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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.

sayak maji , 8 Years ago

Grade 11

1 Answers

Manas Shukla

Last Activity: 8 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

$e^{-2\Pi } = R - v_{0}t$

$t= \frac{R-e^{-2\Pi }}{v_{0}}$

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