Question

Match the column. [Given: mass of sun = $M_s$ | mass of earth = $M_e$ | Radius of earth = R | Distance between the Sun and the Earth = a]

List-I		List-II
(A)	Kinetic energy of Earth	(I)	$-\frac{GM_sM_e}{a}$
(B)	Potential energy of Earth and Sun	(II)	$\frac{GM_sM_e}{2a}$
(C)	Total energy of Earth and Sun	(III)	$\frac{GM_e}{ R}$
(D)	Escape energy from surface of Earth per unit mass	(IV)	$\frac{GM_sM_e}{2a}$

Choose the correct answer from the options given below:

• A. A →3,B →1,C→ 2, D → 4
• B. A → 1,B → 2,C→ 4,D→ 3
• C. A → 2, B →1, C → 4,D → 3
• D. A → 2, B → 1, C → 3,D → 4

Answer 1

The Correct Option is C

Let's analyze the problem and establish the correct relationships using physics concepts, specifically in the context of celestial mechanics and gravitational physics.

1. We start by identifying each term given in List-I and aligning it with the correct expression from List-II:
2. Kinetic Energy of Earth (A): The kinetic energy of a body in orbit can be given by the formula: T = \frac{1}{2}\frac{GM_sM_e}{a} This matches with option (II) \frac{GM_sM_e}{2a}.
3. Potential Energy of Earth and Sun (B): The gravitational potential energy between two masses is given by: U = -\frac{GM_sM_e}{a} This directly matches with the expression in (I) -\frac{GM_sM_e}{a}.
4. Total Energy of Earth and Sun (C): The total mechanical energy of an orbiting system is: E = T + U = -\frac{GM_sM_e}{2a} This corresponds with (IV) -\frac{GM_sM_e}{2a}.
5. Escape Energy from Surface of Earth per unit mass (D): The escape velocity energy per unit mass from a celestial body is described by: E_{\text{escape}} = \frac{GM_e}{R} This relationship corresponds with (III) \frac{GM_e}{R}.

By matching the expressions with the correct terms, we arrive at the pairing:

A → 2, B → 1, C → 4, D → 3

Thus, the correct choice is A → 2, B →1, C → 4, D → 3.

For deeper insight, remember these concepts:

• Gravitational potential energy is typically negative because it's calculated with the assumption that the force is attractive.
• In a stable orbit, the kinetic energy is determined by balancing gravitational attraction with rotational motion.
• The escape energy is the energy needed per unit mass to leave a celestial body's gravitational influence.