Question

A ball of mass 5kg experiences a force \[F = 2{x^2} + x\] . Work done in displacing the ball by $2m$ is then
A) $\dfrac{{22}}{3}J$
B) \[\dfrac{{44}}{3}J\]
C) $\dfrac{{32}}{3}J$
D) $\dfrac{{16}}{3}J$

Answer 1

Hint: Displacement of a body can be defined as the distance between two different positions of an object in motion. It is a vector quantity as it also has a direction. An object is displaced from its position if force is applied to it. Force can be a push, pull, or some twist acting on the body.

Complete step by step answer:
Step I:
Given that the mass of the ball is, $m = 5kg$
Force applied is, $F = 2{x^2} + x$
Displacement of the ball is, $d = 2m$
Work is done W =?
Step II:
Work is said to be done if the force is applied on the body and there is displacement in the body. It can be calculated as,
$W = F.d$ ------------(i)
Since,
Work done =\[W = \int\limits_0^2 {F.dx} \]
(∵ the ball is displaced from 0 to 2 m)
$\Rightarrow W = \int\limits_0^2 {(2{x^2} + x)} dx$
On evaluating the above integral,
\[\Rightarrow W = \int\limits_0^2 {2{x^2}dx + \int\limits_0^2 {xdx} } \]
$\Rightarrow W = 2.[\dfrac{{{x^3}}}{3}]_0^2 + [\dfrac{{{x^2}}}{2}]_0^2$
On simplification,
$\Rightarrow W = 2(0 - \dfrac{{{2^3}}}{3}) + (0 - \dfrac{{{2^2}}}{2})$
$\Rightarrow W = 2(\dfrac{{ - 8}}{3}) - \dfrac{4}{2}$
On simplification,
$\Rightarrow W = \dfrac{{ - 16}}{3} - 2$
$\Rightarrow W = \dfrac{{ - 22}}{3}$

The work done in displacing the ball is $\dfrac{{22}}{3}J$. Hence option (A) is the right answer.

Note:
It is clear that the work done is negative in this case. A negative work done shows that the object is moving in the opposite direction from that of the force applied. If the force and displacement are acting in the same direction, the work is said to be positive. Work done can also be zero. If the force applied and the displacement are perpendicular to each other than the value of work done will be zero. Work done can also be zero if either the force applied is zero or the displacement of the body is zero.