Question

The following figure depict a circular motion. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated on the figure. The simple harmonic motion of the $x$ - projection of the radius vector of the rotating particle $P$ can be shown as:

• A. $x(t)=a \cos \left(\frac{2 \pi t}{4}+\frac{\pi}{4}\right)$
• B. $x ( t )= a \cos \left(\frac{\pi t }{4}+\frac{\pi}{4}\right)$
• C. $x ( t )= a \sin \left(\frac{2 \pi t }{4}+\frac{\pi}{4}\right)$
• D. $x ( t )= a \cos \left(\frac{\pi t }{3}+\frac{\pi}{2}\right)$

Answer 1

Answer: (A)

In figure
Ampliterde $=a$
At time, $t=0 \quad(\phi=\pi / 4)$
At time $t,\left(\phi=\frac{2 \pi}{T} \times t\right)$
$[\phi=$ phase $] .$
since initial phese $(\phi=1 / 4)$
$\Rightarrow \phi=\frac{2 \pi}{T} x t+\frac{\pi}{4}$
Now, Representing it into vector.
$x$ co ordinate $=a \cos \phi$
$=a \cos \left[\frac{2 \pi}{T} t+\frac{\pi}{4}\right]$
$T=4 s .$
$x(t)=a \cos \left[\frac{\pi t}{2}+\frac{\pi}{4}\right]$