1. Chapter 14
Periodic Motion
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2. Getting Ready for Physics 4C
Read (or re-read!) Chapter 14
Periodic Motion (14.1)
Simple Harmonic Motion (14.2)
Energy in SHM (14.3)
Simple Pendulums (14.5)
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3. What causes periodic motion?
Consider a spring acting on a sliding mass
Force from stretching AND compressing the spring causes oscillation of the system, or periodic motion.
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4. What causes periodic motion?
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5. What causes periodic motion?
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6. What causes periodic motion?
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7. What causes periodic motion?
A “restoring” force
Always acts opposite to displacement
Since F = ma, acceleration a(x) has opposite sign to displacement (x)
a(x) ~ - w2x
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8. Characteristics of periodic motion
Amplitude, A, is maximum magnitude of displacement (from equilibrium.)
Period, T, is the time for one cycle.
Frequency, f, is number of cycles per unit time.
Angular frequency, w is 2π times frequency:
The frequency and period are reciprocals of each other:
f = 1/T and T = 1/f.
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9. Simple harmonic motion (SHM)
When the restoring force is directly proportional to the displacement from equilibrium, the resulting motion is called simple harmonic motion (SHM).
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10. Simple harmonic motion (SHM)
In many systems the restoring force is approximately proportional to displacement if the displacement is sufficiently small.
That is, if the amplitude is small enough, the oscillations are approximately simple harmonic.
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11. Characteristics of SHM
For a body of mass m vibrating by an ideal spring with a force constant k:
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12. Characteristics of SHM
The greater the mass m in a tuning fork’s tines, the lower the frequency of oscillation, and the lower the pitch of the sound that the tuning fork produces.
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13. Displacement as a function of time in SHM
The displacement as a function of time for SHM is:
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14. Displacement as a function of time in SHM
Increasing m with the same A and k increases the period of the displacement vs time graph.
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15. Displacement as a function of time in SHM
Increasing k with the same A and m decreases the period of the displacement vs time graph.
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16. Displacement as a function of time in SHM
Increasing A with the same m and k does not change the period of the displacement vs time graph.
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17. Displacement as a function of time in SHM
Increasing ϕ with the same A, m, and k only shifts the displacement vs time graph to the left.
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18. Graphs of displacement and velocity for SHM
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19. Graphs of displacement and acceleration for SHM
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20. E = 1/2 mvx2 + 1/2 kx2 = 1/2 kA2 = constant
Energy in SHM
The total mechanical energy E = K + U is conserved in SHM:
E = 1/2 mvx2 + 1/2 kx2 = 1/2 kA2 = constant
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21. Energy diagrams for SHM
The potential energy U and total mechanical energy E for a body in SHM as a function of displacement x.
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22. Energy diagrams for SHM
The potential energy U, kinetic energy K, and total mechanical energy E for a body in SHM as a function of displacement x.
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23. Vertical SHM
If a body oscillates vertically from a spring, the restoring force has magnitude kx. Therefore the vertical motion is SHM.
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24. Vertical SHM
If the weight mg compresses the spring a distance Δl, the force constant is k = mg/Δl .
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25. Angular SHM
A coil spring exerts a restoring torque is called the torsion constant of the spring.
The result is angular simple harmonic motion.
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26. Potential energy of a two atom system
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27. Vibrations of molecules
Shown are two atoms having centers a distance r apart, with the equilibrium point at r = R0.
If they are displaced a small distance x from equilibrium, the restoring force is approximately
Fr = –(72U0/R02)x
So k = 72U0/R02, and the motion is SHM.
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28. The simple pendulum
A simple pendulum consists of a point mass (the bob) suspended by a massless, unstretchable string.
If the pendulum swings with a small amplitude with the vertical, its motion is simple harmonic.
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29. Tyrannosaurus rex and the physical pendulum
We can model the leg of Tyrannosaurus rex as a physical pendulum.
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