Transcript Potential Energy - McMaster University

Harmonic Motion (II)
• Mass and Spring
• Energy in SHM
Physics 1D03 - Lecture 33
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SHM:
x  A cos(t   )
x(t)
A
T
A = amplitude
t
 = phase constant
 = angular frequency
-A
A is the maximum value of x (x ranges from +A to -A)
 gives the initial position at t=0: x(0) = A cos
 is related to the period T and the frequency f = 1/T
T (period) is the time for one complete cycle (seconds).
Frequency f (cycles per second or hertz, Hz) is the number of
complete cycles per unit time.
Physics 1D03 - Lecture 33
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Example - frequency
What is the oscillation period of a FM radio station with a signal at
100MHz ?
Example - frequency
A mass oscillating in SHM starts at x=A and has a period of T.
At what time, as a fraction of T, does it first pass through x=A/2?
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Example
The block is at its equilibrium position and
is set in motion by hitting it (and giving it
an initial velocity) at time t = 0. Its motion
is SHM with amplitude 5 cm and period 2
seconds. Write the function x(t).
Result:
v0
M
x
x(t) = (5 cm) cos[π t – π/2]
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Example
The block is at x0 = +5 cm, with positive
velocity v0, at time t = 0. Its motion is SHM
with amplitude 10 cm and period 2 s. If
x(t) = A cos (t  ), the phase constant 
should be:
A)
B)
C)
D)
E)
M
v0
x0
0o
30o
60o
-30o
-60o
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Velocity and Acceleration
x(t )  A cos(t   )
dx
v(t ) 
  A sin(t   )
dt
dv
2
2
a(t ) 
  A cos(t   )   x
dt
a(t)   2 x(t)
Not e: vMAX  A
aMAX  A 2
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Position, Velocity and Acceleration
x(t)
t
v(t)
t
a(t)
t
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Question:
Where in the motion is the velocity largest?
Where in the motion is acceleration largest?
When do these happen ?
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When do we have Simple Harmonic Motion ?
Position:
x  A cos(t   )
Differentiate:
dx
dv
v , a
dt
dt
and we find that acceleration is proportional to
displacement:
a(t) =   2 x(t)
SHM is also called ‘oscillatory’ motion, and is ‘periodic’.
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