JEE PYQ: Units & Measurements - Question ID e2e36b1b5242 (JEE Main 2021)

Core Formula & Concept:

In dimensional analysis, we express any physical quantity in terms of fundamental (base) units. Here, the fundamental units are:

• Time: $t$
• Velocity: $v$
• Angular momentum: $l$

We need to express the dimension of mass ($m$) in terms of these three fundamental units. Key formulas used:

1. Velocity: $v = \frac{\text{displacement}}{\text{time}} \Rightarrow [v] = \frac{L}{T}$
2. Angular momentum: $l = mvr \Rightarrow [l] = [m] \cdot [v] \cdot [r]$, where $r$ is distance (dimension $L$)
3. Mass is a base quantity in the standard SI system, but here we must express it in terms of $t$, $v$, and $l$.

Step-by-Step Derivation:

Step 1: Express standard dimensions in terms of $t$, $v$, and $l$
We know:

• $[v] = \frac{L}{T} \Rightarrow L = v \cdot T$
• $[l] = m \cdot v \cdot r \Rightarrow [l] = m \cdot v \cdot L$ (since $r$ has dimension $L$)

Substitute $L = v \cdot t$ into the angular momentum dimension: $[l] = m \cdot v \cdot (v \cdot t) = m \cdot v^2 \cdot t$ Step 2: Solve for mass $m$
Rearrange the equation to isolate $m$: $m = \frac{l}{v^2 \cdot t}$ Step 3: Express dimensionally
Thus, the dimension of mass in terms of $t$, $v$, and $l$ is: $[m] = [t^{-1} \cdot v^{-2} \cdot l^{1}]$ Step 4: Match with given options
Comparing with the options:

• A: $[t^{-1} v^{1} l^{-2}]$ → Incorrect
• B: $[t^{1} v^{2} l^{-1}]$ → Incorrect
• C: $[t^{-2} v^{-1} l^{1}]$ → Incorrect
• D: $[t^{-1} v^{-2} l^{1}]$ → Correct

Common Traps & Exam Tip:

Common Mistakes:

• Students often confuse the dimension of angular momentum ($l = mvr$) and forget that $r$ (distance) must be expressed in terms of $v$ and $t$. This leads to incorrect exponents.
• Some misapply the formula for momentum ($p = mv$) instead of angular momentum, leading to wrong dimensions.
• Sign errors in exponents (e.g., writing $v^2$ instead of $v^{-2}$) are frequent.

Exam Tip:
Always start by writing the defining formula of the given fundamental units (here, $v = \frac{L}{T}$ and $l = mvr$). Then systematically eliminate standard dimensions ($M$, $L$, $T$) in favor of the new fundamental units ($t$, $v$, $l$). Double-check each algebraic step to avoid sign and exponent errors.