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m. Vibrations and Waves. Physics 2053 Lecture Notes. Vibrations and Waves. Vibrations and Waves. Topics. 13.01 Hooke’s Law . 13.02 Elastic Potential Energy. 13.03 Comparing SHM with Uniform Circular Motion . 13.04 Position, Velocity and Acceleration - PowerPoint PPT Presentation
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Vibrations and Waves
Physics 2053Lecture Notes
m
Vibrations and Waves
Vibrations and Waves
13.01 Hooke’s Law 13.02 Elastic Potential Energy13.03 Comparing SHM with Uniform Circular Motion 13.04 Position, Velocity and Acceleration as a Function of Time
Topics
13.05 Motion of a Pendulum
Vibrations and Waves (3 of 33)
Hooke’s Law
If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic (T).
m
We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 ).
x = 0
Vibrations and Waves (4 of 33)
m
x = 0
Hooke’s Law
The minus sign on the force indicates that it is a restoring force – it is directed to restore the mass to its equilibrium position.
The restoring force exerted by the spring depends on the displacement:
m
x
F
[13.1] kxFs
Vibrations and Waves (5 of 33)
m
x
F kxF
(a) (k) is the spring constant
(b) Displacement (x) is measured from the equilibrium point
(c) Amplitude (A) is the maximum displacement
(e) Period (T) is the time required to complete one cycle(f) Frequency (f) is the number of cycles completed per second
(d) A cycle is a full to-and-fro motion
Hooke’s Law
Vibrations and Waves (6 of 33)
If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.
m
xo okxF
mgEquilibrium
Position
Hooke’s Law
Vibrations and Waves (7 of 33)
Any vibrating system where the restoring force is proportional to the negative of the displacement
moves with simple harmonic motion (SHM), and is often called a simple harmonic oscillator.
kxF
Hooke’s Law
Vibrations and Waves (8 of 33)
Potential energy of a spring is given by:
The total mechanical energy is then:
The total mechanical energy will be conserved
2kx
2mvE
22total
Elastic Potential Energy
[13.3] 2
kxPE2
s
Vibrations and Waves (9 of 33)
If the mass is at the limits of its motion, the energy is all potential.
m
A 2kAPE
2
m
x = 0
vmax
If the mass is at the equilibrium point, the energy is all kinetic.
2mv
KE2max
Elastic Potential Energy
Vibrations and Waves (10 of 33)
This can be solved for the velocity as a function of position:
The total energy is, therefore
And we can write: 2
kA2
mvE22
maxtotal
2kx
2mv
2kA 222
where mkAA
mkv 2
max
Elastic Potential Energy
[13.6] xA mkv 22
Vibrations and Waves (11 of 33)
The acceleration can be calculated as function of displacement
m
x
F
kxF
kxma
xmka
Amkamax
Elastic Potential Energy
Vibrations and Waves (12 of 33)
If we look at the projection onto the x axis of an object moving in a circle of radius A at a constant speed vmax, we find that the x component of its velocity varies as:
This is identical to SHM.
A
vmax
x
θsinvv max
22 xAmkv
22 xA
mkAvmax A
xAθsin22
AxA
mkAv
22
v
Comparing Simple Harmonic Motion with Circular Motion
Vibrations and Waves (13 of 33)
2
2max
Ax1vv
Therefore, we can use the period and frequency of a particle moving in a circle to find the period and frequency of SHM:
mkAvmax
mkAvmax
TAπ2
kmπ2T
mk
π21
T1f
Comparing Simple Harmonic Motion with Circular Motion
fA2
Vibrations and Waves (14 of 33)
A mass m at the end of a spring vibrates with a frequency of 0.88 Hz. When an additional 680 g mass is added to m, the frequency is 0.60 Hz. What is the value of m?
Comparing SHM with Uniform Circular Motion (Problem)
Vibrations and Waves (15 of 33)
A mass on a spring undergoes SHM. When the mass passes through the equilibrium position, its instantaneous velocity
A) is maximum.
B) is less than maximum, but not zero.
C) is zero.
D) cannot be determined from the information given.
Vibrations and Waves
Vibrations and Waves
A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a minimum?
A) at either A or B
B) midway between A and B
C) one-fourth of the way between A and B
D) none of the above
Vibrations and Waves
Vibrations and Waves
A 0.60 kg mass at the end of a spring vibrates 3.0 times per second with an amplitude of 0.13 m. Determine the velocity when it passes the equilibrium point,
Comparing SHM with Uniform Circular Motion (Problem)
Vibrations and Waves (18 of 33)
A 0.60 kg mass at the end of a spring vibrates 3.0 times per second with an amplitude of 0.13 m. Determine the velocity when it is 0.10 m from equilibrium,
Comparing SHM with Uniform Circular Motion (Problem) con’t
m/s 45.2vmax
Vibrations and Waves (19 of 33)
A 0.60 kg mass at the end of a spring vibrates 3.0 times per second with an amplitude of 0.13 m. Determine the total energy of the system,
Comparing SHM with Uniform Circular Motion (Problem) con’t
m/s 45.2vmax
Vibrations and Waves (20 of 33)
A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its potential energy is a minimum?
A) at either A or B
B) midway between A and B
C) one-fourth of the way between A and B
D) none of the above
Vibrations and Waves
Vibrations and Waves
Doubling only the amplitude of a vibrating mass-and-spring system produces what effect on the system's mechanical energy?
A) increases the energy by a factor of two
B) increases the energy by a factor of three
C) increases the energy by a factor of four
D) produces no change
Vibrations and Waves
Vibrations and Waves
A mass of 2.62 kg stretches a vertical spring 0.315 m. If the spring is stretched an additional 0.130 m and released, how long does it take to reach the (new) equilibrium position again?
Comparing SHM with Uniform Circular Motion (Problem)
Vibrations and Waves (23 of 33)
Doubling only the spring constant of a vibrating mass-and-spring system produces what effect on the system's mechanical energy?
A) increases the energy by a factor of three
B) increases he energy by a factor of four
C) produces no change
D) increases the energy by a factor of two
Vibrations and Waves
Vibrations and Waves
A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible.
The Simple Pendulum
Vibrations and Waves (25 of 33)
mg
F
s
x
sinmgF
Lxsin
x L
mgF
Small angles x s
s L
mgF
k forSHM
km2T
Lmgm2 (13.15)
gL2T
L
m
The Simple Pendulum
Vibrations and Waves (26 of 33)
A simple pendulum consists of a mass M attached to a weightless string of length L. For this system, when undergoing small oscillations
A) the frequency is proportional to the amplitude.
B) the period is proportional to the amplitude.
C) the frequency is independent of the length L.
D) the frequency is independent of the mass M.
Vibrations and Waves
Vibrations and Waves
The length of a simple pendulum is 0.760 m, the pendulum bob has a mass of 365 grams, and it is released at an angle of 12.0° to the vertical. With what frequency does it vibrate? Assume SHM.
Comparing SHM with Uniform Circular Motion (Problem)
Vibrations and Waves (28 of 33)
The length of a simple pendulum is 0.760 m, the pendulum bob has a mass of 365 grams, and it is released at an angle of 12.0° to the vertical. What is the pendulum bob’s speed when it passes through the lowest point of the swing?
Comparing SHM with Uniform Circular Motion (Problem) con’t
Vibrations and Waves (29 of 33)
The length of a simple pendulum is 0.760 m, the pendulum bob has a mass of 365 grams, and it is released at an angle of 12.0° to the vertical. Assume SHM. What is the total energy stored in this oscillation, assuming no losses?
Comparing SHM with Uniform Circular Motion (Problem) con’t
Vibrations and Waves (30 of 33)
Summary of Chapter 11
For SHM, the restoring force is proportional to the displacement. kxF
The period is the time required for one cycle, and the frequency is the number of cycles per second.
kmπ2T Period for a mass on a spring:
2kx
2mvE
22total
During SHM, the total energy is continually changing from kinetic to potential and back.
Vibrations and Waves (31 of 33)
A simple pendulum approximates SHM if its amplitude is not large. Its period in that case is:
gLπ2T
The kinematics of a mass/spring system:
mkAA
mkv 2
max
Amkamax
22 xAmkv Velocity
xmka
Acceleration
Summary of Chapter 11
Vibrations and Waves (32 of 33)
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