JEE PYQ: Units & Measurements - Question ID ccb29f5e3b19 (JEE Main 2021)

Core Formula & Concept:

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), and time ($T$). The given problem involves determining the dimensions of a quantity $P$ defined by the formula: $P = E L^2 M^{-5} G^{-2}$ where:

• $E$ is energy,
• $L$ is angular momentum,
• $M$ is mass,
• $G$ is the gravitational constant.

To find the dimensions of $P$, we first express each of these quantities in terms of $M$, $L$, and $T$.

Key dimensional formulas:

• Energy ($E$): $[E] = [M L^2 T^{-2}]$
• Angular momentum ($L$): $[L] = [M L^2 T^{-1}]$
• Mass ($M$): $[M] = [M]$
• Gravitational constant ($G$): $[G] = [M^{-1} L^3 T^{-2}]$

Step-by-Step Derivation:

Step 1: Write the dimensional formula for each quantity in the expression for $P$.
Given: $P = E \cdot L^2 \cdot M^{-5} \cdot G^{-2}$ Substitute the dimensions: $[P] = [E] \cdot [L]^2 \cdot [M]^{-5} \cdot [G]^{-2}$ Step 2: Substitute the known dimensions. $[P] = [M L^2 T^{-2}] \cdot [M L^2 T^{-1}]^2 \cdot [M]^{-5} \cdot [M^{-1} L^3 T^{-2}]^{-2}$ Step 3: Expand the exponents. $[P] = [M L^2 T^{-2}] \cdot [M^2 L^4 T^{-2}] \cdot [M]^{-5} \cdot [M^{2} L^{-6} T^{4}]$ Note: $[G]^{-2} = [M^{-1} L^3 T^{-2}]^{-2} = [M^{2} L^{-6} T^{4}]$ Step 4: Combine the dimensions by adding exponents for each fundamental dimension ($M$, $L$, $T$).

• For mass ($M$): $1 + 2 - 5 + 2 = 0$
• For length ($L$): $2 + 4 + (-6) = 0$
• For time ($T$): $-2 - 2 + 4 = 0$

Step 5: Write the final dimensional formula for $P$. $[P] = [M^0 L^0 T^0]$ Conclusion: The dimensions of $P$ are $[M^0 L^0 T^0]$, which corresponds to a dimensionless quantity. This matches option D. Common Traps & Exam Tip:

Students often make the following mistakes in such problems:

1. Incorrect dimensional formulas: Confusing the dimensions of angular momentum ($L$) with linear momentum ($M L T^{-1}$). Angular momentum has dimensions $[M L^2 T^{-1}]$. Always verify the dimensions of each quantity before substitution.
2. Sign errors in exponents: Forgetting that negative exponents in the formula (e.g., $M^{-5}$ or $G^{-2}$) reverse the sign of the dimensions. For example, $G^{-2}$ becomes $[M^2 L^{-6} T^4]$, not $[M^{-2} L^6 T^{-4}]$.
3. Arithmetic errors in combining exponents: Adding or subtracting exponents incorrectly when combining dimensions. Double-check each step to avoid simple calculation mistakes.
4. Misinterpreting dimensionless quantities: Assuming that a quantity with dimensions $[M^0 L^0 T^0]$ is invalid or trivial. In reality, such quantities are dimensionless and can represent ratios or pure numbers (e.g., fine-structure constant).

Exam Tip: Always write down the dimensions of each quantity explicitly before substituting. This minimizes errors and makes the derivation clearer. If the final dimensions simplify to $[M^0 L^0 T^0]$, the quantity is dimensionless, which is a valid and important result.