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.
To calculate the distance at which the gravitational forces exerted by the Earth and the Moon are equal, we need to use the equation for gravitational force:
F = (G * m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant (approximately 6.674 * 10^-11 N*m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the objects.
In this case, we want to find the distance where the forces are equal. Assuming the object has a negligible mass compared to the Earth and the Moon, we can set the forces equal to each other:
G * m1 * mEarth / r^2 = G * m2 * mMoon / r^2.
Since G and r^2 are common factors on both sides of the equation, we can cancel them out:
m1 * mEarth = m2 * mMoon.
Since the mass of the object is negligible, we can ignore it, so the equation simplifies to:
mEarth = m2 * mMoon.
The mass of the Earth (mEarth) is approximately 5.972 × 10^24 kg, and the mass of the Moon (mMoon) is approximately 7.348 × 10^22 kg.
Now we can calculate the distance (r) at which the gravitational forces are equal:
r = √(G * mEarth / mMoon).
Substituting the values, we get:
r = √(6.674 * 10^-11 N*m^2/kg^2 * 5.972 × 10^24 kg / 7.348 × 10^22 kg).
Simplifying the expression:
r = √(9.81 * 10^6 m^2).
Therefore, the distance at which the gravitational force exerted by the Earth on an object becomes equal in magnitude to the force exerted by the Moon is approximately 9,900 kilometers.