JEE Physics 2023 Question with Solution

Given: Young's modulus $Y = 3.2 \times 10^{11} \, Nm^{-2}$ and density $\rho = 8 \times 10^3 \, kg \, m^{-3}$.

Find: The velocity of longitudinal wave in the rod.

For a solid rod, the velocity of longitudinal waves is given by

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

Substituting the given values,

$$v = \sqrt{\frac{3.2 \times 10^{11}}{8 \times 10^3}}$$ $$v = \sqrt{4 \times 10^7}$$ $$v = 6.32 \times 10^3 \, \text{m/s}$$

Therefore, the correct option is D.

Given: The rod has Young's modulus $3.2 \times 10^{11} \, Nm^{-2}$ and density $8 \times 10^3 \, kg \, m^{-3}$.

Find: Speed of the longitudinal wave.

The speed of a longitudinal wave in a solid depends on the elastic property and inertia of the material. Hence,

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

Now evaluate the ratio first:

$$\frac{Y}{\rho} = \frac{3.2 \times 10^{11}}{8 \times 10^3} = 4 \times 10^7$$

Taking square root,

$$v = \sqrt{4 \times 10^7} = 6.32 \times 10^3 \, \text{ms}^{-1}$$

Thus, the velocity of longitudinal wave in the rod is $6.32 \times 10^3 \, \text{ms}^{-1}$, so the correct option is D.