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Home Vibration and Waves Vibrations Proof  Sinusoidal Motion  
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Proof  Sinusoidal MotionIn this section we prove
The basic idea of the proof can be understood by imagining that you are watching a child on a merrygoround from far away. Because you are in the same horizontal plane as her motion, she appears to be moving from side to side along a line. Circular motion viewed edgeon doesn't just look like any kind of backandforth motion, it looks like motion with a sinusoidal xt graph, because the sine and cosine functions can be defined as the x and y coordinates of a point at angle q on the unit circle. The idea of the proof, then, is to show that an object acted on by a force that varies as F= kx has motion that is identical to circular motion projected down to one dimension. The equation
will also fall out nicely at the end. For an object performing uniform circular motion, we have a = v^{2}/r . The x component of the acceleration is therefore
where θ is the angle measured counterclockwise from the x axis. Applying Newton's second law,
Since our goal is an equation involving the period, it is natural to eliminate the variable v = circumference/T = 2πr/T, giving
The quantity r cos θ is the same as x, so we have
Since everything is constant in this equation except for x, we have proven that motion with force proportional to x is the same as circular motion projected onto a line, and therefore that a force proportional to x gives sinusoidal motion. Finally, we identify the constant factor of 4π ^{2 }m /T ^{2} with k, and solving for T gives the desired equation for the period,
Since this equation is independent of r, T is independent of the amplitude.


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