Question

A force $F = 2i - 3j + 7k$ acts on a particle undergoing displacement $r = 7i + 3j - 2k$. Calculate the work done by the force.
A) $37\;{\text{J}}$
B) $- 9\;{\text{J}}$
C) $49\;{\text{J}}$
D) $14\;{\text{J}}$

Answer 1

In this question, the concept of the dot product is used that is the work done by the force is the dot product of the force and the displacement when the quantities are in vector form and as we know that the dot product of the two vector quantities always provide scalar quantity.

Complete step by step solution:
Method 1:

We know that the force is a vector quantity, i.e. a quantity having magnitude and direction both. Also, displacement is another vector quantity which has both attributes.
The given force $F{\text{ }} = {\text{ }}2i - 3j + 7k$, has $2$ units of it in the x-direction, $3$units of it in the negative y- direction and $7$ units of it in the z-direction and its magnitude can be calculated as $\Rightarrow F = \sqrt {\left( {{2^2} + {3^2} + {7^2}} \right)}$
$\Rightarrow F = \sqrt {62} \;{\text{N}}$

Again if we see the given displacement vector, i.e. $r{\text{ }} = 7i + 3j - 2k$, it has $7$ units of it in the x-direction, $3$units of it in the y direction and $2$ units of displacement in the negative z-direction and its magnitude is calculated as,
$\Rightarrow r = \sqrt {\left( {{7^2} + {3^2} + {2^2}} \right)}$
$\Rightarrow r = \sqrt {62} \;{\text{m}}$

As we know that the work done is equal to the dot product of the force and the displacement vectors. Also we know that the dot product of two vector quantities is a scalar quantity. Thus, the work done is a scalar quantity.

Now we calculate the work done by the force as,
$W = \vec F \cdot \vec r$
Now, we substitute the vector form of the force and the position vector in the above equation as,
$\Rightarrow W = \left( {2i - 3j + 7k} \right) \cdot \left( {7i + 3j - 2k} \right)$
Now, we calculate the dot product as,
$\Rightarrow W = \left( {2 \times 7 - 3 \times 3 - 7 \times 2} \right)$
After simplification we get,
$\therefore W = - 9\;{\text{J}}$
Hence, the work done by the force is $- 9\;{\text{J}}$.

Thus, the correct option is (B).

Note: If the vectors are not given and only magnitudes are given, then work done is simply the product of the force, the distance moved and the cosine of the angle between the line of action of the force and the way towards which the object/system is moved or displaced because of the force.