Both the sun and the moon exert gravitational forces on the oceans of the earth, causing tides.

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Both the sun and the moon exert gravitational forces on the oceans of the earth, causing tides.

(a) Show that the ratio of the force exerted by the sun to that exerted by the moon is MSr2m/Mmr2S, where MS and Mm are the masses of the sun and moon and rs and rm are the distances from the earth to the sun and to the moon. Evaluate this ratio.

(b) Even though the sun exerts a much greater force on the oceans than the moon does, the moon has a greater effect on the tides because it is the difference in the force from one side of the earth to the other that is important. Differentiate the expression F = Gm1m2/r2 to calculate the change in F due to a small change in r. Show that dF/F = (-2 dr)/r.

(c) During one full day, the rotation of the earth can cause the distance from the sun or moon to an ocean to change by at most the diameter of the earth. Show that for a small change in distance, the change in the force exerted by the sun is related to the change in the force exerted by the moon by

M.r3 ΔΕ. AF M,

and calculate this ratio.

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