Final answer:
The point at which the gravitational field between Earth and the Moon is zero is at a distance of d/10 from the center of Earth.The correct option is option d.
Step-by-step explanation:
To find the point at which the gravitational field between the Earth and the Moon is zero, we use the principle of superposition of gravitational forces. Since the mass of the Earth (Me) is 81 times the mass of the Moon (Mm), and the distance between their centers is d, we can set up an equation that equates the gravitational forces due to the Earth and the Moon at a distance x from the Earth's center.
The gravitational force exerted by Earth at distance x is given by Fe = (G*Me*m)/(x2), and the force exerted by the Moon at distance (d-x) from the Moon is given by Fm = (G*Mm*m)/((d-x)2) where m is the mass of the object experiencing the force, and G is the gravitational constant.
Setting these two forces equal, since at the point where the gravitational field is zero they must cancel each other out, yields:
(G*Me*m)/(x2) = (G*Mm*m)/((d-x)2).
Canceling common terms and substituting Mm with Me/81, as given by the ratio of masses, we are left with:
1/x2 = 1/(81*(d-x)2).
Taking the square root of both sides and solving for x yields the solution:
x = d / (1+sqrt(81)) = d / (1+9) = d / 10.
So the correct answer is (d) d/10, meaning that at a distance of d/10 away from the center of the Earth, the gravitational field due to both Earth and the Moon will be zero.