Let\'s say for the purpose of this question that the Specific Mechanical Energy,

ID: 1418296 • Letter: L

Question

Let's say for the purpose of this question that the Specific Mechanical Energy, , of our moon in its orbit around Earth is moon. (We don't care to calculate what moon is exactly and so we won't.)

Now let's also say that our moon is replaced by a soccer ball. Obviously there is an enormous difference in the mass of the moon and the soccer ball. In fact, the mass of the moon is approximately 1,783,000,000 (or 1.783 x 109) times greater than the soccer ball. The orbit of the moon and the orbit of the soccer ball are the same but the moon is almost 2 billion times the mass of the soccer ball.

What is the Specific Mechanical Energy of the soccer ball, ball, while in lunar orbit around the Earth?

Don't tell me the value of ball, instead, select from below the soccer ball's Specific Mechanical Energy in terms of moon, the Specific Mechanical Energy of the moon in its orbit.

Explanation / Answer

assumption: the orbit is circular.

now, if the orbit's radius is R and speed of rotation is v,

then the gravitational attraction force is being balanced by centripetal force.

if mass of earth is M, mass of moon is m1 and mass of soccer ball is m2:

then for the moon,

G*M*m/R^2=m*v^2/R

==>v^2=G*M/R

so the speed of the moon/ball does not depend upon its mass.
and speed of both the objects will be the same.
as specific kinetic energy=0.5*v^2

specific kinetic eenrgy of both moon and the ball are the same.

now, specific potential energy=G*M/R

here also it is independent of mass of the object.

as specific mechanical energy =specific potential energy+specific kientic energy

and both specific potential energy and specific kinetic energy

are independent of mass of the object,

then specific mechanical energy is also independent of mass of the object.

as all other aspects (orbital radius , speed) are same for the moon and the soccer ball,

specific mechanical energy of the ball will be same as speciifc mechanical energy of the moon.

hence option 3 is correct.

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Let\'s say for the purpose of this question that the Specific Mechanical Energy,

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