91Ó°ÊÓ

$$\begin{gathered} \mathrm{W}=900\mathrm{J} \\ \mathrm{speed}=1.0\mathrm{m}/\mathrm{s} \end{gathered}$$

The rate at which force does the work on the object is called as power due to the force. The work done on a particle by a constant force during its displacement is given as

$\mathbf{W}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{F}}\mathbf{.}\stackrel{\mathbf{→}}{\mathbf{d}}$

Formula:

The skier and the rope are moving with constant speed. Hence, the net force on the skier is zero. The force oftherope is balanced by the component of the gravity along the incline. This force is constant and not related to the speed of the rope. Hence, when the same force acts on the rope and moves it throughthesame distance, the work done by the force of the rope isthesame even though the speed changes to $2.0\mathrm{m}/\mathrm{s}$.

The work done is given by

$\mathrm{W}=\stackrel{→}{\mathrm{F}}.\stackrel{→}{\mathrm{d}}$

$\mathrm{W}=\stackrel{→}{\mathrm{F}}.\stackrel{→}{\mathrm{d}}=\mathrm{Fd}$

Hence, we determine the force as

$\mathrm{F}=\frac{\mathrm{W}}{\mathrm{d}}$

Substitute all the value in the above equation.

$\begin{array}{rcl}\mathrm{F}& =& \frac{900\mathrm{J}}{8\mathrm{m}}\\ & =& 112.5\mathrm{N}\end{array}$

Hence the force is, 112.5 N.

Here, both the force and velocity of the rope are in the same direction. Hence, the rate of doing work i.e. power can be calculated as

$$\begin{gathered} {\mathrm{P}}_{\mathrm{avg}}=\frac{\mathrm{W}}{∆\mathrm{t}} \\ {\mathrm{P}}_{\mathrm{avg}}=\mathrm{Fv} \end{gathered}$$

Substitute all the value in the above equation.

$\begin{array}{rcl}{\mathrm{P}}_{\mathrm{avg}}& =& 112.5\mathrm{N}×1.0\mathrm{m}/\mathrm{s}\\ & =& 112.5\mathrm{W}\end{array}$

When the speed of the rope changes, only the power will change since the force of the rope is constant and independent of speed. Hence we get,

$$\begin{gathered} {\mathrm{P}}_{\mathrm{avg}}=\frac{\mathrm{W}}{∆\mathrm{t}} \\ {\mathrm{P}}_{\mathrm{avg}}=\mathrm{Fv} \end{gathered}$$

Substitute all the value in the above equation.

$\begin{array}{rcl}{\mathrm{P}}_{\mathrm{avg}}& =& 112.5\mathrm{N}×2.0\mathrm{m}/\mathrm{s}\\ & =& 225\mathrm{W}\end{array}$

Hence the rate of doing work when rope moves with speed of $1.0\mathrm{m}/\mathrm{s}\mathrm{is}112.5\mathrm{W}$and The rate of doing work when rope moves with speed of $2.0\mathrm{m}/\mathrm{s}\mathrm{is}225\mathrm{W}$.