First slide

Power

Question

A particle of mass m is driven by a machine that delivers a constant power k watts. If the particle starts from rest the force on the particle at time t is

Difficult

A

$(\sqrt{\frac{mk}{2}}) t^{- \frac{1}{2}}$
B

$(\sqrt{mk}) t^{- \frac{1}{2}}$
C

$(\sqrt{2 mk}) t^{- \frac{1}{2}}$
D

$(\frac{1}{2} \sqrt{mk}) t^{- \frac{1}{2}}$

Solution

Power k = F.v = (ma)v = m $\frac{dv}{dt} v = mv \frac{dv}{dt}$
Rearranging we get, $vdv = \frac{k}{m} dt$
Integrating both side,
$\int_{0}^{v} vdv = \frac{k}{m} \int_{0}^{t} dt \Rightarrow \frac{v^{2}}{2} = \frac{k}{m} t$
$\Rightarrow v = \sqrt{\frac{2 kt}{m}}$ ---------------(i)
Force acting on particle F = $m \frac{dv}{dt} - - - - - - (ii)$
Differentiating equation (i) w.r.t time
$\frac{dv}{dt} = (\sqrt{\frac{2 k}{m}}) \frac{{dt}^{- \frac{1}{2}}}{d} = (\sqrt{\frac{2 k}{m}}) \frac{1}{2} t^{- \frac{1}{2}}$
= $(\sqrt{\frac{k}{2 m}}) t^{- \frac{1}{2}}$
Substituting the value of $\frac{dv}{dt}$ in equation (ii) we get
$F = m \frac{dv}{dt} = m (\sqrt{\frac{k}{2 m}}) t^{- \frac{1}{2}} = (\sqrt{\frac{mk}{2}} t^{- \frac{1}{2}})$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App