JEE PYQ: Units & Measurements - Question ID 0f160a34a357 (JEE Main 2021)

In dimensional analysis, we express any physical quantity in terms of the chosen fundamental quantities. Here, the fundamental quantities are force ($F$), length ($L$), and time ($T$). The goal is to express the dimension of density in terms of $F$, $L$, and $T$.

Density ($\rho$) is defined as mass per unit volume: $\rho = \frac{\text{mass}}{\text{volume}}.$

To express density in terms of $F$, $L$, and $T$, we must first express mass in terms of these fundamental quantities. We know from Newton’s second law that: $F = ma \implies \text{mass} = \frac{F}{a}.$ Acceleration ($a$) is the rate of change of velocity, and velocity is the rate of change of displacement. Thus: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2} \implies [a] = \frac{L}{T^2}.$ Substituting this into the expression for mass: $\text{mass} = \frac{F}{a} \implies [\text{mass}] = \frac{F}{L/T^2} = F L^{-1} T^2.$

Volume is a derived quantity in terms of length: $[\text{volume}] = L^3.$ Thus, the dimension of density is: $[\rho] = \frac{[\text{mass}]}{[\text{volume}]} = \frac{F L^{-1} T^2}{L^3} = F L^{-4} T^2.$

1. Express mass in terms of $F$, $L$, and $T$: From Newton’s second law, $F = ma$, we get: $m = \frac{F}{a}.$ Since acceleration $a$ has dimensions $\frac{L}{T^2}$, we substitute: $[m] = \frac{F}{L/T^2} = F L^{-1} T^2.$
2. Express volume in terms of $L$: Volume is length cubed: $[\text{volume}] = L^3.$
3. Express density in terms of $F$, $L$, and $T$: Density is mass per unit volume: $[\rho] = \frac{[m]}{[\text{volume}]} = \frac{F L^{-1} T^2}{L^3} = F L^{-4} T^2.$
4. Match with the given options: The derived dimension of density is $F L^{-4} T^2$, which corresponds to option A.

Students often make the following mistakes in this question:

• Incorrect expression for mass: Some students forget that mass is derived from $F = ma$ and incorrectly assume mass is a fundamental quantity. This leads to errors in expressing density.
• Miscounting exponents: A common error is miscounting the exponents of $L$ when dividing mass by volume. For example, writing $L^{-3}$ instead of $L^{-4}$.
• Ignoring time dimension: Some students overlook the $T^2$ term in the dimension of mass, leading to incorrect options like $F L^{-4}$.

Exam Tip: Always start by expressing all derived quantities in terms of the given fundamental quantities. Double-check each step, especially when dealing with exponents, to avoid careless mistakes.