In a binary star system consisting of two stars of equal mass, where is the gravitational potential equal to zero? a) exactly halfway between the stars b) along a line bisecting the line connecting the stars c) infinitely far from the stars d) none of the above

The gravitational potential at a point in space due to a single point mass is given by: \(V(r) = \frac{-GM}{r}\) Where \(V(r)\) is the gravitational potential, \(G\) is the gravitational constant, \(M\) is the mass of the point mass, and \(r\) is the distance from the mass to the point where the potential is being measured. The gravitational potential due to a system of multiple point masses can be found using the principle of superposition, by simply adding the potentials due to each mass at the point of interest.

In the case of a binary star system with two stars of equal mass, we have: \(V_{total}(r) = V_1(r) + V_2(r)\) Where \(V_1(r)\) and \(V_2(r)\) are the gravitational potentials due to the two stars. Since the stars have equal mass, \(M_1 = M_2 = M\), and we can write: \(V_{total}(r) = \frac{-GM}{r_1} + \frac{-GM}{r_2}\) We are looking for regions where the total gravitational potential is equal to zero, so we set \(V_{total}(r)\) to 0: \(0 = \frac{-GM}{r_1} + \frac{-GM}{r_2}\)

In the equation \(0 = \frac{-GM}{r_1} + \frac{-GM}{r_2}\), the total gravitational potential will be equal to zero if and only if the individual potentials are equal: \(\frac{-GM}{r_1} = \frac{-GM}{r_2}\) This implies that \(r_1 = r_2\), which means that the point where the potential is zero must be equidistant from both stars. Let's compare this conclusion to the given answer choices: a) exactly halfway between the stars: This choice aligns with our conclusion that the point must be equidistant from both stars, which means it would be halfway between the stars. b) along a line bisecting the line connecting the stars: It is not enough to say that the point must lie along the line connecting the stars since this line contains points that are not equidistant from the stars. c) infinitely far from the stars: This choice does not align with our conclusion that the point must be equidistant from both the stars, and it would not be physically possible for the potential to be zero at an infinite distance from them. d) none of the above: Since option (a) matches our conclusion, this answer choice is incorrect. Overall, the correct answer is: a) exactly halfway between the stars