JEE PYQ: Units & Measurements - Question ID a9a1dea5f41a (JEE Main 2026)

Core Formula & Concept:

In a solid metallic bar, the speed \( v \) of a longitudinal wave is governed by the elastic and inertial properties of the material. The fundamental relation is:

$v = \sqrt{\frac{Y}{\rho}}$ where

• \( Y \) is the Young’s modulus of the material (a measure of its stiffness),
• \( \rho \) is the density of the material (a measure of its inertia).

This formula tells us that the wave speed increases with the square root of the stiffness and decreases with the square root of the density.

Step-by-Step Derivation:

Step 1: Express the original speed

Given the original speed \( v_0 = 400 \) m/s, we have $v_0 = \sqrt{\frac{Y_0}{\rho_0}}.$

Step 2: Introduce the fractional changes

The density is increased by 0.5 %, so the new density is $\rho = \rho_0 \times (1 + 0.005) = 1.005\,\rho_0.$ The Young’s modulus is increased by 1 %, so the new modulus is $Y = Y_0 \times (1 + 0.01) = 1.01\,Y_0.$

Step 3: Compute the new speed

The new speed \( v \) is $v = \sqrt{\frac{Y}{\rho}} = \sqrt{\frac{1.01\,Y_0}{1.005\,\rho_0}} = \sqrt{\frac{1.01}{1.005}}\;\sqrt{\frac{Y_0}{\rho_0}} = \sqrt{\frac{1.01}{1.005}}\;v_0.$

Step 4: Approximate the square root

We use the binomial approximation for small \( x \): $\sqrt{1 + x} \approx 1 + \frac{x}{2}.$ Write $\frac{1.01}{1.005} = \frac{1 + 0.01}{1 + 0.005} \approx (1 + 0.01)\,(1 - 0.005) = 1 + 0.01 - 0.005 - 0.00005 \approx 1 + 0.00495.$ Hence $\sqrt{\frac{1.01}{1.005}} \approx \sqrt{1 + 0.00495} \approx 1 + \frac{0.00495}{2} = 1 + 0.002475.$ Therefore $v \approx (1 + 0.002475)\,v_0 = 1.002475 \times 400\;\text{m/s} \approx 400.99\;\text{m/s}.$ Rounding to the nearest integer gives approximately 401 m/s.

Common Traps & Exam Tip:

1. Incorrect sign in the density change: Some students mistakenly subtract the 0.5 % instead of adding it, leading to an overestimate of the speed.
2. Forgetting the square root: A direct ratio of the percentage changes (1 % minus 0.5 % = 0.5 %) without taking the square root gives 0.25 % change, which is wrong.
3. Approximation errors: Using a linear approximation for the ratio without expanding the denominator can introduce small errors. Always expand \( \frac{1+x}{1+y} \) as \( (1+x)(1-y) \) for small \( x,y \).