JEE PYQ: Units & Measurements - Question ID 323c2024fd2d (JEE Main 2019)

Young’s modulus ($Y$) is defined as the ratio of tensile stress to tensile strain within the elastic limit of a material. Mathematically, $Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} = \frac{F \cdot L}{A \cdot \Delta L}$ where

• $F$ = applied force,
• $A$ = cross-sectional area,
• $L$ = original length,
• $\Delta L$ = change in length.

In this problem, however, we are asked to express the dimensions of $Y$ in a new system where speed ($V$), acceleration ($A$), and force ($F$) are treated as fundamental units. We must therefore express each of the SI base dimensions ($M$, $L$, $T$) in terms of $V$, $A$, and $F$.

1. Express mass ($M$), length ($L$), and time ($T$) in terms of $V$, $A$, and $F$.
   We know the following relations:
   • $F = M \cdot A$ ⇒ $M = \frac{F}{A}$
   • $A = \frac{V}{T}$ ⇒ $T = \frac{V}{A}$
   • $V = \frac{L}{T}$ ⇒ $L = V \cdot T = V \cdot \frac{V}{A} = \frac{V^2}{A}$
2. Write the SI dimensions of Young’s modulus in terms of $M$, $L$, $T$.
   $[Y] = [M L^{-1} T^{-2}]$
3. Substitute the expressions for $M$, $L$, and $T$ into $[Y]$.
   $$\begin{aligned} [Y] &= \Bigl(\frac{F}{A}\Bigr) \cdot \Bigl(\frac{V^2}{A}\Bigr)^{-1} \cdot \Bigl(\frac{V}{A}\Bigr)^{-2} \\ &= \frac{F}{A} \cdot \frac{A}{V^2} \cdot \frac{A^2}{V^2} \\ &= F \cdot A^{1-1+2} \cdot V^{-2-2} \\ &= F \cdot A^{2} \cdot V^{-4} \end{aligned}$$
4. Compare with the given options.
   The derived dimension is $V^{-4} A^{2} F$, which matches option C.

1. Incorrect substitution of $L$ or $T$.
   Many students forget that $L = V \cdot T$ and then substitute $T = V/A$ to get $L = V^2/A$. Skipping this step leads to wrong exponents on $V$ and $A$.
2. Sign errors in exponents.
   When inverting terms like $(V^2/A)^{-1}$, it is easy to misplace a negative sign. Always double-check each exponent algebraically.
3. Confusing acceleration $A$ with area $A$.
   In the problem statement, $A$ denotes acceleration, not area. Students sometimes mix these up, especially when the same symbol appears in the formula for Young’s modulus.

Exam Tip: Before jumping into the algebra, write down the three fundamental relations ($M = F/A$, $T = V/A$, $L = V^2/A$) clearly. This keeps the substitution error-free.