Question

Which of the following pairs of physical quantities have the same dimensions?
A. Force and work
B. Torque and energy
C. Torque and power
D. Force and torque

Answer 1

Basically we have seven physical quantities i.e. mass, length, time, temperature, luminous intensity, electric current, and mole. But mostly we use only first three physical quantity i.e. Mass, Length, Time. Dimension represents the powers of fundamental physical quantities for example $\left[ {{{\rm{M}}^a}{{\rm{L}}^b}{{\rm{T}}^c}} \right]$ where a, b, c are dimensions.

Complete step by step answer:
We know that Force = mass × acceleration
$\therefore {\rm{F = }}\left[ {\rm{M}} \right] \times \left[ {{\rm{L}}{{\rm{T}}^{ - 2}}} \right]$
$\Rightarrow {\rm{F = }}\left[ {{\rm{ML}}{{\rm{T}}^{ - 2}}} \right]$
Now, work = Force $\times$ Displacement
$\therefore {\rm{W = }}\left[ {{\rm{ML}}{{\rm{T}}^{ - 2}}} \right] \times \left[ {\rm{L}} \right]$
$\Rightarrow {\rm{W = }}\left[ {{\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}} \right]$
Here work and energy have the same unit, so that the dimensional formula is also the same.
Energy
$\therefore {\rm{E = }}\left[ {{\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}} \right]$

Dimensions of power = Energy/Time
$\therefore {\rm{P = }}\dfrac{{\left[ {{\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}} \right]}}{{\left[ {\rm{T}} \right]}}$
(We put here dimensional formula of energy from above energy dimensional formula)
$\Rightarrow {\rm{P = }}\left[ {{\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 3}}} \right]$
And the dimensional formula of torque = Force × Perpendicular distances
$\therefore \tau = \left[ {{\rm{ML}}{{\rm{T}}^{ - 2}}} \right] \times \left[ {\rm{L}} \right]$
$\Rightarrow \tau = \left[ {{\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}} \right]$

When we see above all the dimensional formulas, we find out that the (B) option is correct because torque and energy have same dimensional formula.

Additional information:
The expression which tells us how the fundamental quantities which are present in a physical quantity is known as a dimensional formula. It also helps us to derive units from one system of the unit to another system. Dimensional formulae are always being written in square brackets.
If X be the physical quantity here, than the equation is represented as ${\rm{X = }}\left[ {{{\rm{M}}^a}{{\rm{L}}^b}{{\rm{T}}^c}} \right]$ this is the dimensional formula and the dimensions are (a, b, c.)

Note:
There are some limitations of dimensional analysis are:
i) This method cannot be applicable to a trigonometric, exponential, and logarithmic function.
ii) If the physical quantities are dependent on more than three physical quantities than it is difficult to find the dimensional formula.
iii) Dimensionless quantities cannot be determined by this method. It can be found by theory or some experimental work.