In the Earth?Moon system, there is a point where the gravitational forces balanc
ID: 2010325 • Letter: I
Question
In the Earth?Moon system, there is a point where the gravitational forces balance. This point is known as the L1 point where the L stands for Lagrange, a famous French mathematician. Assume that the mass of the Moon is 1/81 that of the Earth. (a) At what point, on a line between the Earth and the Moon, is the gravitational force exerted on an object by the Earth exactly balanced by the gravitational force exerted on the object by the Moon? (The distance between the Earth and the Moon is 3.844 105 km.) (b) Is this point stable or unstable? (c) Calculate the ratio of the force of gravity due to the Sun, acting on an object at this point, to the force of gravity due to Earth and, separately, to the force of gravity due to the Moon. F_sun/F _moon = ? F_SUN/F_Earth = ?
Explanation / Answer
a) G M0 m/R0^2 = G Me m/Re^2 e = earth and o = moon Re / Ro = (Me / Mo)^1/2 = (81.43)^1/2 = 9.02 Ro = .111 Re Re + .111 Re = 3.844 * 10E5 km (assuming distance between centers) c) [G Ms m / Rs^2 / G Me m / Re^2] = (Ms/Me) * (Re^2/Rs^2) [(G Ms m / Rs^2) / (G Mo m / Ro^2)] = (Ms/Mo) * (Ro^2/Rs^2) Mo = .01228 Me and Ms = 329390 Me Rs = 1.49 * 10E8 km b) from gravitational forces alone (unstable), however I would have to guess stable due to kinematics. When the moon is between the earth and the sun, the gravitational force of the sun is greater than that of the earth - but we still have the moon.