Question

Two planets having equal masses are in circular orbit around a star. Planet A has a smaller orbital radius than planet B. Which statement is true?Planet A has more kinetic energy, less potential energy, and less mechanical energy (potential plus kinetic) than planet B.Planet A has more kinetic energy, less potential energy, and more mechanical energy (potential plus kinetic) than planet B.Planet A has more kinetic energy, more potential energy, and more mechanical energy (potential plus kinetic) than planet B.Planet A and planet B have the same amount of mechanical energy (potential plus kinetic).

Answer 1

Answer:

Explanation:

To solve this, we must know two things.

First, the force of gravity acting on an orbiting object is equal to its mass times centripetal acceleration.

Second, the force of gravity between two objects is defined by Newton's law of universal gravitation: Fg = mMG/r², where Fg is the force of gravity, m and M are the masses of the objects, G is the universal constant of gravitation, and r is the distance between the objects.

Therefore:

Fg = m v²/r

mMG/r² = m v²/r

v² = MG/r

The potential energy of each planet is:

PE = mgr = m (MG/r²) r = mMG/r

The kinetic energy of each planet is:

KE = 1/2 mv² = 1/2 m (MG/r) = 1/2 mMG/r

The total mechanical energy is:

ME = PE + KE = 3/2 mMG/r

Since both planets have the same mass, the only difference is the orbital radius.  Since planet A has a smaller orbital radius, it has more potential energy, more kinetic energy, and more mechanical energy.