Physics 101: Lecture 22 Simple Harmonic Motion

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Physics 101 Lecture 22Simple Harmonic Motion

• Todays lecture will cover Textbook Sections 10.4
  - 10.6

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Review Ideal Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed
  position.
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality. (often called spring
  constant)

relaxed position
FX - kx lt 0
x
x gt 0
x0
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Review Simple Harmonic Motion
Uniform circular motion lt-gt Motion of an object
attached to an ideal spring
Period T (seconds per cycle) Frequency f
1/T (cycles per second) Angular frequency
2f 2/T
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Simple Harmonic MotionQuick Review
x(t) Acos(t) v(t) -Asin(t) a(t)
-A2cos(t)
x(t) Asin(t) v(t) Acos(t) a(t)
-A2sin(t)
OR
Period T (seconds per cycle) Frequency f
1/T (cycles per second) Angular frequency
2f 2/T
xmax A vmax A amax A2
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Review Period of a Spring
For simple harmonic oscillator 2f 2/T
For mass M on spring with spring constant k
Demos Amk dependence
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Concept Question

• If the amplitude of the oscillation (same block
  and same spring) was doubled how would the
  period of the oscillation change (The period is
  the time it takes to make one complete
  oscillation)
• 1. The period of the oscillation would double.2.
  The period of the oscillation would be halved3.
  The period of the oscillation would stay the same

x
2A
t
-2A
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Potential Energy of a Spring
Where x is measured fromthe equilibrium position
PES
x
0
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Same thing for a vertical spring
y
y0
Where y is measured fromthe equilibrium position
PES
y
0
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In either case...
Etotal 1/2 Mv2 1/2 kx2 constant
KE PE
KEmax 1/2Mv2max1/2M2A2 1/2kA2 PEmax
1/2kA2 Etotal 1/2kA2
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Concept Question

• In Case 1 a mass on a spring oscillates back and
  forth. In Case 2 the mass is doubled but the
  spring and the amplitude of the oscillation is
  the same as in Case 1. In which case is the
  maximum kinetic energy of the mass the biggest
• 1. Case 12. Case 23. Same

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Concept Question
PE 0 KE KEMAX
PE 1/2kx2 KE 0
same for both
samefor both
xA
x0
xA
x-A
x0
x-A
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Concept Question

• The same would be true for vertical springs...

PE 1/2k y2
Y0
PE 1/2k y2
Y0
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Pendulum

• For small oscillation period does not depend
  on
• mass
• amplitude

Demos MAL dependence
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Concept Question

• 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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Concept Question

• A pendulum is hanging vertically from the ceiling
  of an elevator. Initially
• the elevator is at rest and the period of the
  pendulum is T. Now the pendulum accelerates
  upward. The period of the pendulum will now be
• 1. greater than T
• 2. equal to T
• 3. less than T

Effective g is larger when accelerating
upward (you feel heavier)