Question

If the moon is standing in the zenith (directly over us), the effective gravitational acceleration g is reduced on the surface of the earth. If the moon is in the nadir (directly under us), will g be larger?

Answer 1

NO. If the moon is in the nadir(directly under us), the effective acceleration on the earth is lesser.

Explaination.....

Let us consider M be the mass of the earth, m be the mass of the moon, R be the center-to-center distance between earth and moon, and r the radius of the earth. Let us also subscript N refer to the condition in which the moon is at the nadir, and Z the one in which the moon is at the zenith.

Let the moon and earth be on the x axis, with the moon lying to the right of the earth.

The acceleration of the earth due to the moon's gravitational attraction is g=Gm/R²

. The acceleration of a test mass in the laboratory in the two conditions is g=±GM/r²+Gm/(R±r)², where + is for N, −for Z. Subtraction gives the acceleration of the test mass relative to the laboratory,

gR=g−gE=±GMr²+Gm(R±r)2−GmR²

which becomes, with the approximation 1/(1+ϵ)²−1≈−2ϵ

,gR≈±[g0−2GmrR³]

In condition N this quantity is positive, while in condition Z it is negative. Since the apparatus rotates 180 degrees in 12 hours due to the rotation of the earth, what we actually measure is |gR|, which is the same in both cases. So finally moon either on zenith or on nadir, the effective acceleration is lesser .

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