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

To solve this problem, we need to find out the force acting on a particle of mass m driven by a machine that delivers constant power k watts, given that the particle starts from rest.

1. Understanding Power and Kinetic Energy:

Power is the rate at which work is done or energy is transferred. For a constant power k, it can be expressed as:

P = \frac{dW}{dt} = F \cdot v

where F is the force and v is the velocity of the particle.

The kinetic energy KE of the particle is given by:

KE = \frac{1}{2}mv^2

2. Relating Kinetic Energy and Power:

The work done on the particle changes its kinetic energy. Thus from the equation for power:

k = Fv = \frac{d}{dt}\left(\frac{1}{2}mv^2\right)

3. Since the particle starts from rest, its initial velocity is zero:

Hence, using P = Fv, and knowing that power is constant:

v = \frac{1}{2m}\left(\frac{k}{v}\right)^2

v = \left(\frac{2k}{m}\right)^{1/3} t^{1/3}

4. Finding Force:

Using P = Fv again and substituting for v from above:

F = \frac{k}{v} = k \cdot \left(\frac{m}{2k}\right)^{1/3} t^{-1/3}

Simplifying the expression for force:

F = \sqrt{\frac{mk}{2}} t^{-1/2}

5. Conclusion:

The force on the particle at time t is given by F = \sqrt{\frac{mk}{2}} t^{-1/2}, matching the correct option.

Thus, the correct answer is the third option: $\sqrt{\frac{mk}{2}}\,t^{-1/2}$.