Transcript Potential Energy - McMaster University

Harmonic Motion (II)
(Serway 15.2, 15.3)
• Mass and Spring
• Energy in SHM
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Velocity and Acceleration
x(t )  A cos(t   )
dx
  A sin(t   )
v(t ) 
dt
dv
  A 2 cos(t   )   2 x
a(t ) 
dt
a(t)   2 x(t)
Not e: vMAX  A
aMAX  A 2
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Energy in SHM
Look again at the block & spring
M
K  12 mv 2  12 m 2 A2 sin 2 (t   )
U  12 kx2  12 kA2 cos 2 (t   )
 k!
K  U  12 A2 m 2 sin 2 (t   )  k cos 2 (t   ) 
 12 kA2 sin 2 (t   )  cos 2 (t   ) 
 12 kA2  a constant (total mechanical energy)
We could also write E = K+U = ½ m(vmax )2
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Question:
Suppose U=½ kA2 at t = 0, and T=1 sec.
When is U=K during the cycle?
(i.e. when do their curves cross?)
When does
Put in
1
2
1
2
kx 2  12 m v2 ?
x  A cos(t ), v  A sin(t )
kA2 cos2 (t )  12 m2 A2 sin 2 (t )
 cos(t )   sin(t )
t  45, 135, 225, 315
  4 , 3 4 , 5  4 , 7  4 radians
So if T  1sec  T    2 rad
and U  K at t  18 sec, 3 8 sec, 5 8 sec, 7 8 sec
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E
U
K
t
U, K oscillate back and
forth “out of phase” with
each other; the total E is
constant.
T
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Armed with the equation of motion:
and the expressions for energy:
x  A cos(t  )
U  12 kx2 K  12 mv2
We can solve a large range of problems in SHM.
Example 1: A 0.25kg block is oscillating on a spring with
k=4N/m. At t=0, v=-0.2m/s and a=+0.5m/s2.
Find its total energy and the equation of
motion.
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Solution
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Example 2: A 50.0g object connected to a spring k=35.0N/m
oscillates on a horizontal, frictionless surface with an amplitude
of 4.0cm. Find:
a) the total energy of the system
b) the speed of the object when the position is 1.0cm
c) the kinetic energy when x=3.0cm
d) the potential energy when x=3.0cm.
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Example 3: A 200 g mass is attached to a horizontal
spring and executes SHM with a period of 0.25s. The total
energy of the system is 2.0J.
a) Find the force constant of the spring
b) Find the amplitude of the motion
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Example 4: A car having a mass of 1000kg is driven into a
brick wall. The car’s bumper behaves like a spring of
constant 5x106 N/m and compresses 3.16 cm as the car is
brought to rest. What was the speed of the car before
impact, assuming no mechanical energy loss with the wall?
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Example 5: A particle executes SHM with an amplitude of
3.0 cm. At what position does its speed equal one half of
its maximum speed?
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