Question

The dimensional formula for moment of couple is:
\[\begin{align}
  & A.[{{M}^{1}}{{L}^{2}}{{T}^{-1}}] \\
 & B.[{{M}^{1}}{{L}^{2}}{{T}^{-2}}] \\
 & C.[{{M}^{-1}}{{L}^{2}}{{T}^{-2}}] \\
 & D.[{{M}^{1}}{{L}^{1}}{{T}^{-2}}] \\
\end{align}\]

Answer 1

Hint: The moment of a force is the perpendicular force, called torque, which causes the object to rotate. Also, torque is defined as the perpendicular force acting on the object which is at a distance. If there are two such moments which act on a body simultaneously, then it is called a couple.

Complete step-by-step answer:
We know that a couple is given by $F\times d$ where $F$ is the force and $d$ is the perpendicular distance between the two forces. The SI unit of a couple is \[Nm\].
The couple is a pseudo-vector quantity, which has both direction and magnitude. The magnitude of the net couple is given by taking the net torque from taking the clockwise or anticlockwise moments which are acting on any arbitrary point of the rotation. The direction of the couple is given by the right-hand screw rule. If the thumb points the direction of force then the curling of the fingers give the direction of torque. Also since a couple is a cross product of the force and the perpendicular distance , the couple is perpendicular to the plane consisting both the force and the perpendicular distance.
We know the dimension of force is given as $[M^{1}L^{1}T^{-2}]$ and the dimension of distance is \[\left[ L \right]\].
We know from dimensional analysis that when two physical quantities are multiplied, then we can add their power.
Then the dimensions of couple is given as $[M^{1}L^{1}T^{-2}][L]=ML^{2}T^{-2}$

So, the correct answer is “Option B”.

Note: A couple is a special case of moment, where two anti-parallel torques separated by a distance $d$ rotate an object. A few examples of couples include the force acting on a steering wheel of a car when both the hands are held on the steering wheel.