A particle is moving in a circle of radius R in such a way that any insant the total acceleration makes an angle of `45^(@)` with radius. Initial speed of particle is `v_(0)`. The time taken to complete the first revolution is:

Options

A. `(2R)/(v_(0))(1-e^(-2pi))`
B. `(R)/(v_(0))(1-e^(-2pi))`
C. `R/(v_(0))`
D. `(2R)/(v_(0))`

Answer

b

Explanation

Total acceleration makes an anlge of `45^(@)` with radius i.e., tangential acceleration =radial acceleration.
`R alpha=R omega^(2)`
or `alpha=omega^(2)`
or `(d omega)/(dt)=omega^(2)`
or `(d omega)/(omega^(2))=dt`
or `int_(omega_(0))^(omega) (domega)/(omega^(2))=int_(0)^(t)dt`
`omega=(omega_(0))/(1-omega_(0)t)`
or `(d theta)/(dt)=(omega_(0))/(1-omega_(0)t)`
`int_(0)^(2pi)d theta=int_(0)^(t) (omega_(0)dt)/(1-omega_(0)t)`
or `t= 1/(omega_(0))(1-e^(-2pi))`
`=R/(v_(0))(1-e^(-2pi))`