Physics 123 11. Vibrations and Waves 11.1 Simple Harmonic Motion 11.2 Energy in SHM 11.3 Period and...
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Transcript of Physics 123 11. Vibrations and Waves 11.1 Simple Harmonic Motion 11.2 Energy in SHM 11.3 Period and...
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
11.11 Reflection and Interference of Waves
11.12 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 kx2
• The kinetic energy of the mass is
1/2 mv2
• The total energy is the sum of the KE of the mass and the PE of the spring.
KE + PE is constant
Problem 11.1 . . . 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 , v0 , in m/s, is most nearly
A. 0.1
B. 1
C. 10
D. 100
Solution 11.1 . . . Speed of SHM
KE = PE
Equating the two gives 1/2 mv02 = 1/2 kA2 so
v0 = A (k/m)1/2
Plugging in the values gives v0 = 1 m/s
Correct choice is B
Which graph represents position vs time?
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?
Which graph represents speed vs time?
Either A or D
If time starts when displacement is maximum then D and v = v0 sin 2 f t
If time starts when displacement is zero (equilibrium) then A and v = v0 cos 2 f t
Note: Velocity and position graphs are out of step (out of phase) by 900 or /2
Extreme calculus says (trust me!) . . .
v0 = 2 f A
Also we know that v0 = A (k/m)1/2
Put two and two together:
f = (k/m)1/2/ 2
Which graph represents acceleration vs time?
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 11.2 . . . 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 11.2 . . . 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
Node
AntinodeNode
Node
Antinode
= 2L/3
= L
= 2LL
Problem 11.3 . . . 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 11.3 . . . 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 T1/2 will be a straight line
3. The slope of the line will be 1 / [f (m/L)1/2]
That’s all folks!