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A particle moving in a circular path of radius R in such a way that at any instant tangential and radial acceleration are equal. If at t=0 the speed of particle is v find time period to complete first revolution.

A particle moving in a circular path of radius R in such a way that at any instant tangential and radial acceleration are equal. If at t=0 the speed of particle is v find time period to complete first revolution.

Grade:11

1 Answers

Manas Shukla
102 Points
7 years ago
Lets assume initial velocity of particle to be = {v_{0}} , so as not to be confused with variable velocity...
Its given a_{t} = v^{2}/ R
Now \frac{\partial v}{\partial t} = \frac{v^{2}}{R}
Integrating we get -\frac{1}{v} = \frac{t}{r} + C
substituting values at t = 0 we find C = -\frac{1}{v_{0}}
Now we know v =\partial x/\partial t
And we know for 1 revolution dx will vary from 0 to 2\Pi R
\frac{\partial x}{\partial t} = \frac{Rv_{0}}{R-v_{0}t}
\int_{0}^{2\Pi R} \partial x = \int_{0}^{t} \frac{Rv_{0}}{R-v_{0}t} \partial t
Solving we get
2\Pi R = \frac{R v_{0}}{-v_{0}} ln (R-v_{0}t)
Cancelling common terms
e^{-2\Pi } = R - v_{0}t
t= \frac{R-e^{-2\Pi }}{v_{0}}
Ask if you have any problems , Best of luck

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