Question

A particle moves with a velocity $\overrightarrow{v}=\left( 5\widehat{i}-3\widehat{j}+6\widehat{k} \right)m{{s}^{-1}}$ under the influence of a constant force $\overrightarrow{F}=\left( 10\widehat{i}+10\widehat{j}+20\widehat{k} \right)N$. The instantaneous power applied to the particle is
A. $200 W$
B. $320 W$
C. $140 W$
D. $170 W$

Answer 1

Power is the amount of work done per unit time. Use the formula for work done on a particle and derive an expression for power in terms of applied force and velocity. Then substitute the given values and calculate the power applied on the particle.

Formula used:
$P=\dfrac{dW}{dt}$
$W=\overrightarrow{F}.\overrightarrow{s}$
$\dfrac{d\overrightarrow{s}}{dt}=\overrightarrow{v}$

Complete step by step answer:
Let us first understand what is power.Power is defined as the rate a work done on a particle. In other words, power is the amount of work done per unit time. Therefore, power (P) can be written as $P=\dfrac{dW}{dt}$ …. (i), where W is the work done on the particle and t is time.

We also know that work done is equal to the dot product of the applied force and the displacement of the particle, i.e. $W=\overrightarrow{F}.\overrightarrow{s}$.
Substitute this value of W in (i).
$P=\dfrac{d}{dt}\left( \overrightarrow{F}.\overrightarrow{s} \right)$ …. (ii)
If the applied force is constant then (ii) can be written as $P=\overrightarrow{F}.\dfrac{d\overrightarrow{s}}{dt}$ …. (iii)
And the derivative of displacement with respect to time is the velocity of the particle.
Therefore, $\dfrac{d\overrightarrow{s}}{dt}=\overrightarrow{v}$.
Substitute this in (iii).
$P=\overrightarrow{F}.\overrightarrow{v}$.

With this, we derived an expression for instantaneous power (P) in terms of the applied force and velocity of the particle. Now, substitute the given values of applied force and velocity.
$\Rightarrow P=\left( 10\widehat{i}+10\widehat{j}+20\widehat{k} \right).\left( 5\widehat{i}-3\widehat{j}+6\widehat{k} \right)$
$\Rightarrow P=\left( (10\times 5)+(10\times (-3))+(20\times 6) \right)W$
$\therefore P=\left( 50-30+120 \right)=140W$.
Therefore, instantaneous power applied on the particle is $140 W$.

Hence, the correct option is C.

Note: The expression that we derived for power is instantaneous power. This means that the power applied on the particle at a given point of time. It sometimes happens that the power delivered is not constant because a force accelerates a particle and when a particle accelerates, its velocity changes with time.