Question

The dimensional formula for torque is
$
  A.{\text{ M}}{{\text{L}}^2}{T^{ - 2}} \\
  B.{\text{ M}}{{\text{L}}^{ - 1}}{T^{ - 1}} \\
  C.{\text{ }}{{\text{L}}^2}{\text{ }}{{\text{T}}^{ - 1}} \\
  D.{\text{ }}{{\text{M}}^2}{T^{ - 2}}{K^{ - 1}} \\
 $

Answer 1

Hint:Here we will proceed by writing the formula of Torque that is Angular reaction $ \times $ Moment of Inertia. After expanding the formula, write the dimensional formulas of both angular reaction and moment of inertia , we can easily calculate the dimensional formula for torque.

Complete step-by-step answer:
Force causes acceleration and Torque causes angular accelerations. Torque is proportional to the distance between the rotation axis and the point of application of the force (the point at which the force is applied).
Mathematically, Torque is defined as a product of Angular acceleration and Moment of Inertia. Its formula is
Torque = Angular Acceleration $ \times $ Moment of Inertia …….(1)
Since, Moment of Inertia (M.O.I)= Radius of Gyration$^2$ $ \times $ Mass
Dimension formula of radius of gyration is ${L^2}$ and mass is ${M^1}$
Therefore, The dimensional formula of Moment of Inertia = ${M^1}{L^2}{T^0}$ …..(2)
And, Angular acceleration = Angular velocity $ \times $ Time$^{ - 1}$
Dimension formula of angular velocity is ${L^0}{T^{-1}}$ and time is ${T^{-1}}$
Therefore, the dimensional formula of Angular Acceleration = ${M^0}{L^0}{T^{ - 2}}$ …..(3)
On substituting equation (2) and (3) in equation (1) we get,
Torque= Moment of Inertia $ \times $ Angular Acceleration
Or, I =
 $
  \left[ {{M^1}{L^2}{T^0}} \right] \times \left[ {{M^0}{L^0}{T^{ - 2}}} \right] \\
   = \left[ {M{\text{ }}{{\text{L}}^2}{T^{ - 2}}} \right] \\
 $
Therefore, the torque is dimensionally represented as $\left[ {M{\text{ }}{{\text{L}}^2}{T^{ - 2}}} \right]$
Where, M= Mass
L = Length
T = Time
$\therefore $ C is the correct option.

Note- Dimensional formula is an expression for the unit of a physical quantity in terms of the fundamental quantities. The fundamental quantities are mass (M), length (L), and time (T). A dimensional formula is expressed in terms of powers of M,L and T.Whenever we come up with this type of problem where we are asked to find out the dimensional formula for torque. Then by using moment of inertia that is ${M^1}{L^2}{T^0}$ and angular acceleration that is ${M^0}{L^0}{T^{ - 2}}$ we can easily derive the dimensional formula for torque.