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Name |
Acceleration (simple harmonic motion) |
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Equation |
a = - ω 2 x |
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| a = - ω 2 r sin ( ωt ) |
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Definition of terms |
a = acceleration, x = position, r = amplitude of oscillation, t = time,
ω = frequency of oscillation |
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Comments |
A very important dynamical problem is that of a mass attracted to a point by a force proportional to the distance from the point.
In one dimension, we write this is as F= −kx, which on substituting F= ma, |
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k |
Redefining |
k |
= ω2 we recover the first |
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m |
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equation.
Since acceleration is the second derivative of position with respect to time, we see that this equation is a second-order differential equation, written |
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d2x |
+ ω2x = 0, the solution of which is x = r sin(ωt + f), |
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where f is an arbitrary constant. Setting f = 0 we recover the second equation above,
a = −ω2r sin ωt. |
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Any system governed by such a force displays oscillatory behaviour, the position and acceleration varying according to the equations above. Such motion is called simple harmonic motion, and any body obeying the simple harmonic equation of motion is called a harmonic oscillator.
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References |
Newtonian Mechanics , A P French, Thomas Nelson & Sons, 1971 |
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