Simple%20Harmonic%20Motion

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Simple Harmonic Motion
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Simple Harmonic Motion

• Vibrations
• Vocal cords when singing/speaking
• String/rubber band
• Simple Harmonic Motion
• Restoring force proportional to displacement
• Springs F -kx

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Question II

• A mass on a spring oscillates back forth with
  simple harmonic motion of amplitude A. A plot of
  displacement (x) versus time (t) is shown below.
  At what points during its oscillation is the
  magnitude of the acceleration of the block
  biggest?
• 1. When x A or -A (i.e. maximum displacement)
• 2. When x 0 (i.e. zero displacement)
• 3. The acceleration of the mass is constant

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Potential Energy in Spring

• Force of spring is Conservative
• F -k x
• W -1/2 k x2
• Work done only depends on initial and final
  position
• Define Potential Energy PEspring ½ k x2

Force
work
x
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Energy in SHM

• A mass is attached to a spring and set to motion.
  The maximum displacement is xA
• Energy PE KE constant!
• ½ k x2 ½ m v2
• At maximum displacement xA, v 0
• Energy ½ k A2 0
• At zero displacement x 0
• Energy 0 ½ mvm2
• Since Total Energy is same
• ½ k A2 ½ m vm2
• vm sqrt(k/m) A

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Question 3

• A mass on a spring oscillates back forth with
  simple harmonic motion of amplitude A. A plot of
  displacement (x) versus time (t) is shown below.
  At what points during its oscillation is the
  speed of the block biggest?
• 1. When x A or -A (i.e. maximum displacement)
• 2. When x 0 (i.e. zero displacement)
• 3. The speed of the mass is constant

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Question 4

• A spring oscillates back and forth on a
  frictionless horizontal surface. A camera takes
  pictures of the position every 1/10th of a
  second. Which plot best shows the positions of
  the mass.

1 2 3
EndPoint
Equilibrium
EndPoint
EndPoint
Equilibrium
EndPoint
EndPoint
Equilibrium
EndPoint
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Springs and Simple Harmonic Motion
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What does moving in a circle have to do with
moving back forth in a straight line ??
x
8
8
q
R
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7
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SHM and Circles
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Simple Harmonic Motion
x(t) Acos(?t) v(t) -A?sin(?t) a(t)
-A?2cos(?t)
x(t) Asin(?t) v(t) A?cos(?t) a(t)
-A?2sin(?t)
OR
Period T (seconds per cycle) Frequency f
1/T (cycles per second) Angular frequency ?
2?f 2?/T For spring ?2 k/m
xmax A vmax A? amax A?2
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Example

• A 3 kg mass is attached to a spring (k24 N/m).
  It is stretched 5 cm. At time t0 it is released
  and oscillates.
• Which equation describes the position as a
  function of time x(t)
• A) 5 sin(wt) B) 5 cos(wt) C) 24 sin(wt)
• D) 24 cos(wt) E) -24 cos(wt)

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Example

• A 3 kg mass is attached to a spring (k24 N/m).
  It is stretched 5 cm. At time t0 it is released
  and oscillates.
• What is the total energy of the block spring
  system?

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Example

• A 3 kg mass is attached to a spring (k24 N/m).
  It is stretched 5 cm. At time t0 it is released
  and oscillates.
• What is the maximum speed of the block?

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Example

• A 3 kg mass is attached to a spring (k24 N/m).
  It is stretched 5 cm. At time t0 it is released
  and oscillates.
• How long does it take for the block to return to
  x5cm?

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Pendulum Motion

• For small angles
• T mg
• Tx -mg (x/L) Note F proportional to x!
• S Fx m ax
• -mg (x/L) m ax
• ax -(g/L) x
• Recall for SHO a -w2 x
• w sqrt(g/L)
• T 2 p sqrt(L/g)
• Period does not depend on A, or m!

L
T
m
x
mg
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Example Clock

• If we want to make a grandfather clock so that
  the pendulum makes one complete cycle each sec,
  how long should the pendulum be?

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Question 1

• Suppose a grandfather clock (a simple pendulum)
  runs slow. In order to make it run on time you
  should
• 1. Make the pendulum shorter
• 2. Make the pendulum longer

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Summary

• Simple Harmonic Motion
• Occurs when have linear restoring force F -kx
• x(t) A cos(wt)
• v(t) -Aw sin(wt)
• a(t) -Aw2 cos(wt)
• Springs
• F -kx
• U ½ k x2
• w sqrt(k/m)
• Pendulum (Small oscillations)
• w sqrt(L/g)

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