Question

The moment of force \[\overrightarrow{F}=2\widehat{i}+3\widehat{j}\] at the point (2,2) about the origin is given by?
A) \[2\widehat{k}\]
B) \[-2\widehat{k}\]
C) \[\widehat{k}\]
D) None of the above

Answer 1

Hint: Moment of force is also called torque. Torque is defined as the force which acts on a body and then the body starts rotating around some axis. So, the torque makes the body rotate. Torque is a vector quantity and it has both the magnitude and the direction.

Complete step by step solution:
We are given the force as \[\overrightarrow{F}=2\widehat{i}+3\widehat{j}\] and it is acting at the point (2,2). We need to find the torque about the origin. So the perpendicular distance is between the points P(2,2) and Q(0,0)
We know from the vector algebra position vector from the origin can be directly written as , \[\overrightarrow{r}=2\widehat{i}+2\widehat{j}\]
torque is given as \[\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}\]
\[\begin{align}
  & \Rightarrow (2\widehat{i}+2\widehat{j})\times (2\widehat{i}+3\widehat{j}) \\
 & =\left( \begin{matrix}
   \widehat{i} & \widehat{j} & \widehat{k} \\
   2 & 2 & 0 \\
   2 & 3 & 0 \\
\end{matrix} \right) \\
 & =0\widehat{i}+0\widehat{j}+2\widehat{k} \\
 & \Rightarrow 2\widehat{k} \\
\end{align}\]

So, the correct answer is “Option A”.

Additional Information:
First of all, let us define what is a moment of force. The moment of force is a measure of the ability of the force acting on a particular body to rotate it around a given axis.

Note:
While calculating the torque we have to keep in our mind that it is a vector quantity and is given by the cross product of force acting on the body and the perpendicular distance from the axis of rotation. So, we have to consider the angle between the two vectors. But here we had calculated the cross product by using vector algebra, and the result is also a vector, so we do not need the angle between the two vectors.