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

$\sqrt{2 mk} t^{- 1 / 2}$
B

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

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

$\sqrt{mk} t^{- 1 / 2}$

Solution

Constant power acting on the particle of mass m is k watt.
or P = k
$\frac{dW}{dt} = k; dW = kdt$
$Integrating both sides, \int_{0}^{W} dW = \int_{0}^{t} kdt$
$\Rightarrow W = kt$
Using work energy theorem,
$W = \frac{1}{2} {mv}^{2} - \frac{1}{2} m (0)^{2}$
$kt = \frac{1}{2} {mv}^{2} [Using equation (i)]$
$v = \sqrt{\frac{2 kt}{m}}$
$Acceleration of the particle, a = \frac{dv}{dt}$
$a = \frac{1}{2} \sqrt{\frac{2 k}{m}} \frac{1}{\sqrt{t}} = \sqrt{\frac{k}{2 mt}}$
$Force on the particle, F = ma = \sqrt{\frac{mk}{2 t}} = \sqrt{\frac{mk}{2}} t^{- 1 / 2}$

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