Question

Consider that \(M\), \(L\), and \(T\) indicate mass, length and time, respectively. If \([ML^2T^{-n}]\) is the dimensional formula for the physical quantity Torque, then the value of \(n\) is _ _ _. (in integer)

Hint:

Torque has the same dimensions as work and energy: \[ ML^2T^{-2} \] but torque is a vector quantity while work is scalar.

Answer 1

Correct Answer: 2

Step 1: Recall the definition of torque.
Torque is the turning effect of a force acting at a perpendicular distance from the axis of rotation.
Mathematically,
\[ \text{Torque} = \text{Force} \times \text{Distance} \]

Step 2: Write the dimensional formula of force.
From Newton's second law,
\[ F=ma \] where acceleration has dimensions
\[ [a]=LT^{-2} \] Thus, dimensions of force are
\[ [F]=MLT^{-2} \]

Step 3: Include the distance factor.
Torque is force multiplied by distance.
Distance has dimensions
\[ [L]=L \]

Step 4: Find dimensions of torque.
\[ [\text{Torque}] = [Force]\times [Distance] \] \[ =(MLT^{-2})(L) \] \[ =ML^2T^{-2} \]

Step 5: Compare with the given formula.
Given dimensional formula is
\[ [ML^2T^{-n}] \] Actual dimensional formula is
\[ [ML^2T^{-2}] \]

Step 6: Equate the powers of time.
Comparing powers of \(T\), we get
\[ n=2 \]

Step 7: Final conclusion.
Therefore, the value of \(n\) is
\[ \boxed{2} \]