How Do Mass Differences Affect Satellites with Identical Orbits?

[Temper888]

Q) Suppose two satellites are orbiting the Earth with the same orbital radius and the same period.However, one satellite is 20 times more massive than the other.This is possible, because.
A) the gravitational force is inversely proportional to r^2.
B) the gravitational force is proportional to mass.

The answer is B. The explanation as follows:B says that the mass of a body increases, the force of gravity on that body increases proportionately. However, since F=ma, the net force on the satellite also increases proportionality with mass, and these two effects cancel each other out.

I do not understand where did the F=ma come from and how does it cancel the force of gravity?

 

[gneill]

Q) Suppose two satellites are orbiting the Earth with the same orbital radius and the same period.However, one satellite is 20 times more massive than the other.This is possible, because.
A) the gravitational force is inversely proportional to r^2.
B) the gravitational force is proportional to mass.

The answer is B. The explanation as follows:B says that the mass of a body increases, the force of gravity on that body increases proportionately. However, since F=ma, the net force on the satellite also increases proportionality with mass, and these two effects cancel each other out.

I do not understand where did the F=ma come from and how does it cancel the force of gravity?

Hi Temper888. Welcome to Physics Forums.

It's not a matter of the force of gravity being canceled so much as the effects inertia scaling with the effects of gravitational force. This is known as the equivalence principle.

As the mass of the satellite grows larger so does its inertia. As you may recall, inertia is the resistance of a given mass to acceleration. When you apply some force F to mass M, the result is acceleration A. F = M*A.

In the equation, f = m*a, the m represents the inertial mass. It is the constant of proportionality between the force applied to a given object and the resulting acceleration. When m is large it takes more force to produce a given acceleration. When m is smaller, less force is required.

In the law of universal gravitation we have f = G*M*m/r². In this case the m's represent the gravitational masses of the objects. It just so happens that the gravitational mass is equal to the inertial mass (this has been measured and confirmed in laboratory experiments). This is the basis of the Equivalence Principle.

Since gravitational mass and inertial mass have the same numerical values, an object of some mass m falling in the gravitational field of another object of mass M will have an inertia (resistance to acceleration) that scales exactly with m. These are the effects that cancel. While there is a larger gravitational force when m is larger, the resistance to acceleration (f = m*a) grows by the same amount. The result is that objects fall at the same rate regardless of their mass (see the story of Galileo dropping objects of different masses from the Tower of Piza).

 

[Temper888]

Thank you so much.I completely understand it now:)