Physics 123

11. Vibrations and Waves 11.1 Simple Harmonic Motion 11.2 Energy in SHM 11.3 Period and sinusoidal nature of SHM 11.4 The Simple Pendulum 11.6 Resonance 11.7/8 Wave motion and types of waves Reflection and Interference of Waves Standing Waves

Simple Harmonic Motion

Amplitude The amplitude of this periodic motion is the distance between all of the following except 1. A and F 2. A and B 3. D and F 4. B and F

Amplitude Amplitude is the maximum displacement from the equilibrium position. All choices are measures of amplitude except D which is twice the amplitude. The correct answer is D

Terms of Endearment A complete round trip is called an oscillation. The period is the time to execute an oscillation. We use the symbol T to denote the period. The frequency is the number of oscillations per second. We use the symbol f to denote frequency. T = 1/f

SHM governed by F = -kx The equilibrium position represents the relaxed length of the spring. The spring has a tendency to return to this position if it is stretched or compressed. The restoring force is given by Hooke's Law: F = - kx Periodic Motion governed by F = -kx is called Simple Harmonic Motion (SHM)

Energy in SHM The potential energy stored in the spring is 1/2 kx 2 The kinetic energy of the mass is 1/2 mv 2 The total energy is the sum of the KE of the mass and the PE of the spring. KE + PE is constant

Problem Speed of SHM A 100 g mass is attached to a spring whose spring constant k = 50 N/m. The amplitude of oscillation A = 5 cm. The maximum speed of the mass, v 0, in m/s, is most nearly A.0.1 B.1 C.10 D.100

Solution Speed of SHM KE = PE Equating the two gives 1/2 mv 0 2 = 1/2 kA 2 so v 0 = A (k/m) 1/2 Plugging in the values gives v 0 = 1 m/s Correct choice is B

Which graph represents position vs time?

Either A or D If time starts when displacement is maximum then A and x = A cos 2  f t If time starts when displacement is zero (equilibrium) then D and x = A sin 2  f t

Which graph represents speed vs time?

Either A or D If time starts when displacement is maximum then D and v = v 0 sin 2  f t If time starts when displacement is zero (equilibrium) then A and v = v 0 cos 2  f t Note: Velocity and position graphs are out of step (out of phase) by 90 0 or  /2

Extreme calculus says (trust me!)... v 0 = 2  f A Also we know that v 0 = A (k/m) 1/2 Put two and two together: f = (k/m) 1/2 / 2 

Which graph represents acceleration vs time?

Suppose the position is given by x = A cos 2  f t We know that F = -kx So a = - k x /m So a = - (kA /m) cos 2  f t Note: Graph of a vs t will look like the one for x vs t except for the negative sign

Simple Pendulum

Period of a Pendulum The formula for the period of a simple pendulum is A.T = 2  (m/L) 1/2 B.T = 2  (L/g) 1/2 C.T = 2  (g/L) 1/2 D.T = 2  (A/m) 1/2

Period of a Pendulum We know that the formula for any SHM is: T = 2  (m/k) 1/2 The question is what is the "k" referring to in the absence of any spring? The figure indicates that for a simple pendulum our k = mg/L. So T = 2  (L/g) 1/2

Problem What about Bob? Pendulum 1 is 60 cm long and the mass of the bob is 10 g. Pendulum 2 is also 60 cm long but the mass of its bob is 20 g. Which pendulum oscillates faster (higher frequency (f) or smaller time period (T)?

Solution What about Bob? Both have the same frequency. T does not depend on the mass of the bob!

What is a Wave? A wave is the propagation of a disturbance

Making Waves 1. Pluck a string. A pulse (disturbance) travels down the string. 2. Ripples move outward in water Note: The disturbance travels but the medium simply oscillates back-and-forth (SHM)

Three Waves! A wave is crest to crest or trough to trough Wavelength = v = / T v = f

What does the speed of waves depend on? The speed depends on the properties of the medium. Wave speed in water is different than the speed on a string. Also the type of string and the tension matter

Speed of waves on a string v = [Tension / Linear Mass Density] 1/2 v = [T / (m/L)] 1/2

Types of Waves Transverse Waves:The disturbance travels in a direction perpendicular to the back-and- forth SHM motion of the particles of the medium (string,water) Longitudinal Waves:The disturbance travels in the same direction as the back-and-forth SHM motion of the particles of the medium (sound)

Reflection of a Wave Reflection: When a wave runs into an obstacle it bounces back. Example: Echo is the refection of sound waves

Resonance Resonance: If an external force is applied at the same frequency as the natural frequency the oscillations increase in amplitude. Example 1: Pushing a child on a swing Example 2: Tacoma Narrows Bridge Collapse Example 3: Caruso and the shattered wine glass

Interference Constructive Interference: When two waves run into each other in step (in phase). The outcome is increased amplitude Destructive Interference: When two waves run into each other out of step (out of phase). The outcome is decreased amplitude

Standing Waves on a String A combination of reflection, interference, and resonance makes standing waves on a string!

Standing Waves on a String

Problem Standing Waves A vibrator excites a string at a fixed frequency. By using different weights we put the string under different amounts of tension. This makes the string oscillate with different wavelengths (different number of loops). What is the equation relating T and

Solution Standing Waves We know these two equations: v = f v = [T / (m/L)] 1/2 So f= [T / (m/L)] 1/2 = (1/f ) [T / (m/L)] 1/2

What does = (1/f ) [T / (m/L)] 1/2 mean? It means that ….. 1. The wavelength is proportional to the square root of the tension in the string 2. A graph of vs T 1/2 will be a straight line 3. The slope of the line will be 1 / [f (m/L) 1/2 ]

That’s all folks!