Question

A force $(3i + 4j)$ Newton acts on a body and displaces it by $(3i + 4j)$ meters. The work done by this force is:
A. 10 J
B. 5 J
C. 25 J
D. 12 J

Answer 1

In the question, we need to determine the work done by the force to displace the body. For this, we will use the relation between the work, force, and displacement, which is given as $W = F \cdot d$

Complete step by step answer:

The product (dot product if vectors are involved) of the force applied to the body and the displacement of the body due to the applied force is known as the work done by the force.

Here, the applied force is given as $F = (3i + 4j)$ , and the displacement of the body to the applied force is given as $d = (3i + 4j)$.

The square root of the sum of the squares of the absolute values of the vectors gives the equivalent scalar value of the vector.

$F = (3i + 4j) \\\ \left| F \right| = \left| {3i + 4j} \right| \\\ = \sqrt {{3^2} + {4^2}} \\\ = \sqrt {9 + 16} \\\ = \sqrt {25} \\\ = 5 \\\$

Similarly,

$d = \left( {3i + 4j} \right) \\\ \left| d \right| = \sqrt {3i + 4j} \\\ = \sqrt {{3^2} + {4^2}} \\\ = \sqrt {9 + 16} \\\ = \sqrt {25} \\\ = 5 \\\$

So, substitute $\left| F \right| = 5{\text{ Newton}}$ and $|d| = 5{\text{ meters}}$ in the formula $W = Fd$ to determine the work done by the force on the object.
$W = Fd \\\ = 5 \times 5 \\\ = 25{\text{ Newton - Meter}} \\\ = 25{\text{ joules}} \\\$

Hence, the work done by the force $(3i + 4j)$ Newton acted on a body and displaced it by $(3i + 4j)$ meters is 5 joules.

Option C is correct.

Note: It is to be noted here that the dot product gives the equivalent vector, which is aligned in the same direction. Alternatively, this question can also be solved by following the dot product of the vectors for the force and the displacement.

$$W = F \cdot d \\\ = (3i + 4j)(3i + 4j) \\\ = \left( {3i} \right)\left( {3i} \right) + \left( {4j} \right)\left( {4j} \right) \\\ = 9 + 16 \\\ = 25{\text{ joules}} \\\$$

Here, we have used the concept that $i \cdot i = j \cdot j = 1$ and $i \cdot j = j \cdot i = 0$ as the angle between the coordinate axes is ${90^0}$ and $\cos {90^0} = 0$.