Question

Please solve the following question and also tell the Dimensions of b ?

Answer 1

Given that the mass of the rotating particle is `m` and the resistive frictional force acting on it is `F_("fric")=3bv^2`, where `v` represents instantaneous speed of the particle,

Hence its instantaneous retardation will be

`-(dv)/(dt)=F_"fric"/m=(3bv^2)/m`

`=>-(dv)/v^2=(3b)/mdt`

`=>-int(dv)/v^2=(3b)/m intdt`

`=>1/v=(3b)/m t+c`, where c= integration constant

Now it is given that at `" "t= 0 ,v=v_0`

So `c=1/v_0`

So the relation becomes

`=>1/v=(3b)/m t+1/v_0`

It is also given that at `t=t_0, v=v_0/3`

`=>3/v_0=(3b)/m t_0+1/v_0`

So `t_0=(2m)/(3bv_0)`

Again `b==(2m)/(3t_0v_0)`

So dimension of `b="dimension of mass"/("dimension of time"xx "dimension of velocity"`

`=([M])/([T][LT^-1])=[ML^-1]`