Given: $f = 500 \text{ Hz}$, $v_S = 20 \text{ m/s}$, $v_L = 0 \text{ m/s}$ (stationary observer), $v = 340 \text{ m/s}$.

Formula (source moving towards stationary listener): $f' = f \left( \frac{v}{v - v_S} \right)$.

Apparent frequency: $f' = 500 \text{ Hz} \left( \frac{340 \text{ m/s}}{340 \text{ m/s} - 20 \text{ m/s}} \right) = 500 \text{ Hz} \left( \frac{340}{320} \right) = 500 \times 1.0625 = 531.25 \text{ Hz}$.

Given: $v_0 = 332 \text{ m/s}$ at $T_0 = 0^\circ\text{C} = 273 \text{ K}$.

We want $v_T = 1.5 v_0$.

Formula: $v \propto \sqrt{T}$ (where $T$ is absolute temperature).

$\frac{v_T}{v_0} = \sqrt{\frac{T}{T_0}} \Rightarrow 1.5 = \sqrt{\frac{T}{273 \text{ K}}}$.

Squaring both sides: $(1.5)^2 = \frac{T}{273} \Rightarrow 2.25 = \frac{T}{273}$.

$T = 2.25 \times 273 = 614.25 \text{ K}$.

Converting to Celsius: $T_C = 614.25 - 273 = 341.25^\circ\text{C}$.

Set 2 - Solutions

Q1. Answer any THREE (6 marks)

Derivation of Newton's formula for the velocity of sound in a gas and Laplace's correction:
Newton's Formula: Newton assumed that when sound travels through a gas, the compressions and rarefactions occur slowly enough for the temperature to remain constant (isothermal process). The bulk modulus for an isothermal process is $E_T = P$. So, the velocity of sound $v = \sqrt{\frac{E_T}{\rho}} = \sqrt{\frac{P}{\rho}}$.

Laplace's Correction: Laplace pointed out that compressions and rarefactions in a sound wave occur rapidly, so there is insufficient time for heat exchange with the surroundings. Thus, the process is adiabatic, not isothermal. For an adiabatic process, the bulk modulus is $E_A = \gamma P$, where $\gamma = C_p/C_v$ is the ratio of specific heats.

Therefore, the corrected formula for the velocity of sound is $v = \sqrt{\frac{\gamma P}{\rho}}$. This formula gives values that match experimental observations accurately.
Concept of beats and conditions for their formation:
Beats: When two sound waves of slightly different frequencies traveling in the same direction interfere, they produce a periodic variation in the intensity of the resultant sound. This periodic rise and fall in intensity is called beats.

Conditions for formation:
1. Two sound sources must have slightly different frequencies (frequency difference should be small, typically less than $10 \text{ Hz}$).
2. The amplitudes of the two waves should be approximately equal.
3. The waves must travel in the same direction and interfere.
Intensity of sound and its relation to amplitude:
Intensity of sound: The intensity of sound is defined as the average rate of flow of sound energy per unit area perpendicular to the direction of propagation of the wave. Its SI unit is Watts per square meter ($W/m^2$).

Relation to amplitude: The intensity of a sound wave is directly proportional to the square of its amplitude ($I \propto A^2$) and also directly proportional to the square of its frequency ($I \propto f^2$).
Stationary waves and their characteristics:
Stationary waves (or standing waves): These are formed when two identical progressive waves (same amplitude, frequency, and wavelength) traveling in opposite directions superpose. They appear to be stationary, with fixed positions of maximum and minimum displacement.

Characteristics:
1. There are points of zero displacement called nodes, and points of maximum displacement called antinodes.
2. Energy is not transferred from one region to another; it remains localize