Question

Find the work done by a force $\vec F = x\hat i + xy\hat j$ acting a particle to displace it from point $O\left( {0,0} \right)$ to$C\left( {2,2} \right)$.

Answer 1

Work done is obtained by performing the dot product of Force and displacement of the object. Work done is a scalar quantity. The work done is equal to the magnitude of the force that is multiplied by the distance an object moves in the direction of the force.

Formula Used:
$d\vec w = \vec F.d\vec s$
Where:
$d\vec w$= The amount of work done
$\vec F$= The force applied
$d\vec s$= The amount of displacement caused due to the applied force.

Complete step by step solution:
Work done can be defined as work is done, when a force acts on an object and it causes a displacement.
In order to calculate the amount of work done, we require three quantities namely, force, displacement and the angle between force and displacement.
In the given question:
Force, $\vec F = x\hat i + xy\hat j$
Let us consider the displacement caused by the object be denoted by$d\vec s$, such that
$d\vec s = dx\hat i + dy\hat j$
Let us consider, force $\vec F$causes a small displacement of $d\vec s$as a result of which$d\vec w$.
We know:
$d\vec w = \vec F.d\vec s$
Putting the values of $\vec F$and$d\vec s$, we find:
$d\vec w = \left( {x\hat i + xy\hat j} \right).\left( {dx\hat i + dy\hat j} \right)$
By solving the dot product, we obtain the following equation:
$d\vec w = x dx + xy dy$
Now, we need to integrate$d\vec w$, to get the work done W, thus on the right hand side, we integrate $d\vec x$and $d\vec y$within their respective limits.
The points $O\left( {0,0} \right)$and $C\left( {2,2} \right)$are given, integrating $d\vec x$ from $0to2$and integrating $d\vec y$from$0to2$, in order to get the total work done, we obtain:
$W = _0^2\smallint xdx + _0^2\smallint xydy$
Thus, we get:
$W = 6J$
This is our required answer.

Note: The SI unit for work done is in joules$(J)$ as work done is a measure of transfer of energy. The integration done above, is done within the two points O and C, owing to their respective coordinate values. Work being a scalar quantity has only magnitude and no direction.