Question

A particle is revolving in a circle of radius R with initial velocity u . Its starts retarding with constant retardation v²/4πR . The number of revolutions it makes in time 8πR/u is

Answer 1

Answer:2  

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Explanation:

                      v²-u²=2as

                      0²-v²=-2as

                      v²=2as

                      a=v²/2s

                      S=v²/2a=v²×4πR/2v²

                         =2πR

                   

                         v=u-at

                         0=v-v²t/4πR

                         t=4πR/v

         time left for particle to move

                     =8πR/v-4πR/v

                     =4πR/v

                         

                    s=ut+1/2at²

                      =1/2 at²  (as u=0)

angle moved ∅=2πR/R

                         =2π

                         =One revolution

So, total number of rotation is 2

Answer 2

The total number of rotation is 2.

Explanation:

Given that,

Radius = R

Time $t'=\dfrac{8\pi R}{v}$

Retardation $a=\dfrac{v^2}{4\pi R}$

We need to calculate the distance moved by the particle along circle before coming to rest

Using equation of motion

$v^2=u^2+2as$

Here, u = 0

$v^2=2as$

$s=\dfrac{v^2}{2a}$

Put the value of a into the formula

$s=\dfrac{v^2\times4\pi R}{2\times v^2}$

$s=2\pi R$

We need to calculate the angle moved

Using formula of angle moved

$\theta=\dfrac{s}{r}$

Where, s = distance covered

r = radius

Put the value into the formula

$\theta=\dfrac{2\pi R}{R}$

$\theta=2\pi$

Which is equal to one revolution

We need to calculate the time taken by particle stop

Using equation of motion

$v=u+at$

$v=0+at$

Put the value into the formula

$v=\dfrac{v^2t}{4\pi R}$

$t=\dfrac{4\pi R}{v}$

Time left for the particle to move is

$t'= t'-t$

Put the value into the formula

$t''=\dfrac{8\pi R}{v}-\dfrac{4\pi R}{v}$

$t''=\dfrac{4\pi R}{v}$

We need to calculate the distance covered by the particle in this time

Using equation of motion

$s=ut+\dfrac{1}{2}at^2$

$s=\dfrac{1}{2}at^2$

Put the value into the formula

$s=\dfrac{1}{2}\times\dfrac{v^2}{4\pi R}\times(\dfrac{4\pi R}{v})^2$

$s=2\pi R$

We need to calculate the angle

Using formula of angle

$\theta=\dfrca{s}{r}$

Put the value into the formula

$\theta=\dfrac{2\pi R}{R}$

$\theta=2\pi$

Which is equal to the one revolution

Hence, The total number of rotation is 2.

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Topic : revolution

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