Question

Which of the following pairs of physical quantities does not have same dimensional formula?

• A. Work and torque.
• B. Impulse and linear momentum.
• C. Angular momentum and Planck’s constant.
• D. Tension and surface tension.

Answer 1

The correct option is D Tension and surface tension.

Step:1 Find the dimensional formula for work and torque.

Formula used: $\tau =\overrightarrow{r}\times \overrightarrow{F}$
Work, $W=\overrightarrow{F}.\overrightarrow{d}$ $W=Fd\cos \theta$
Dimension formula for work,
$=\left[M^1{L}^1{T}^{-2}\right]\left[L^1\right]=\left[M^1{L}^2{T}^{-2}\right]$
Torque,$\tau =\overrightarrow{r}\times \overrightarrow{F}$ $\tau =rF\sin \theta$

Dimensional formula for torque, $=\left[M^1{L}^1{T}^{-2}\right]\left[L^1\right]=\left[M^1{L}^2{T}^{-2}\right]$

Step 2: Find the dimensional formula for angular momentum and Plank’s constant.

Formula used: $h=\frac{E}{v}$
Angular momentum, $L=mvr$
Dimensional formula for angular momentum,
= $\left[M^1\right]\left[L^1{T}^{-1}\right]\left[L^1\right]=\left[M^1{L}^2{T}^{-1}\right]$

Plank's constant, $h=\frac{E}{v}$
Dimensional formula for Plank’s constant,
$=\frac{\left[M^1{L}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[M^1{L}^2{T}^{-1}\right]$

Step 3: Find the dimensional formula for tension and surface tension.

Tension, F = ma
Dimensional formula for tension,
= $\left[M^1\right]\left[L^1{T}^{-2}\right]=\left[M^1{L}^1{T}^{-2}\right]$
Surface tension, $T=\frac{F}{l}$
Dimensional formula for surface tension,
$=\frac{\left[M^1{L}^1{T}^{-2}\right]}{\left[L^1\right]}=\left[M^1{L}^0{T}^{-2}\right]$

Step 4: Find the dimensional formula for impulse and linear momentum.

impulse, $J=F\Delta t$
Dimensional formula for impulse,
$=\left[M^1{L}^1{T}^{-2}\right]\left[T^1\right]=\left[M^1{L}^1{T}^{-1}\right]$
Linear momentum, $P=mv$
Dimensional formula for linear momentum,
= $\left[M^1\right]\left[L^1{T}^{-1}\right]=\left[M^1{L}^1{T}^{-1}\right]$

Final answer: (c) Tension and surface tension