Question

an object in space is placed between two planets as shown. planet a has a mass m and the distance from the center of planet a to the object is d. planet b has a mass 2m and the distance from the center of planet b to the object is 2d. upon releasing the object from rest, where will the object move and how will the potential energy of the two-planet - object system change?
a the object will move toward planet a and the gravitational potential energy of the system will decrease.
b the object will move toward planet a and the gravitational potential energy of the system will increase.
c the object will move toward planet b and the gravitational potential energy of the system will decrease.
d the object will move toward planet b and the gravitational potential energy of the system will increase.

Explanation:

Step1: Calculate the gravitational force from each planet

According to Newton's law of universal gravitation \(F = G\frac{Mm}{r^{2}}\). Let the mass of the object be \(m_{0}\).
The gravitational force exerted by Planet \(A\) on the object is \(F_{A}=G\frac{mm_{0}}{d^{2}}\).
The gravitational force exerted by Planet \(B\) on the object is \(F_{B}=G\frac{(2m)m_{0}}{(2d)^{2}}=G\frac{2mm_{0}}{4d^{2}} = G\frac{mm_{0}}{2d^{2}}\).
Since \(F_{A}>F_{B}\), the net force is towards Planet \(A\), so the object will move towards Planet \(A\).

Step2: Analyze the change in gravitational potential energy

The gravitational potential energy of a two - body system is \(U=-G\frac{Mm}{r}\).
When the object moves towards Planet \(A\) (a more massive body in the direction of the net force), work is done by the gravitational force.
According to the work - energy theorem \(W =-\Delta U\). When work is done by the gravitational force (\(W> 0\)), \(\Delta U=U_{f}-U_{i}<0\), so the gravitational potential energy of the system decreases.

Answer:

A. The object will move toward Planet A and the gravitational potential energy of the system will decrease.