Kinetic Theory

1. Key Concept: According to the Kinetic Theory of Gases, the average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas and depends only on the temperature.
2. Formula: The average translational kinetic energy per molecule of any ideal gas is given by: where is the Boltzmann constant and is the absolute temperature in Kelvin.
3. Analysis: The formula shows that the average kinetic energy per molecule depends solely on the temperature ( ). It does not depend on the type of gas, the mass of the molecules, or the molar mass.
4. Given Information:
   • Gas 1: Helium (He)
   • Gas 2: Hydrogen (H₂)
   • Temperature: 300 K (same for both gases)
   • Mass Information: Helium is two times heavier than H₂ (Molar mass of He ≈ 4 g/mol, Molar mass of H₂ ≈ 2 g/mol). This information is irrelevant for the average kinetic energy per molecule.
5. Comparison: Since both Helium and H₂ are at the same temperature (300 K), their average kinetic energy per molecule must be the same. Therefore, .

The correct option is (b) same as H₂.

Let's analyze each part based on the physics described:

Part 1: The Rope Wave

1. Wave Type Identification: The oscillation is applied horizontally *along* the direction of the rope (x-axis). The wave travels along the rope. When the particle oscillation direction is parallel to the wave propagation direction, the wave is defined as longitudinal.
2. Superposition and Reflection: The longitudinal wave travels to the left end, reflects (at a fixed end, a longitudinal wave typically reflects with a phase shift, but the key point is reflection occurs), and travels back to the right. The incident wave (traveling left) and the reflected wave (traveling right) have the same frequency and are confined between two fixed ends.
3. Resultant Wave: The superposition (interference) of two identical waves traveling in opposite directions in a confined medium results in a standing wave, also known as a stationary wave. This wave pattern does not propagate; it has fixed points of zero amplitude (nodes) and maximum amplitude (antinodes).

Part 2: The Simple Pendulum (Inferred from options for D and E)

1. Pendulum Period: The formula for the period ( ) of a simple pendulum (for small oscillations) is , where is the length and is the acceleration due to gravity. Notice that the mass ( ) of the pendulum bob does not appear in this formula.
2. Period vs. Mass: Therefore, the period of a simple pendulum is independent from its mass.
3. Pendulum Frequency: Frequency ( or ) is the reciprocal of the period: . The frequency depends on and . Specifically, the frequency squared ( ) is directly proportional to the ratio . While the frequency itself is proportional to , among the given options, is the quantity that fundamentally determines the frequency (along with constants).

Filling in the Blanks:

• (A) BLANK-1: The resultant wave pattern is a standing wave. Correct choice: stationary
• (B) BLANK-2: The wave involves oscillations parallel to propagation. Correct choice: longitudinal
• (C) BLANK-3: Classifying the relationship between oscillation and propagation direction. Correct choice: longitudinal
• (D) BLANK-4: The relationship between a pendulum's period and its mass. Correct choice: independent from
• (E) BLANK-5: The quantity related to pendulum frequency dependence on and . Correct choice: