Answer to Question #249468 in Physics for Jubin

Question #249468
A spacecraft is going from the earth to the moon. Find such a position from the earth over there The gravitational force is zero. Given
Earth mass = 6.0 × 10^24 kg. Moon mass = 74 × 10^22 kg; The distance between the center of the earth and the center of the moon = 3.8 x 10^8m
1
Expert's answer
2021-10-11T08:55:19-0400

Lets set the origin at the Earth. Then the force at distance rr from the Earth is given as follows:


F1=GMEmr2F_1 = G\dfrac{M_Em}{r^2}

where G=6.67×1011m3kgs2G = 6.67\times 10^{-11}\dfrac{m^3}{kg\cdot s^2} is the gravitational constant, MEM_E is the mass of the Earth, mm is the mass of the ship.

The force from the Moon at the same point is given as follows:


F2=GMMm(dr)2F_2 = G\dfrac{M_Mm}{(d-r)^2}

where dd is the distance between the center of the Earth and the center of the Moon. The total gravitational force is 0 if:


F1=F2F_1=F_2\\GMEmr2=GMMm(dr)2MEr2=MM(dr)2G\dfrac{M_Em}{r^2} = G\dfrac{M_Mm}{(d-r)^2}\\ \dfrac{M_E}{r^2} = \dfrac{M_M}{(d-r)^2}

Expressing rr, find:


(MEMM)r22dMEr+MEd2=0(M_E-M_M)r^2 - 2dM_Er + M_Ed^2 = 0

Substituting the values and solving the quadratic equation, obtain:


r2.8×108mr\approx 2.8\times 10^8m

Answer. 2.8×108m2.8\times 10^8m.


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