The Earth has mass M E and average radius R E . The Moon has mass M M and the av

Question

The Earth has mass ME and average radius RE. The Moon has mass MM and the average distance from the center of mass of the moon to the center of mass of the Earth is DM.

a) find an expression for the moment of inertia I_E of the earth on its axis in terms of the given parameters

b) find an expression for the moment of inertia of the moon I_M in its orbit about the earth, you may assume the diameter of the moon is very small relative to the earth-moon distance

c) calculate for distance D-S in meters that the moon would have to be from the earth so that the moon's moment of inertia I_M about the earth is equal to the moment of inertia I_E of the earth on its axis

Explanation / Answer

a)Moment of inertial of eath is same as that of a sphere of radius R_E and mass M_E

therefore its 2/5 (M_E)*(R_I)^2
b)Considering the question here. we must take moon as a point object . Thus moment of inertial becomes I_M= M_M * (D_M)^2.
This is just the simple case of point object revolving around an axis.

c)Given M_M *(D_M)^2=2/5 M_E * (R_I)^2. Therefore

D_M = (2/5 * M_E/M_M)^0.5 *R_I

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