The mass of the earth is 6.00 ×10^24 kg and that of the moon is 7.40 ×10^22 kg. The constant o

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The mass of the earth is 6.00 ×10^24 kg and that of the...

The mass of the earth is $6.00 \times 10^{24} \mathrm{~kg}$ and that of the moon is $7.40 \times 10^{22} \mathrm{~kg}$. The constant of gravitation $\mathrm{G}=6.67 \times 10^{-11} \mathrm{Nm}^{2} \mathrm{~kg}^{-2}$. The potential energy of the system is $-7.79 \times 10^{28}$ joules the mean distance between the earth and moon is $=$ meter. (A) $3.80 \times 10^{8}$ (B) $3.37 \times 10^{8}$ (C) $7.60 \times 10^{8}$ (D) $1.90 \times 10^{2}$

The mass of the earth is $6.00 \times 10^{24} \mathrm{~kg}$ and that of the moon is $7.40 \times 10^{22} \mathrm{~kg}$. The constant of gravitation $\mathrm{G}=6.67 \times 10^{-11} \mathrm{Nm}^{2} \mathrm{~kg}^{-2}$. The potential energy of the
system is $-7.79 \times 10^{28}$ joules the mean distance between the earth and moon is $=$ meter.
(A) $3.80 \times 10^{8}$
(B) $3.37 \times 10^{8}$
(C) $7.60 \times 10^{8}$
(D) $1.90 \times 10^{2}$

Key Concepts

Two-Body Systems in Gravitational Contexts

When analyzing systems involving two interacting masses, such as planetary bodies, the interaction is simplified by treating the system as a two-body problem. This approach allows the use of gravitational potential energy and force equations to determine parameters like separation distance, orbital characteristics, and the energy dynamics of the system.

Solving for Distance Using Energy Equations

In many gravitational problems, knowing the total energy of the system allows one to solve for the distance between interacting bodies. By rearranging the gravitational potential energy formula, one can determine the separation distance when the masses and the energy of the system are known. This technique is widely used in astrophysics and orbital mechanics to infer spatial parameters from energetic measurements.

Gravitational Potential Energy

This is the energy associated with the configuration of two masses under the influence of gravity. It is given by the formula U = -G m1 m2 / r, where G is the gravitational constant, m1 and m2 are the masses, and r is the separation distance. The negative sign indicates that work would be required to separate the masses to an infinite distance, reflecting their attractive interaction.

Newton's Law of Universal Gravitation

This fundamental law describes the gravitational attraction between any two masses. It states that the force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them. This law underpins the derivation of the gravitational potential energy expression and is essential for solving problems involving celestial bodies.

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