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

• A. $\sqrt{2mk}\,t^{-1/2}$
• B. $\frac{1}{2}\sqrt{mk}\,t^{-1/2}$
• C. $\sqrt{\frac{mk}{2}}\,t^{-1/2}$
• D. $\sqrt{mk}\,t^{-1/2}$

Answer 1

The Correct Option is C

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 \limits_0^w dw= \int \limits_0^t k\, dt $
$\Rightarrow \, \, \, \, \, \, w=kt ...(i)$
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)]
$v=\sqrt{\frac{2kt}{m}}$
Acceleration of the particle, $a = \frac{dv}{dt}$
$a=\frac{1}{2}\sqrt{\frac{2k}{m}}\frac{1}{\sqrt t}=\sqrt{\frac{k}{2mt}}$
Force on the particle, F = ma = $\sqrt{\frac{mk}{2r}}=\sqrt{\frac{mk}{2}}t^{-1/2}$