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 $\underset{{\omega }_o}{\overset{\omega }{\int }}\frac{d\omega }{{\omega }^2}=\underset{0}{\overset{t}{\int }}$ dt

$\omega =\frac{{\omega }_o}{1-{\omega }_{o}t}$

or $\frac{d\theta }{dt}=\frac{{\omega }_o}{1-{\omega }_{o}t}$

or $\underset{0}{\overset{2\pi }{\int }}d\theta =\underset{0}{\overset{t}{\int }}\frac{{\omega }_{0}dt}{1-{\omega }_{0}t}$

or $t=\frac{1}{{\omega }_0}\left(1-e^{-2\pi }\right)$

$=\frac{R}{v_0}\left(1-r^{-2\pi }\right)$