Question

A particle moves \[x - \]axis under the action of a position dependent force\[F = \left( {5{x^2} - 2x} \right)N\]. Work done forces on the particle when it moves from origin to \[x = 3m\] is?
\[(A)45J\]
\[(B)36J\]
\[(C)32J\]
\[(D)42J\]

Answer 1

Hint: Work Done on an object depends on the amount of force \[F\] causing the work, the displacement energy by an object. The S.I. unit of work is Joule \[(J)\]. The position-dependent force is directed toward or away from the start position and also their magnitude increases with the distance from the starting position. The force to be variable, work should be calculated by the Integration. The sum rule is the integration of the sum of two functions is equal to the sum of integral of each function.


Complete answer:
Since the Force acting on the body is variable, we cannot directly use the formula,
\[Work = F.S\]
For the force to be variable, work should be calculated by the Integration.
\[ \Rightarrow Work = \int\limits_0^3 {\left( {5{x^2} - 2x} \right)} dx\]
\[ \Rightarrow Work = \int\limits_0^3 {\left( {5{x^2}} \right)} dx - \int\limits_0^3 {\left( {2x} \right)} dx\]
\[ \Rightarrow Work = \dfrac{5}{3} \times {3^2} - \dfrac{2}{2} \times {3^2}\]
\[ \Rightarrow Work = 45 - 9\]
\[ \Rightarrow Work = 36J\]
Hence, Work done by the forces on the particle when it moves from origin to \[x = 3m\] is \[36J\].

Note: A work done is the quantity of energy transferred by the force to move an object is termed.
The SI unit for work and energy is the joule \[(J)\], which is equivalent to a force of one newton exerted through a distance of one meter \[(m)\].
Joule is used to measure the capacity to do work or generate heat.
The work done by a force of one newton \[(N)\] acting over a distance of one meter equals one joule.