Two planets of mass m are orbiting a star of mass M, as shown. The planets are i

Question

Two planets of mass m are orbiting a star of mass M, as shown. The planets are in the same orbit, with radius r, but are always at opposite ends of a diameter. Find an expression for the net gravitational force on one of the planets. Given the orbital period T, find an expression for the acceleration of one planet as it circles around the star at a radius r. Using the results of parts a) and b), find an exact expression for the orbital period T in terms of only the variables given (and some constants). In the limit of the second planet becoming smaller and smaller, does your answer for the orbital period T of the first planet in part c) make sense? Please explain.

Explanation / Answer

(a)The force acting on one of the masses m by the other two is
[GMm /r^2] + [Gmm / 4r^2]

(b) so the net force= Fnet= [GMm /r^2] + [Gmm / 4r^2]

This acts as the centripetal force and hence = Fnet= ma

ma= [GMm /r^2] + [Gmm / 4r^2]

   a= [GM /r^2] + [Gm/ 4r^2]

(c) The net force Fnet as the centripetal force and hence Fnet = mr^2

[GMm /r^2] + [Gmm / 4r^2] = mr^2

r ^3 * ^2 = G [4 M + m] / 4

= 2 / T and hence

T = 4 r ^ (3/2) / { G [4 M + m]}

(d) yes the answer still does make sense.we treat te planets as point masses. so their size doesnot change T.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.