Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance

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1 Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1

2 Revision problem Please try problem #31 on page 480 A pendulum clock keeps time by the swinging of a uniform solid rod

3 Pendulums Waves, tides Springs Simple Harmonic Motion 3

4 Simple Harmonic Motion Requires a force to return the system back toward equilibrium Spring Hooke s Law Pendulum and waves and tides gravity Oscillation about an equilibrium position with a linear restoring force is always simple harmonic motion (SHM) 4

5 Springs Hooke s Law F=-kx 5

6 Hooke s Law F=-kx Springs 6

7 For a small angle, the force is proportional to angle of deflection, θ. Pendulum F return mgsin 7

8 For a small angle, the return force is proportional to the distance from the equilibrium point: Pendulum sin s L F return mg mg L s 8

9 Kinematics of SHM Simple Harmonic motion can be described by a sinusoidal wave for displacement, velocity and acceleration: 9

10 The angle for the sinusoidal wave changes with time. Kinematics of SHM It goes full circle 0 to π radians in one period of revolution, T. x( t) A cos t T 10

11 We define the frequency of revolution as Kinematics of SHM f 1 T x( t) A cos ft Frequency, f, has units s -1 or Hertz, Hz 11

12 Kinematics of SHM Velocity is 90 o or π/ radians out of phase: v( t) vmax sin ft 1

13 Kinematics of SHM Acceleration is 180 o or π radians out of phase a( t) amax cos ft 13

14 Kinematics of SHM SHM equations of motion x( t) A cos( ft ) v( t) a( t) v a max max sin( cos ft ) ft 14

15 A circular motion when looked end-on gives us a velocity like: Calculating v max v v sin(ft max ) 15

16 The velocity around the circle will be Calculating v max D A v max T T v max fa 16

17 For circular motion, we know about acceleration and forces Calculating a max mv F ma, F r a max v max A 17

18 Kinematics of SHM SHM equations of motion x( t) A cos( ft ) v( t) fa sin( ft ) a( t) (f ) A cos ft 18

19 Energy is conserved: Bounces between kinetic and potential energy SHM and Energy E E E total kinetic potential E kinetic 1 mv 1 kx E potential 19

20 The max KE must equal the max PE: SHM and Energy 1 m( v max ) 1 ka k v max m A 0

21 Finding the period of oscillation for a spring We now have equations for v max : v max k m A fa f 1 k m, T m k Period of oscillation is independent of the amplitude of the oscillation. 1

22 Finding the period of oscillation for a pendulum Consider the acceleration using the equation for the return force, and the relation between acceleration and displacement: a F m 1 m a (f ) max A mg L s g L A

23 Finding the period of oscillation for a pendulum We can calculate the period of oscillation f 1 g L, T L g Period is independent of the mass, and depends on the effective length of the pendulum. 3

24 Damped Oscillations All the oscillating systems have friction, which removes energy, damping the oscillations 4

25 Damped Oscillations We have an exponential decay of the total amplitude x ( t) max Ae t / 5

26 Damped Oscillations The time constant, τ, is a property of the system, measured in seconds x ( t) max Ae t / A smaller value of τ means more damping the oscillations will die out more quickly. A larger value of τ means less damping, the oscillations will carry on longer. 6

27 Damped Oscillations under-damped τ>>t critically-damped τ~t over-damped τ<<t 7

28 Driven Oscillations and Resonance An oscillator can be driven at a different frequency than its resonance or natural frequency. The amplitude can be large if the system is undamped. 8

29 Ocean tides are produced from the Moon (and Sun) gravitational pull on the oceans to make a 0cm wave. Moon drives the wave at 1 hours 5 minutes Tidal resonances 9

30 The natural resonance of local geography can affect this: e.g. Bay of Fundy in Canada where the tidal range is amplified from the 0cm wave to 16m. Tidal resonances 30

31 Natural geography can also make double tides: Tidal resonances 31

32 Undamped driven resonance Tacoma Narrows Bridge, Washington State,

33 Summary Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 33

34 Homework problems Chapter 14 Problems 48, 49, 50, 5, 54, 59, 6, 63 34

Energy in a Simple Harmonic Oscillator. Class 30. Simple Harmonic Motion

Energy in a Simple Harmonic Oscillator. Class 30. Simple Harmonic Motion Simple Harmonic Motion Class 30 Here is a simulation of a mass hanging from a spring. This is a case of stable equilibrium in which there is a large extension in which the restoring force is linear in