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Periodic Motion Chapter 13 Periodic Motion PowerPoint ® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman Lectures by James Pazun Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Goals for Chapter 13 • To outline periodic motion • To quantify simple harmonic motion • To explore the energy in simple harmonic motion • To consider angular simple harmonic motion • To study the simple pendulum • To examine the physical pendulum • To explore damped oscillations • To consider driven oscillations and resonance Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Homework Read 476 to 486 3, 5, 9, 15 Read 486 to 494 25, 27 Read 495 to 505 (Skip Molecular Vibration) 31, 35, 45, 49, 51, 61, 69 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Introduction • If you look to the right, you’ll see a time-lapse photograph of a simple pendulum. It’s far from simple, but it is a great example of the regular oscillatory motion we’re about to study. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Describing oscillations Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Describing oscillations Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Describing oscillations Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley In Oscillations . Displacement : the distance the body is from its equilibrium position. Restoring Force : the force that tends to restore the body to its equilibrium position. Amplitude : the maximum displacement from equilibrium Period : (T) the time for one cycle. Frequency : f( ) the number of cycles per unit of time. Angular Frequency : ( ω) 2 π times the frequency. = 1 = 1 (13.1) f T T f ω = π = 2π 2 f T (13.2) Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 13.1 Summary PERIODIC MOTION: PERIODIC MOTION IS MOTION THAT REPEATS ITSELF IN A DEFINITE CYCLE. IT OCCURS WHENEVER A BODY HAS A STABLE EQUILIBRIUM POSITION AND A RESTORING FORCE THAT ACTS WHEN IT IS DISPLACED FROM EQUILIBRIUM. PERIOD T IS THE TIME FOR ONE CYCLE. FREQUENCY IS THE NUMBER OF CYCLES PER UNIT TIME. ANGULAR FREQUENCY IS TIMES THE FREQUENCY. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Period, Frequency, and Angular Frequency An ultrasonic transducer (loudspeaker) used for medical diagnosis oscillates at a frequency of 6.7 MHz = 6.7 x 10 6 Hz. How much time does each oscillation take, and what is the angular frequency? Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Simple harmonic motion • An ideal spring responds to stretch and compression linearly, obeying Hooke’s Law. • For a real spring, Hookes’ Law is a good approximation. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Definition of Simple Harmonic Motion The easiest way to view SHM is with a mass on a spring. It obeys Hooke’s Law: Fx = -kx (13.3) When the restoring force is directly proportional to the displacement from equilibrium, as given by Eq. (13.3), the oscillation is called simple harmonic motion abbreviated SHM. 2 2 The acceleration ax = d x/dt = Fx/m of a body in SHM is given by: d 2 x k a = = − x (13.4) x dt 2 m Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Simple harmonic motion viewed as a projection If you illuminate uniform circular motion (say by shining a flashlight on a ball placed on a rotating lazy-Susan spice rack), the shadow projection that will be cast will be undergoing simple harmonic motion, like in figures 13.4 below. The SHM and circular motion analogy should hold true if the amplitude, A, equals the disk radius and the angular frequency equals the disk’s angular velocity. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Comparing acceleration of shadow at P to SHM The circle the ball moves in so that its shadow matches SHM is called the circle of reference as we call Q the reference point. At a certain time, t, the vector OQ makes an angle,θ, with the horizontal. As the point Q moves around the reference circle with a constant angular speed ω, the vector rotates with the same speed, called a phasor . The x-component of the phasor at time t is just the x-coordinate of the point Q: x = A cos θ This position also refers to the position of the shadow at P, which is the projection of Q onto the x-axis. The acceleration of P is just the x-component of the acceleration of Q. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Comparing acceleration of shadow at P to SHM ω2 From centripetal acceleration, the acceleration of Q is a Q = A, directed toward O, as shown below. θ The x-component is just a x = -aQ cos ω2 θ Combining these two equations gives a x = - A cos θ ω2 From before x = A cos , so a x = - x This gives exactly what is to be expected from equation 13.4, if the angular speed of the reference point Q is related to the force constant k and mass m of the oscillating body by k k ω 2 = or ω = m m Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Equations on 481 Both omegas, one for angular velocity and one for angular frequency are the same thing. ω = k m ω 1 k f = = 2π 2π m 1 2π m T = = = 2π f ω k Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Angular frequency, frequency, and period in SHM A spring is mounted horizontally, with its left end held stationary. By attaching a spring balance to the free end and pulling toward the right, we determine that the stretching force is proportional to the displacement and that a force of 6.0 N causes a displacement of 0.030 m. We remove the spring balance and attach a 0.50 kg body to the end, pull it a distance of 0.020 m, release it, and watch it oscillate. Find the: Force constant of the spring. Angular frequency. Frequency. Period of the oscillation. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley X versus t for SHO If at time t = 0 the phasor OQ makes an angle Φ with the x-axis then at any later time t the angle is ( ωt + Φ). The position of the oscillating object can be found by the equation: x = Acos( ωt +φ) (13.13) Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Variations on a theme—write equations on board Changing either k or m changes the period of oscillation. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley SHM phase and motion • SHM can occur with various phase angles , the angle at which t = 0. • For a given phase we can examine position, velocity, and acceleration. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley SHM equations x = Acos( ωt +φ) When t = 0: = −ω ω +φ vx Asin( t ) = φ xo Acos = −ω 2 ω +φ ax Acos( t ) v = −ωAsinφ ox Do these equations fit with equation 13.4: a = −ω 2 Acosφ 2 ox = d x = − k ax 2 x 2 dt m vo A = x2 + x o ω 2 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Watch variables change for a glider example Keep problems solving strategy on 485 in mind when solving the following problem. Let’s return to the system of mass and horizontal spring we considered previously, with k = 200 N/m and m = 0.50 kg. This time we give the body an initial displacement of +0.015 m and an initial velocity of +0.40 m/s. a) Find the period, amplitude, and phase angle of the motions. b) Write equations for the displacement, velocity, and accelerations as functions of time. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 13.2 Summary Simple harmonic motion: If the restoring force in periodic motion is directly proportional to the displacement x, the motion is called simple harmonic motion (SHM). In many cases this condition is satisfied if the displacement from equilibrium is small. The angular frequency, frequency, and period in SHM do not depend on the amplitude, but only on the mass m and force constant k. The displacement, velocity, and acceleration in SHM are sinusoidal functions of time; the amplitude A and phase angle of the oscillation are determined by the initial position and velocity of the body. (See Examples 13.2, 13.3, 13.6, and 13.7.) Homework: Read pages 486 to 492 On page 507: 3, 5, 9, 15 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy in SHM • Energy is conserved during SHM and the forms (potential and kinetic) interconvert as the position of the object in motion changes. 12+ 1 2 = 1 2 (13.21) E= 2mvx 2 kx 2 kA • Page 487 goes through a proof of this relationship. • We can rewrite equation 13.21 for a velocity at any displacement: k v = ±A2 − x 2 x m (13.22) • Using this equation we can find the maximum velocity (when displacement is zero): k k (13.23) v =AAA2 = = ω max m m Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy in SHM • You can use equation 13.22 to calculate the velocity at each point in the figure below. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy in SHM II • Figure 13.13 shows the interconversion of kinetic and potential energy with an energy versus position graphic.
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