Question

Which of the following physical quantities do not have the same dimensional formula?
(A). Work and torque
(B). Angular momentum and planck's constant
(C). Tension and surface tension
(D). Impulse and linear momentum

Answer 1

Hint: For every pair of quantities, find the dimensional formula and compare them. Find a pair for which the formulae don’t match. Use the dimensional formulae of the basic quantities that are used to derive the given quantities.

Formula used: Formula for work, torque, angular momentum, planck's constant, impulse, linear momentum, tension and surface tension are used from their basic definitions:
$\begin{align}
  & \text{work}=\text{force}\times \text{displacement} \\
 & \text{torque}=\text{Force}\times \text{momentum arm} \\
 & \text{angular momentum}=\text{Moment of Inertia}\times \text{angular velocity} \\
 & \text{Plank }\!\!'\!\!\text{ s constant}=\dfrac{\text{Energy}}{\text{Frequency}} \\
 & \text{impulse}=\text{Force}\times \text{time} \\
 & \text{linear momentum}=\text{mass}\times \text{velocity} \\
 & \text{tension}=\text{force} \\
 & \text{surface tension}=\dfrac{\text{Force}}{\text{Length}} \\
\end{align}$

Complete step by step answer:
Every quantity can be expressed in the terms of the following seven dimensions
Dimension Symbol
Length L
Mass M
Time T
Electric charge Q
Luminous intensity C
Temperature K
Angle None

For option A.:
The dimensional formula for work is
$\begin{align}
  & \text{work}=\text{force}\times \text{displacement} \\
 & \left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right] \\
\end{align}$
The dimensional formula for torque is
\[\begin{align}
  & \text{torque}=\text{Force}\times \text{momentum arm} \\
 & \left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right] \\
\end{align}\]
These match, therefore this is not the correct answer.
For option B.:
The dimensional formula for angular momentum is
$\begin{align}
  & \text{angular momentum}=\text{Moment of Inertia}\times \text{angular velocity} \\
 & \left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right] \\
\end{align}$
The dimensional formula for planck's constant is
\[\begin{align}
  & \text{Plank }\!\!'\!\!\text{ s constant}=\dfrac{\text{Energy}}{\text{Frequency}} \\
 & \left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right] \\
\end{align}\]
These match, therefore this is not the correct answer.
For option D.:
The dimensional formula for impulse is
\[\begin{align}
  & \text{impulse}=\text{Force}\times \text{time} \\
 & \left[ {{M}^{1}}{{L}^{1}}{{T}^{-1}} \right] \\
\end{align}\]
The dimensional formula for work is
\[\begin{align}
  & \text{linear momentum}=\text{mass}\times \text{velocity} \\
 & \left[ {{M}^{1}}{{L}^{1}}{{T}^{-1}} \right] \\
\end{align}\]
These match, therefore this is not the correct answer.
Whereas, for option .C:
The dimensional formula for tension is
\[\begin{align}
  & \text{tension}=\text{force} \\
 & \left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right] \\
\end{align}\]
The dimensional formula for surface tension is
\[\begin{align}
  & \text{surface tension}=\dfrac{\text{Force}}{\text{Length}} \\
 & \left[ {{M}^{1}}{{L}^{0}}{{T}^{-2}} \right] \\
\end{align}\]
Therefore, the correct answer to this question is option C. Tension and surface tension.

Note: A mistake that a student can commit in a rush, is to think the dimensions of surface tension and tension are the same because their names suggest that. Whereas, that is not true. As seen in the solution, tension has the unit of force. Whereas, surface tension has the unit of force per unit length.