Question

A particle is moving in a circle of radius $R$ in such a way that at any instant the total acceleration makes an angle of $45^{\circ}$ with radius. Initial speed of particle is $v_0$. The time taken to complete the first revolution is:

• A. $\frac{2R}{v_{o}}\left(1-e^{-2\pi}\right)$
• B. $\frac{R}{v_{o}}\left(1-e^{-2\pi}\right)$
• C. $\frac{R}{v_{o}}$
• D. $\frac{2R}{v_{o}}$

Answer 1

Answer: (B)

Total acceleration makes an angle of $45^{\circ}$ with radius, i.e., tangential acceleration = radial acceleration.
$R\alpha=R\omega^{2}$
or $\alpha=\omega^{2}$
or $\frac{d\omega}{dt}=\omega^{2} $
or $\frac{d\omega}{\omega^{2}}=dt $
or $\int\limits_{\omega_o}^{\omega}\frac{d\omega}{\omega^{2}}=\int\limits^{t}_{0}$ dt
$\omega= \frac{\omega_{o}}{1-\omega_{o}t}$
or $\frac{d\theta}{dt}= \frac{\omega_{o}}{1-\omega_{o}t} $
or $ \int\limits^{2\pi}_{0} d\theta=\int\limits^{t}_{0} \frac{\omega_{0}dt}{1-\omega_{0}t} $
or $t=\frac{1}{\omega_{0}}\left(1-e^{-2\pi}\right) $
$ =\frac{R}{v_{0}}\left(1-r^{-2\pi}\right)$