Question

What is the moment about the point $\hat i + 2\hat j - \hat k$ of a force represented by $3\hat i + \hat k$ acting through the point $2\hat i - \hat j + 3\hat k$?
${\text{A}}{\text{. }} - 3\hat i + 11\hat j + 9\hat k$
${\text{B}}{\text{. }}3\hat i + 2\hat j + 9\hat k$
${\text{C}}{\text{. }}3\hat i + 4\hat j + 9\hat k$
${\text{D}}{\text{. }}\hat i + \hat j + \hat k$

Answer 1

Hint- Moment of force is given by $\vec T = \vec r{\text{ }} \times {\text{ }}\vec F$ where $\vec F$ is the force acting perpendicular to the point and r is perpendicular distance between the force acting and axis of rotation.

As it is given in the question that the force acts through the points. So, let first point be represented by vector $\vec A = \hat i + 2\hat j - \hat k$ and other point by vector $\vec B = 2\hat i - \hat j + 3\hat k$ then,
According to the rule, we calculate $\vec r$,where $\vec r$ is the resultant vector of $\vec B$ and $\vec A$ given as
$
  \vec r = \vec B - \vec A \\
   \Rightarrow 2\hat i - \hat j + 3\hat k - \left( {\hat i + 2\hat j - \hat k} \right) \\
   \Rightarrow \hat i - 3\hat j + 4\hat k \\
$
Now, as per the question we have to find the moment of a force $\vec F = 3\hat i + \hat k$.
So, we have a formula for calculating moment of force that is Moment $\vec T = \vec r{\text{ }} \times {\text{ }}\vec F$
we will put the value of $\vec r$ and $\vec F$ in the formula, we get
$ \Rightarrow \left( {\hat i - 3\hat j + 4\hat k} \right) \times \left( {3\hat i + \hat k} \right)$
 Writing the above equation in matrix form
$
   \Rightarrow \left( {\begin{array}{*{20}{c}}
  i&j&k \\
  1&{ - 3}&4 \\
  3&0&1
\end{array}} \right) \\
   \Rightarrow - 3\hat i + 11\hat j + 9\hat k \\
$
Therefore moment $\vec T = - 3\hat i + 11\hat j + 9\hat k$
So, option “A” is correct.

Note- In this type of question basic concept of Moment and knowledge of vectors is required. As it is given that force is acting on a point so first find a vector $\vec r$ and after that using the cross product find moment of force $\vec T$.