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**Simple Harmonic Motion**

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**What is an Oscillation? Vibration**

Goes back and forth without any resulting movement SHM - Simple Harmonic Motion

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**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

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**Describe restoring force**

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

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**Examples of SHM: Simple pendulum Mass on a spring Bungee jumping**

Diving board Object bobbing in the water Earthquakes Musical instruments

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Simple Pendululm Equation: Time is independent of amplitude or mass

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**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

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**Mass on a Spring Time is independent gravitational acceleration**

Equation: Time is independent gravitational acceleration

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**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

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**Restoring Force is proportional to (-) displacement**

Sketch: Negative sign means force is in the opposite direction of the displacement

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**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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**Relationships between variables**

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Other relationships:

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Diagrams

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Graphs:

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**Kinetic and Potential Energies in SHM**

since

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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

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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

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