Topic C.1 Simple harmonic motion

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15 Terms

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What is the definition of simple harmonic motion?

Acceleration is directly proportional to displacement

Acceleration acts in opposite direction to displacement

a = -(ω2)x

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How do kinetic energy, potential energy, acceleration. and velocity change during SHM

At maximum displacement:

Potential energy and acceleration are at maximum

Velocity is zero and kinetic energy are zero,

At equilibrium:

Potential energy and acceleration are zero

Velocity and kinetic energy are at their max

<p><strong>At maximum displacement:</strong></p><p>Potential energy and acceleration are at maximum</p><p>Velocity is zero and kinetic energy are zero, </p><p><strong>At equilibrium: </strong></p><p>Potential energy and acceleration are zero</p><p>Velocity and kinetic energy are at their max</p>
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Graph for total, kinetic, potential energy against displacement

Kinetic is maximum at x=0, potential is maximum at edges (max displacement), total is straight line at top (Ek+Ep), smooth curve connecting

<p>Kinetic is maximum at x=0, potential is maximum at edges (max displacement), total is straight line at top (Ek+Ep), smooth curve connecting</p>
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Graph for total, kinetic, potential energy against time

Potential starts at max value, kinetic starts at zero (as x=0 at t=0), total is straight line (Ek+Ep)

<p>Potential starts at max value, kinetic starts at zero (as x=0 at t=0), total is straight line (Ek+Ep)</p>
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What is angular frequency (ω)

2pi/T, similar to how frequency is 1/T

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How to get the frequency from acceleration-displacement graph?

Find gradient (square of angular frequency, ω2)

Rearrange for T

Equate to 1/F

Solve F

<p>Find gradient (square of angular frequency, ω<sup>2</sup>)</p><p>Rearrange for T</p><p>Equate to 1/F</p><p>Solve F</p>
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Displacement equation for SHM + graph

Displacement starts at 0 (because sine graph) assuming no phase angle shift

<p>Displacement starts at 0 (because sine graph) assuming no phase angle shift</p>
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Phase angle

Graph is shifted by portion of cycle equivalent to the portion of 2pi that the phase angle is

Example of how phase angle works: if T = 8s, then pi/4 phase angle would cause it to shift by 1s, as pi/4 is 1/8th of 2pi and 1s is 1/8th of 8s

A useful phase angle to know is pi, because it causes the output of the function to be negative.

Positive phase angle is leftwards shift, negative is rightwards (opposite of what is expected)

<p>Graph is shifted by portion of cycle equivalent to the portion of 2pi that the phase angle is </p><p>Example of how phase angle works: if T = 8s, then pi/4 phase angle would cause it to shift by 1s, as pi/4 is 1/8th of 2pi and 1s is 1/8th of 8s</p><p>A useful phase angle to know is pi, because it causes the output of the function to be negative.</p><p>Positive phase angle is leftwards shift, negative is rightwards (opposite of what is expected)</p>
9
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Forces acting during pendulum motion

Concept: T must be greater than mg when bob is moving in circle, as centripetal force is required

Equilibrium position: T larger than mg, centripetal force is maximum

Halfway: T still larger than mg, centripetal force is greater than 0

Max displacement: T is equal to mg, centripetal force is 0

Restoring force: mgsinθ is the restoring force, restoring force always acts perpendicular to tension

<p><strong>Concept:</strong> T must be greater than mg when bob is moving in circle, as centripetal force is required</p><p><strong>Equilibrium position:</strong> T larger than mg, centripetal force is maximum</p><p><strong>Halfway:</strong> T still larger than mg, centripetal force is greater than 0</p><p><strong>Max displacement:</strong> T is equal to mg, centripetal force is 0</p><p><strong>Restoring force: </strong>mgsinθ is the restoring force, restoring force always acts perpendicular to tension</p>
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Max displacement, velocity, acceleration values

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Kinetic energy formula derivation

Substitute formula for v in terms of amplitude and displacement into kinetic energy formula

<p>Substitute formula for v in terms of amplitude and displacement into kinetic energy formula</p>
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Max kinetic energy/total energy derivation

Substitute 0 into kinetic energy formula, as kinetic energy is maximum at 0 displacement (equilibrium).

This is also equal to total energy.

<p>Substitute 0 into kinetic energy formula, as kinetic energy is maximum at 0 displacement (equilibrium). </p><p>This is also equal to total energy.</p>
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How does particle oscillate with travelling wave?

Particle position on x-axis is fixed

If wave is moving right, then particle will follow wave as if it is moving leftwards, and viceversa.

<p>Particle position on x-axis is fixed</p><p>If wave is moving right, then particle will follow wave as if it is moving leftwards, and viceversa.</p>
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Transverse waves

Direction of oscillation is perpendicular to direction of travel

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Longitudinal waves

Direction of oscillation is parallel to direction of travel