Prove an equivalent equation of motion for a particle moving with simple harmonic motion

by RoRi

January 31, 2016

Assume a particle is moving with simple harmonic motion with its position governed by the equation

$y = C \sin (kx+\alpha).$

The velocity of the particle is defined to be the derivative $y'$ . We define the frequency of the motion to be the reciprocal of the period.

Prove that the equation of motion can be written as:

$y = A \cos (mx + \beta).$

Find equations relating the constants $A,m,\beta$ and $C, k, \alpha$ .

Proof. We are given $y = C \sin (kx + \alpha)$ , so using the co-relations of sine and cosine we have

$y = C \cos \left(kx + \alpha - \frac{\pi}{2} \right) = A \cos (mx + \beta),$

where

$A = C, \quad m = k, \quad \beta = \alpha - \frac{\pi}{2}. \qquad \blacksquare$

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