Question

The mismatch between the physical quantity and the dimensional formula in the following is

• A. Force: $MLT^{-2}$
• B. Torque: $ML^{2}T^{-1}$
• C. Pressure: $ML^{-1}T^{-2}$
• D. Heat energy: $ML^{2}T^{-2}$
• E. Thermal conductivity: $MLT^{-3}K^{-1}$

Hint:

Logic Tip: Torque and Work (Energy) share the exact same dimensional formula $[ML^2T^{-2}]$. If you know Heat Energy is $[ML^2T^{-2}]$, then Torque must also be $[ML^2T^{-2}]$. The given $T^{-1}$ instantly stands out as wrong.

Answer 1

The Correct Option is B

Concept:
Dimensional analysis requires expressing physical quantities in terms of fundamental dimensions: Mass $[M]$, Length $[L]$, Time $[T]$, and Temperature $[K]$. We must verify each formula to find the incorrect one.
Step 1: Verify Force and Pressure.
Force = Mass $\times$ Acceleration = $[M] \times [LT^{-2}] = [MLT^{-2}]$. (Matches Option A) Pressure = Force / Area = $[MLT^{-2}] / [L^2] = [ML^{-1}T^{-2}]$. (Matches Option C)
Step 2: Verify Heat Energy and Thermal Conductivity.
Heat energy has the same dimensions as Work (Force $\times$ Displacement): Energy = $[MLT^{-2}] \times [L] = [ML^2T^{-2}]$. (Matches Option D) Thermal conductivity ($k$) is defined by the heat transfer equation: $\frac{Q}{t} = k A \frac{\Delta T}{L}$. Solving for dimensions of $k$: $[k] = \frac{[Q] \cdot [L]}{[t] \cdot [A] \cdot [\Delta T]}$ $[k] = \frac{[ML^2T^{-2}] \cdot [L]}{[T] \cdot [L^2] \cdot [K]} = [MLT^{-3}K^{-1}]$. (Matches Option E)
Step 3: Verify Torque.
Torque ($\tau$) is defined as Force $\times$ Perpendicular Distance: $[\tau] = [MLT^{-2}] \times [L] = [ML^2T^{-2}]$. The given option lists $[ML^2T^{-1}]$, which is the dimensional formula for Angular Momentum, not Torque. Therefore, this is the mismatch.