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Chapter 11 Vibrations and Waves Ms. Hanan 11-1 Simple Harmonic Motion Objectives • Identify the conditions of simple harmonic motion. • Explain how force, velocity, and acceleration change as an object vibrates with simple harmonic motion. • Calculate the spring force using Hooke’s law. Vocabulary • • • • • • • • • • Periodic Motion Simple Harmonic Motion Period Amplitude Hooke’s Law Pendulum Oscillation Vibration Spring Constant Displacement Periodic Motion • Motion where a body travels along the same path in a repeated, back and forth manner • Also called oscillation and vibration • We are surrounded by oscillations – motions that repeat themselves (periodic motion) • Grandfather clock pendulum, boats bobbing at anchor, oscillating guitar strings, pistons in car engines • Understanding periodic motion is essential for the study of waves, sound, alternating electric currents, light, etc. • An object in periodic motion experiences restoring forces or torques that bring it back toward an equilibrium position Periodic Motion • Those same forces cause the object to “overshoot” the equilibrium position • Think of a block oscillating on a spring or a pendulum swinging back and forth past its equilibrium position • Examples of periodic motion: R L /5 m r L /2 k m Example 1 Mass-Spring System a a a Equil. position a Example 2 Simple Pendulum a a Equil. position a a Example 3 Floating Cylinder Equil. position a a a a Hooke’s Law Force • LEQ k • x m The force always acts toward the equilibrium position The direction of the restoring force is such that the object Fs=kx is being either pushed or pulled toward the equilibrium position Hooke’s Law Reviewed F kx • When x is positive F is negative ; • When at equilibrium (x=0), F = 0 ; • When x is negative F is positive ; , , Stretched and Equilibrium 11 Equilibrium and Compressed 12 Motion of the Spring-Mass System • Assume the object is initially pulled to a distance A and released from rest • As the object moves toward the equilibrium position, F and a decrease, but v increases • At x = 0, F and a are zero, but v is a maximum • The object’s momentum causes it to overshoot the equilibrium position Graphing x vs. t A T A : amplitude (length, m) T : period (time, s) Sample Problem A – P. 370 If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its equilibrium position, what is the spring constant? Givens: m = 0.55 kg x = -2.0 cm x = -0.02 m g = 9.81 m/s2 Unknowns: k = ? X = -2.0 cm Step 1: Choose the equation/situation: • When the mass is attached to the spring, the equilibrium position changes. • At the new equilibrium point, the net force acting on the mass is zero. • By Hooke’s Law, the Spring Force must be equal and opposite of the weight of the mass FSpring kx Fg mg Fnet FSpring Fg 0 (kx) (mg ) 0 kx mg mg k x mg k x Step 2: Substitute the known values into this equation. m = 0.55 kg x = -0.02 m g = 9.81 m/s2 (0.55kg)(9.81m / s ) k 0.02m 2 k 270 N / m Assignments • Class-work: Practice A , page 371, questions 1, 2, and 3. • Homework: Review and Assess; Page 396: # 8 and 9 Due next class Elastic Potential Energy • The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy PEs = ½kx2 • The energy is stored only when the spring is stretched or compressed • Elastic potential energy can be added to the statements of Conservation of Energy and Work-Energy Energy Transformations • • The block is moving on a frictionless surface The total mechanical energy of the system is the kinetic energy of the block Energy Transformations, 2 • • • The spring is partially compressed The energy is shared between kinetic energy and elastic potential energy The total mechanical energy is the sum of the kinetic energy and the elastic potential energy Energy Transformations, 3 • The spring is now fully compressed • The block momentarily stops • The total mechanical energy is stored as elastic potential energy of the spring Energy Transformations, 4 • • When the block leaves the spring, the total mechanical energy is in the kinetic energy of the block The total energy of the system remains constant Simple Pendulum Restoring force of a pendulum is a Component of the bob’s weight x 2 L2 x F mgsin x x sin x 2 L2 L mg F x L Looks like Hooke’s law (k mg/L) • When oscillations are small, the motion is called simple harmonic motion (shm) and can be described by a simple sine curve. • The pendulum’s potential energy is gravitational, and increases as the pendulum’s displacement increases. • Gravitational potential energy is equal to zero at the pendulum’s equilibrium position. PEg = mgh Assignments • Class-work: Practice section review page 375, questions 1, 2, 3, and 4. • Homework: Vibrations and Waves Problem A, Hooke’s Law Additional Practice Sheet, even questions. Due Sunday 20/2/11