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Question: The dimensional formula for torque is \( A.{\text{ M}}{{\text{L}}^2}{T^{ - 2}} \\\ B.{\te...

The dimensional formula for torque is
A. ML2T−2 B. ML−1T−1 C. L2 T−1 D. M2T−2K−1  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}} \\\

Explanation

Solution

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 Gyration2^2 ×\times Mass
Dimension formula of radius of gyration is L2{L^2} and mass is M1{M^1}
Therefore, The dimensional formula of Moment of Inertia = M1L2T0{M^1}{L^2}{T^0} …..(2)
And, Angular acceleration = Angular velocity ×\times Time−1^{ - 1}
Dimension formula of angular velocity is L0T−1{L^0}{T^{-1}} and time is T−1{T^{-1}}
Therefore, the dimensional formula of Angular Acceleration = M0L0T−2{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 =
[M1L2T0]×[M0L0T−2] =[M L2T−2]  \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 [M L2T−2]\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 M1L2T0{M^1}{L^2}{T^0} and angular acceleration that is M0L0T−2{M^0}{L^0}{T^{ - 2}} we can easily derive the dimensional formula for torque.