Q. A particle is moving in a circle of radius R in such a way that at any instant that `a_r` and `a_t` are equal.If the speed at t=0 is `v_0` ,the time taken to complete the first revolution:
A) `(R/v_0) e^(-2pi)`
B) ` v_0R`
For circular motion, the radial acceleration, `a_r` is given by `v^2/R` , and the tangential acceleration, `a_t` is given by `(dv)/(dt)` .
Since these accelerations are equal, so `v^2/R=(dv)/(dt)`
Hence, `(R/v^2)dv = dt`
`int(R/v^2)dv = intdt`
To evaluate integration constant, C, put` v=v_o at, t=0`
The relation between v and t is,
Now, `v=(ds)/(dt)` , where, s is the length of the arc covered by the particle as it moves in the circle.
putting at t=0, s=0
Therefore, the relation takes the form,
For a complete revolution, putting `s=2piR, t=T` (the time period),
hence correct answer is option D).