the same location oscillate with the The period is same period 2 L T 2 g Clicker Question Physical Pendulum A grandfather clock depends on the period of a pendulum to keep correct time.