PreClass Notes: Chapter 13, Sections

Transcription

1 PreClass Notes: Chapter 13, Sections From Essential University Physics 3 rd Edition by Richard Wolfson, Middlebury College 2016 by Pearson Education, Inc. Narration and extra little notes by Jason Harlow, University of Toronto This video is meant for University of Toronto students taking PHY131. Outline Pushing a child on a swing, you can build up a large amplitude by giving a relatively small push once each oscillation cycle. If your pushing were not in step with the swing s natural oscillatory motion, then the same force would have little effect. R.Wolfson Simple Pendulum Circular motion and S.H.M. Energy in S.H.M. Damped Harmonic Motion Driven Oscillations and Resonance. 1

2 Simple Harmonic Motion Simple Harmonic Motion (S.H.M.) results whenever the following equation applies: Double-time derivative of position = negative constant position If position is represented by x, then: d 2 x dt 2 = ω2 x where ω 2 is a positive constant, and the angular frequency of the oscillations is ω. Almost every stable equilibrium will exhibit SHM for small disturbances from equilibrium. Simple pendulum Point mass on massless cord of length L. The tension force acts directly toward the pivot, so it provides no torque. The torque due to gravity causes the angular acceleration. 2

3 Simple Pendulum Simple Pendulum 3

4 Got it? What happens to the period of a pendulum if its length is quadrupled? A. The period is halved. B. The period is doubled. C. The period is quadrupled. D. The period is quartered. Simple harmonic motion can be viewed as one component of uniform circular motion. Angular frequency in SHM is the same as angular velocity in circular motion. 4

5 Energy in Simple Harmonic Motion Energy in Simple Harmonic Motion 5

6 Energy in Simple Harmonic Motion In the absence of nonconservative forces, the energy of a simple harmonic oscillator does not change. But energy is transfered back and forth between kinetic and potential forms. Energy in Simple Harmonic Motion E = K max = 1 2 mv max 2 E = U max = 1 2 kx max 2 = 1 2 ka2 6

7 Got it? If the total energy of a harmonic oscillator is reduced by a factor of 3, the amplitude of the oscillations A. increases by a factor of 3. B. decreases by a factor of 3. C. increases by a factor of 3. D. decreases by a factor of 3. E. remains unchanged. Simple Harmonic Motion is Everywhere! That s because most systems near stable equilibrium have potential-energy curves that are approximately parabolic. Ideal spring: U 1 2 kx2 1 m 2 x 2 2 Typical potential-energy curve of an arbitrary system: 7

8 Damped Harmonic Motion With nonconservative forces present, SHM gradually damps out: 2 d x dx m kx b 2 dt dt Amplitude declines exponentially toward zero: x t Ae t bt 2 m ( ) cos( ) For weak damping b, oscillations still occur at approximately the undamped frequency With stronger damping, oscillations cease. Critical damping brings the system to equilibrium most quickly. Damped Harmonic Motion bt 2 m x( t) Ae cos( t ) 8

9 Damped Harmonic Motion (a) underdamped (b) critically damped, and (c) overdamped oscillations. Driven Oscillations When an external force acts on an oscillatory system, we say that the system is undergoing driven oscillation. Suppose the driving force is F 0 cosω d t, where ω d is the driving frequency, then Newton s law is The solution is where and 2 d x dx 2 0 cos d m kx b F t dt dt A( ) 0 x( t) Acos( t ) k m d F m ( ) b / m is the natural frequency d 0 d 9

10 Resonance When a system is driven by an external force at near its natural frequency, it responds with largeamplitude oscillations. This is the phenomenon of resonance. The size of the resonant response increases as damping decreases. The width of the resonance curve (amplitude versus driving frequency) also narrows with lower damping. Resonance Resonance curves for several damping strengths; 0 is the undamped natural frequency k/m. 10

11 Resonance Musical instruments are all based on the phenomenon of resonance. A string of a particular length and tension will have certain frequencies for which it resonates at large amplitude and produces a certain frequency of sound. A column of air of a certain length will have certain resonance frequencies as well. 11