JEE PYQ: Units & Measurements - Question ID d96777fd9e66 (JEE Main 2023)

Core Formula & Concept:

In dimensional analysis, we express any physical quantity in terms of fundamental (base) quantities. Normally, mass ($M$), length ($L$), and time ($T$) are treated as fundamental. However, in this problem, force ($F$), velocity ($V$), and time ($T$) are given as the fundamental quantities. Our goal is to express the dimensional formula of density in terms of $F$, $V$, and $T$.

Key formulas and concepts used:

• Density ($\rho$) is defined as mass per unit volume: $\rho = \frac{m}{V}$. Its usual dimensional formula is $M L^{-3}$.
• Force ($F$) is related to mass and acceleration by Newton’s second law: $F = m a$. Since acceleration is velocity per unit time, $a = \frac{V}{T}$, so $F = m \frac{V}{T}$.
• Volume ($V$) is length cubed: $V = L^3$. Since velocity is length per unit time, $V = \frac{L}{T}$, we can express length as $L = V T$.

Step-by-Step Derivation:

Step 1: Express mass ($m$) in terms of $F$, $V$, and $T$.

From $F = m \frac{V}{T}$, we solve for mass: $m = \frac{F T}{V}$ Thus, the dimensional formula of mass in terms of $F$, $V$, and $T$ is: $[m] = F T V^{-1}$

Step 2: Express volume ($V$) in terms of $V$ and $T$.

Volume is length cubed: $V = L^3$. Since velocity is $V = \frac{L}{T}$, we get $L = V T$. Therefore: $V = L^3 = (V T)^3 = V^3 T^3$ Thus, the dimensional formula of volume is: $[V] = V^3 T^3$

Step 3: Express density ($\rho$) in terms of $F$, $V$, and $T$.

Density is mass per unit volume: $\rho = \frac{m}{V}$ Substitute the expressions for $m$ and $V$: $\rho = \frac{F T V^{-1}}{V^3 T^3} = F T V^{-1} V^{-3} T^{-3} = F V^{-4} T^{-2}$ Thus, the dimensional formula of density in terms of $F$, $V$, and $T$ is: $[\rho] = F V^{-4} T^{-2}$

Step 4: Match with the given options.

The derived formula $F V^{-4} T^{-2}$ matches option D.

Common Traps & Exam Tip:

Students often make the following mistakes:

• Incorrect expression for mass: Some forget to express mass in terms of force and velocity, leading to wrong substitutions.
• Volume miscalculation: Confusing velocity ($V$) with volume ($V$) can cause errors. Always use distinct symbols or clarify context.
• Sign errors in exponents: Misapplying negative exponents while simplifying $V^{-1} V^{-3}$ or $T T^{-3}$ is common.
• Overcomplicating the problem: Trying to introduce unnecessary quantities like acceleration or energy instead of sticking to the given fundamental quantities.

Exam Tip: Always start by expressing the usual dimensional formula of the quantity (here, density as $M L^{-3}$), then substitute the new fundamental quantities step-by-step. This systematic approach minimizes errors.