Question

A particle of mass 2 kg travels along a straight line with velocity $v = a\sqrt X $​, where a is a constant. The work done by net force during the displacement of the particle from x=0 to x=4 m is:
A) \[{a^2}\]
B) \[2{a^2}\]
C) \[4{a^2}\]
D) \[\sqrt 2 {a^2}\]

Answer 1

Hint: Whenever a force moves an object over a distance, we can say that work is done. Thus we can calculate the work done by multiplying the force by the distance moved. Work is defined as a force that causes the displacement or the movement of a body. It is a scalar quantity.

Complete step by step solution:
Given data:
Mass of the particle, m = 2 kg
Velocity, $v = a\sqrt X $
Then work done by the net force =?
Initial Velocity, u = 0
It is given that $v = a\sqrt X $. When x=4,
$ \Rightarrow v = a\sqrt X = a\sqrt 4 = 2a$
We know the formula, ${v^2} - {u^2} = 2as$
Where u is the initial velocity, v is the final velocity and s is the displacement.
\[ \Rightarrow {v^2} = 2as\] $\left( {\because u = 0} \right)$
\[ \Rightarrow a = \dfrac{{{v^2}}}{{2s}} = \dfrac{{{{\left( {2a} \right)}^2}}}{{2 \times 4}} = \dfrac{{4{a^2}}}{8} = \dfrac{{{a^2}}}{2}\]
We know that work done is given by the formula, \[W = f \times s\]
\[ \Rightarrow W = m \times a \times s\] $\left( {\because f = ma} \right)$
Substituting the values of m, a, and, s, we get,
\[ \Rightarrow W = m \times a \times s = 2 \times \dfrac{{{a^2}}}{2} \times 4 = 4{a^2}\]
Hence work done, \[W = 4{a^2}\]

Hence the correct option is C.

Note: 1. Work done involves both the force applied and the displacement of the object.
2. Work done is positive, when the force displaces the object in its direction. Work done is negative, If the force and the displacement are in opposite directions. If the direction of the displacement and the force are perpendicular to each other, then the work done is zero.
3. Energy is associated with the work. The capacity of the body to do work is called energy. Thus we can say that the body is said to possess the energy when it is having the capacity to do the work. The unit of energy is also the same as that of work.