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Simple Harmonic Motion
This model applies to a single point particle constrained to move in one dimension whose position is a sinusoidal function of time. Simple harmonic motion is sometimes abbreviated SHM.
Any object that experiences a linear restoring force or torque so that the equation of motion takes the form
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x ]\end
or
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\theta] \end
will experience simple harmonic motion with angular frequency ω. The prototypical example is an object of mass m attached to a spring with force constant k, in which case, by [Hooke's Law]:
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]\end
giving simple harmonic motion with angular frequency
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{m}}$
.
Models
A single point particle (or, for the angular version of SHM, a single rigid body).
The system must be subject to a one-dimensional restoring force (or torque) that varies linearly with the displacement (or angular displacement) from an equilibrium position.
Initial Conditions
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{\omega^{2}}\qquad\qquad\qquad]
[ v_
= v(t=0)]\end
Amplitude of Motion
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^
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+ \left(\frac{v_{0}}
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\right)^{2}}\qquad\qquad\qquad]\end
Phase
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\right) = \sin^{-1}\left(\frac{v_{0}}
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\right)]\end
Position
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\cos(\omega t) + \frac{v_{0}}
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\sin(\omega t)\qquad\qquad\qquad]\end
or, equivalently
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Velocity
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\cos(\omega t)\qquad\qquad\qquad]\end
or, equivalently:
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Acceleration
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\sin(\omega t) = -\omega^
x(t) \qquad]\end
or, equivalently:
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x(t)]\end
- Acceleration versus time graph.
- Velocity versus time graph.
- Position versus time graph.
Relevant Examples
None yet.
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