Example

Question:

The figure depicts two circular motions. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated in the figures. Obtain the simple harmonic motions of the x-projection of the radius vector of the rotating particle \(P\) in each case.

Solution:

(a) At \(t=0\), \(OP\) makes an angle \(45^\circ = \pi/4\) rad with the x-axis. After time \(t\), it covers an angle \(\frac{2\pi}{T}t\) in the anticlockwise sense, making an angle \(\frac{2\pi}{T}t + \frac{\pi}{4}\) with x-axis.
The x-projection is: \[ x(t) = A \cos\left(\frac{2\pi}{T}t + \frac{\pi}{4}\right) \] For \(T=4\) s: \[ x(t) = A \cos\left(\frac{\pi}{2}t + \frac{\pi}{4}\right) \] This is SHM of amplitude \(A\), period 4 s, initial phase \(\frac{\pi}{4}\).

(b) Here at \(t=0\), the angle is \(90^\circ = \frac{\pi}{2}\) with the x-axis. After time \(t\), in the clockwise sense, it makes an angle \(\frac{\pi}{2} - \frac{2\pi}{T}t\).
The x-projection of OP on x-axis at time \(t\) is: \[ x(t) = B \cos\left(\frac{\pi}{2} - \frac{2\pi}{T}t\right) = B \sin\left(\frac{2\pi}{T}t\right) \] For \(T=30\) s: \[ x(t) = B \sin\left(\frac{\pi}{15}t\right) \] which is SHM of amplitude \(B\), period 30 s, and initial phase \(-\frac{\pi}{2}\).