Newtons Law Question

Question

Calculate the distance between the center of the earth and the center of the moon at which the gravitational force exerted by the earth on an object is equal in magnitude to the force exerted by the moon on the object.

I believe that the answer is 3.46*10^8m but am not completely sure

Earth- Mass- 5.98* 10^24- Radius- 6.38*10^6

Moon-Mass- 7.36*10^22

Distance from Earth to Moon- 3.84*10^24

Answer 1

You are correct, though your distance from Earth to the Moon is incorrect.

GM_E/d² = GM_M/(3.84 x 10⁸-d)²

Divide both sides by GM_M and multiply both sides by d²

G cancels.

M_E/M_M = d²/(3.84 x 10⁸-d)²

81.5 = d²/(3.84 x 10⁸-d)²

Square root both sides

9.03 = d/(3.84 x 10⁸-d)

Multiply both sides by (3.84 x 10⁸-d)

3.47 x 10⁹ - 9.03d = d

3.47 x 10⁹ = 10.03d

[d = 3.46 x 10⁸ m]