Simple Harmonic Motion

What is an Oscillation?

VibrationGoes back and forth without any resulting movement

SHM - Simple Harmonic Motion

An object in SHM oscillates about a fixed point.

This fixed point is called mean position, or equilibrium position

This is the point where the object would come to rest if no external forces acted on it

Describe restoring force

Restoring force, and therefore acceleration, is proportional to the displacement from mean position and directed toward it

Examples of SHM: Simple pendulum Mass on a spring Bungee jumping Diving board Object bobbing in the water Earthquakes Musical instruments

Simple Pendululm Equation:

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T = 2ΠL

g

Time is independent of amplitude or mass

Assumptions: 1. Mass of string is negligible compared to mass of load

2. Friction is negligible 3. Angle of swing is small 4. Gravitational acceleration is constant

5. Length is constant

Mass on a Spring Equation:

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T = 2Πm

kTime is independent gravitational acceleration

Assumptions: 1. Mass of spring is negligible compared to mass of load

2. Friction is negligible 3. Spring obeys Hooke’s Law at all times

4. Gravitational acceleration is constant

5. Fixed end of spring can’t move

Restoring Force is proportional to (-) displacement

Meanpositiondisplacementrestoring force

Sketch:

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F ∝−x

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F = −kxNegative sign means force is in the opposite direction of the displacement

Variables for SHM: x displacement from mean position A maximum displacement (amplitude) Ø phase angle (initial displacement at t = 0)

T period (time for one oscillation) f frequency (number of oscillations per unit time)

angular frequency

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=k

m

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2 =k

m

Relationships between variables

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F = −kx

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a =−kx

m

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=ma

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=− 2x

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T = 2Πm

k

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=2Π(1

ω)

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=2π

T

Other relationships:

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x = xo sinωt

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x = xo cosωt

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v = vo cosωt

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v = vo sinωt

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v = ±ω x02 − x 2

Diagrams

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Graphs:http://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.html

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Kinetic and Potential Energies in SHM

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Ek =1

2mv 2

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v = ±ω x02 − x 2since

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Ek =1

2mω2(x0

2 − x 2)

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Ekmax =1

2mω2(x0

2)

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ET =1

2mω2(x0

2)€

E p =1

2mω2x 2

Damping

Energy losses (energy dissipation) due to friction - removes energy from system

For an oscillating object with no damping, total energy is constant - depends on mass, square of initial amplitude, angular frequency

Damping (continued)

Amplitude decreases exponentially - all energy is eventually converted to heat

Critical damping (controlled)- oscillations die out in shortest time possible

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Resonance

System displaced from equilibrium position will vibrate at its natural frequency

System can be forced to vibrate with a driving force at the natural frequency

Examples: musical instruments, machinery, glass, microwave, tuning a radio