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Simple Harmonic Motion
Lecturer: Professor Stephen T. Thornton
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Reading QuizWhich one of the following does not represent simple harmonic motion?
A) Distribution of student exam grades.
B) Automobile car springs.
C) Loudspeaker cone.
D) A mass oscillating at the end of a spring.
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Answer: A
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Last Time
Bernoulli equation/principle
Applications of Bernoulli principle
Read remaining sections of Chapter.
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Today
Oscillations
Simple harmonic motion
Periodic motion
Springs
Energy
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Oscillations
Cone inside loudspeaker Car coil springs
Do demos
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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. The mass and spring system is a useful model for a periodic system.
Oscillations of a Spring
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• Displacement is measured from the equilibrium point.
• Amplitude A is the maximum displacement.
• A cycle is a full to-and-fro motion.
• Period is the time required to complete one cycle.
• Frequency is the number of cycles completed per second.
Oscillations of a Spring
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Oscillations, simple harmonic motion, periodic motion
Start with periodic motion:
T = period of one cycle of periodic motion
f = 1/T = frequency of motion
unit of period: second
unit of frequency: 1 cycle/s = 1 Hz (hertz)
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Displaying Position Versus Time for Simple Harmonic Motion
Chart paper moving up
pen
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Simple Harmonic Motion as a Sine or a Cosine
Note period and amplitude T A
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Simple harmonic motionWe can describe this motion mathematically quite easily:
2cos when at 0x A t x A t
T
We obtain same result for time t and t + T. Look at previous slide.
Math gives same result:
2cos cos 2 cos cos( )x A t A ft A t A t
T
Note: cos at 0x A t
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Note cos at 0x A t
t = 0
t = 0
cos( )x A t
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Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator (SHO).
F = ma = - kx Newton’s second law:
with solutions of the form:
Simple Harmonic Motion
2
2
2
20
d xm kxdt
d x k xmdt
cos( )x A t
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The velocity and acceleration for simple harmonic motion can be found by differentiating the displacement:
Simple Harmonic Motion
2 22
2
cos( )
sin( )
cos( )
x A t
dxv A tdtd x dva A t
dtdta x
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Simple Harmonic Motion
Because then
1Tf
22
2 2
0
0 if
d x k xmdtk kx xm m
km
2 / ,f k m
122
kf mmTk
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Conceptual QuizConceptual Quiz
A) 0A) 0
B) B) AA/2/2
C) C) AA
D) 2D) 2AA
E) 4E) 4AA
A mass on a spring in SHM has
amplitude A and period T. What
is the total distance traveled by
the mass during a time interval T?
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Conceptual QuizConceptual Quiz
A) 0A) 0
B) B) AA/2/2
C) C) AA
D) 2D) 2AA
E) 4E) 4AA
A mass on a spring in SHM has
amplitude A and period T. What
is the total distance traveled by
the mass after a time interval T?
In the time interval time interval TT (the period), the mass goes
through one complete oscillationcomplete oscillation back to the starting
point. The distance it covers is The distance it covers is A + A + A + AA + A + A + A (4 (4AA).).
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Conceptual QuizConceptual Quiz
A) x = A
B) x > 0 but x < A
C) x = 0
D) x < 0
E) none of the above
A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?
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Conceptual QuizConceptual Quiz
A) x = A
B) x > 0 but x < A
C) x = 0
D) x < 0
E) none of the above
A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?
If both If both vv and and aa were zero at were zero at
the same time, the mass the same time, the mass
would be at rest and stay at would be at rest and stay at
rest!rest! Thus, there is NO NO
pointpoint at which both vv and aa
are both zero at the same
time.Follow-up:Follow-up: Where is acceleration a maximum? Where is acceleration a maximum?
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Connection between uniform circular motion and simple harmonic motion.
There is a remarkable relationship between the two.
Do projected uniform circular motion demo.
Let , so that the rate of
circular rotation is constant.
2cos cos( ) cos
t
x A A t A tT
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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.
Oscillations of a Spring
This is the new equilibrium point. The mass oscillates about this level.
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Energy
2 2 2 2
2 2 2
2max
2 2 2 2max
cos( )
1 1sin ( )
2 21 1
cos ( )2 2
Note maximum values of and .
1
21 1 1
2 2 2
E K U x A t
K mv mA t
U kx kA t
K U
U kA
kK mA mA kA
m
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2 2 2 2
2 2 2
2 2 2
1 1cos ( ) sin ( )
2 21
cos ( ) sin ( )21 1 1
2 2 2
E U K kA t kA t
E kA t t
E kA mv kx
E is total mechanical energy = K + U. E will be conserved in this case. We are assuming frictionless motion.
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Energy as a Function of Position in Simple Harmonic Motion
2 2 21 1 1
2 2 2E K U mv kx kA
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Energy as a Function of Time in Simple Harmonic Motion
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Look at simulations
http://physics.bu.edu/~duffy/semester1/semester1.html
Simple harmonic motion
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Spring Oscillation. A vertical spring with spring stiffness constant 305 N/m oscillates with an amplitude of 28.0 cm when 0.260 kg hangs from it. The mass passes through the equilibrium point (y = 0) with positive velocity at t = 0. (a) What equation describes this motion as a function of time? (b) At what times will the spring be longest and shortest?
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Oscillating Mass. A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. It takes 3.6 J of work to compress the spring by 0.13 m. If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 15 m/s2. Find the value of (a) the spring constant and (b) the mass.
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A spring can be stretched a distance of 60 cm
with an applied force of 1 N. If an identical
spring is connected in parallel with the first
spring, and both are pulled together, how
much force will be required to stretch this
parallel combination a distance of 60 cm?
A) 1/4 N
B) 1/2 N
C) 1 N
D) 2 N
E) 4 N
Conceptual QuizConceptual Quiz
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A spring can be stretched a distance of 60 cm
with an applied force of 1 N. If an identical
spring is connected in parallel with the first
spring, and both are pulled together, how
much force will be required to stretch this
parallel combination a distance of 60 cm?
Each spring is still stretched 60 cm, so each spring requires 1 N of force. But because there are two springs, there must be a total of 2 N of force! Thus, the combination of two parallel springs behaves like a stronger spring!!
Conceptual QuizConceptual QuizA) 1/4 N
B) 1/2 N
C) 1 N
D) 2 N
E) 4 N