Question

Which of the following quantities have the same dimension?
A)Force and work done.
B)Momentum and impulse.
C)pressure and force.
D)Surface tension and stress.

Answer 1

Hint: Find the dimension of each and every quantity in all the options to check whether they are the same or not. We can use any one formula of each identity to find its dimension.

Complete step by step solution:
To find the dimension of a quantity, we can use any formula related to that quantity but we will use the easiest ones to save time.
Force-
from Newton’s law of motion,
$F=ma$
Dimension of force $=\left[M\right]\left[L{T}^{-2}\right]=\left[ML{T}^{-2}\right]$
Work done-
$W=F\times s$
Dimension of work$=\left[ML{T}^{-2}\right]\left[L\right]=\left[M{L}^2{T}^{-2}\right]$
Momentum-
$p=mv$
Dimension of momentum$=\left[M\right]\left[L{T}^1\right]=\left[ML{T}^{-1}\right]$
Impulse-
$I=F\times t$
Dimension of impulse$=\left[ML{T}^{-2}\right]\left[T\right]=\left[ML{T}^{-1}\right]$
Pressure-
$P={\displaystyle \frac{F}{A}}$
Dimension of pressure$={\displaystyle \frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}}=\left[M{L}^{-1}{T}^{-2}\right]$
Surface tension-
$T={\displaystyle \frac{F}{l}}$
Dimension of surface tension$={\displaystyle \frac{\left[ML{T}^{-2}\right]}{\left[L\right]}}=\left[M{T}^{-2}\right]$
Stress is the force per unit area so it has the same dimension as pressure.
From here, we can see that the dimension of momentum and impulse is equal.
So, B) is correct.

Additional information:
The dimension of a physical quantity can be expressed as a product of the basic physical dimensions such as length, mass and time, each raised to a rational power. The dimension of a physical quantity is more fundamental than some scale unit used to express the amount of that physical quantity.

Note: The dimension of quantities which are interchangeable into one another have exactly the same dimension. For example, energy can transform from one form to another. So each type of energy whether it is kinetic energy or nuclear energy, they all have the same dimension.