Barron's 2nd Edition PT1 Q42

Barron's Book

42) For this question, we are given a situation where a mass is orbiting Earth. We need to find the affects of tripling the mass on the speed of the orbit. In order to understand what is happening in this situation, let’s make a diagram for the satellite.

Where F[g] is the force of gravity, and v is the speed

Please note that v is not a force, but is drawn to clarify the situation. The object is moving in a circular path while the force of gravity Is directed towards the center of Earth.

At every point, the force of gravity will be directed very close to the center of the orbit. The system is large enough that we can neglect the fact that at specific points gravity may be directed slightly away from the center of orbit. This means our force of gravity acts as our centripetal force and we can equate their equations.

Where F[c] is centripetal force, F[g] force of gravity expressed with the universal force of gravity equation, m is mass, v is speed, r is the radius of the circular orbit, G is the universal gravitation constant, and M[E] is the mass of Earth

Next, we should simplify the equation as far as possible and solve for speed(v).

Where m is mass, v is speed, r is radius of circular orbit, G is the universal gravitational constant, and M[E] is the mass of the earth

This equation clearly shows that the velocity is independent from the mass of the satellite. The velocity will only change if the mass of earth was changed, or if the radius was changed.

Answer: B