Question

A force $(5\hat{i}+4\hat{j})\,N$ acts on a body and produces a displacement $\vec{S}=(6\hat{i}-5\hat{j}+3\hat{k})\,m$. The work done will be-
(1). $10J$
(2). $20J$
(3). $30J$
(4). $40J$

Answer 1

According to the second law of motion, a force is required to change the state of rest or the state of motion of an object. Here, the force vector acts on a body due to which it covers the given displacement. The Work done is the scalar product of force and displacement vectors.

Formulas used:
$W=\vec{F}\cdot \vec{x}$

Complete answer:
The force is an interaction between two bodies which causes a change in the state or the motion of one of the bodies or both of them. Its SI unit is $N$. Force is given by-
$F=ma$
Here,$F$ is the force acting on an object
$m$ is the mass on which the force acts
$a$ is the acceleration of the body
The work done is the product of mass and displacement. Its SI unit is joules ($J$).
$W=Fx$
Here, $W$ is work done by force $F$
$x$is the displacement of the body
Since force and displacement are vector quantities, the work done is the scalar product of force vector and displacement vector.
$W=\vec{F}\cdot \vec{x}$ --- (1)
Substituting the force vector and displacement vector in eq (1), we get,
$\begin{aligned} & W=(5\hat{i}+4\hat{j})\cdot (6\hat{i}-5\hat{j}+3\hat{k}) \\\ & \Rightarrow W=30-20 \\\ \end{aligned}$
$\therefore W=10J$

The work done by the force is $10J$. So, the correct option is (A).

Note:
The scalar product of two vectors is a scalar quantity. The motion of an object is always in the direction of the force except for the frictional force which is opposite to the motion of a body. The work done on a body is stored as its energy. Kinetic energy is possessed by a body when it is in motion.