The mass of the moon is 7.35x10^22 kg. at some point between earth and the moon

Question

The mass of the moon is 7.35x10^22 kg. at some point between earth and the moon the force of the earths gravitational attraction on an object is cancelled by the moons force of gravitational attraction. if the distance between the earth and the moon (centre to centre) is 3.84x10^5 km calculate where this will occur relative to earth. please expain every step because i am having trouble with physics

Explanation / Answer

The distance between moon and earth d = 3.84 x 10^5 km The mass of object = m The mass of Earth = Me = 5.98 x 10^24 kg The mass of moon = Mm = 7.35 x 10^22 kg The distance of the object from earth = x The distance of the moon from object = y Here x + y = d = 3.84 x 10^5 km According to law of gravitation we have F = G m1 * m2 / r^2 So using above equation we have, F1 = G Me * m / x^2 F2 = G Mm * m / y^2 When the object in equlibrium we have F1 = F2 ==> G Me * m / x^2 = G Mm * m / y^2 Me / x^2 = Mm / y^2 5.98 x 10^24 / (d-y)^2 = 7.35 x 10^22 /y^2 ==> (d-y)^2 / y^2 = 81.36 ==> (d-y) / y = 9.02 ==> d-y = 9.02y ==> d = 10.02 y ==> y = d / 10.02 = 3.84 x 10^5 km / 10.02 = 0.3832 x 10^5 km and x + y = 3.84 x 10^5 km ==> x = 3.84 x 10^5 km - 0.3832 x 10^5 km = 3.456 x 10^5 km

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