Question

The dimensional formula for the torque is
(a). \[M{{L}^{2}}{{T}^{-2}}\]
(b). \[M{{L}^{-1}}{{T}^{-1}}\]
(c). \[{{L}^{2}}{{T}^{-1}}\]
(d). \[{{M}^{2}}{{T}^{-2}}{{K}^{-1}}\]

Answer 1

- Hint: Torque is the product of angular acceleration and moment of inertia. We can simply calculate the dimensional formula of torque by putting the dimensional formulas of angular acceleration and moment of inertia in the respective equation.

Complete step-by-step solution -
The term force is used in the linear motions, while torques are used in the rotational motions, but they are having the same basis. Torque is proportional to the lever arm distance. i.e., the distance between the rotation axis and the force applied point. The formula of torque is given by,
Torque = angular acceleration x moment of inertia …………..(1).
The moment of inertia is = Mass x ${radius of gyration^2}$
Dimensional formula of moment of Inertia $=M{{L}^{2}}$
Angular acceleration = Angular velocity x time
Hence, the dimensional formula of angular acceleration will be, $={{T}^{-2}}$
On substituting these formulas in equation (1),
Dimensional formula of Torque will be,
$=\left[ {{M}^{1}}{{L}^{2}} \right]\times \left[ {{T}^{-2}} \right]$
$=\left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right]$
Where M is the mass, L is the length and T is the time
Hence the correct answer is option (A)

Note: When they are asking for dimensional formulas of any quantity, revise the expression of that quantity and put the dimensional formulas of substituent quantities in the solution, and develop the final dimensional formula.