Question

Figure shows the circular motion of a particle. The radius of the circle, the period, since of revolution and the initial position are indicated in the figure. The simple harmonic motion of the x-projection of the radius vector of the rotating particle P is

.

• A. x = 2 cos $\left( 2\pi t + \frac{\pi}{4} \right)$
• B. x = 2 sin $\left( 2\pi t + \frac{\pi}{4} \right)$
• C. x = 2sin$\left( 2\pi t - \frac{\pi}{4} \right)$
• D. x = 2cos$\left( 2\pi t - \frac{\pi}{4} \right)$

Answer 1

Answer: (A)

x = 2 cos $\left( 2\pi t + \frac{\pi}{4} \right)$

Answer 2

Here, T = 1 s

At t = 0 OP makes an $\frac{\pi}{4}( = 45º)$

After a time t, it covers an angle of $\frac{2\pi}{T}t$in the anticlockwise sense and makes an angle of $\left( \frac{2\pi}{T} + \frac{\pi}{4} \right)$with the x-axis.

The projection of OP on the x-axis at time t is given by

$x = 2\cos\left( \frac{2\pi}{T}t + \frac{\pi}{4} \right)$

For, T = 1 s

$\therefore x = 2\cos\left( 2\pi t + \frac{\pi}{4} \right)$