Question

Identify the pair whose dimensions are equal.
A. Torque and work
B. Stress and energy
C. Force and stress
D. Force and work

Answer 1

First we need to know that there are seven fundamental dimensions in physics that are mass, length, time, temperature, electric current, amount of light and amount of matter. All the terms present in physics can be represented in these fundamental dimensions. Second we need to know all the formulas of the given terms so as to write it in its dimensional formula. Then only we can compare the terms and write the correct option.

Complete answer:
As per the problem we need to identify the pair whose dimensions are equal.Let us first check the first option: torque and work are both defined as the product of force and the perpendicular distance.Hence we can write,
$\tau = F \times r$
$\Rightarrow W = F \times r$
Where, Force = $F$, Distance = $r$, Work = $W$ and Torque = $\tau$

Now writing the dimensional formula of the terms we will get,
$F = \left[ {ML{T^{ - 2}}} \right]$
$\Rightarrow r = \left[ L \right]$
Now,
$F \times r = \left[ {ML{T^{ - 2}}} \right]\left[ L \right]$
$\Rightarrow F \times r = \left[ {M{L^2}{T^{ - 2}}} \right]$
So we can write,
$\tau = W = F \times r = \left[ {M{L^2}{T^{ - 2}}} \right]$
From the first option itself we get our correct answer.

Therefore the correct option is A.

Additional:
Stress is defined as forces divided by area.Energy is defined as the force multiplied with displacement.On checking other options:
(B) Stress and energy
Dimension of stress is $\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]$
And dimension of energy is $\left[ {{M^1}{L^2}{T^{ - 2}}} \right]$
Hence, $\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right] \ne \left[ {{M^1}{L^2}{T^{ - 2}}} \right]$

(C) Force and stress
Dimension of force is $\left[ {{M^1}{L^1}{T^{ - 2}}} \right]$
And dimension of stress is $\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]$
Hence, $\left[ {{M^1}{L^1}{T^{ - 2}}} \right] \ne \left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]$

(D) Force and work
Dimension of force is $\left[ {{M^1}{L^1}{T^{ - 2}}} \right]$
And dimension of work is $\left[ {{M^1}{L^2}{T^{ - 2}}} \right]$
Hence, $\left[ {{M^1}{L^1}{T^{ - 2}}} \right] \ne \left[ {{M^1}{L^2}{T^{ - 2}}} \right]$

Note: In second option stress and energy are different stress is related to force and area whereas energy is related to force and displacement, in third option force is related to mass and acceleration of the body whereas the stress is related between to force and area hence there dimension are different and in the last option again force is related to mass and acceleration whereas work is related to force and displacement again they are also not equal.