The Moon can be considered to be a sphere with uniform density with mass MM and radius RM. At the (exact) center of the Moon, the gravitational force on a mass m due to the mass of the Moon is?
I know the answer is zero but want to know why using gauss' law
Assume that there is a certain cavity in the center of the Moon where we are going to place the mass ‘m”. Locate the Gaussian closed surface inside the Moon to enclose its central empty region. There will be nothing inside this surface => the flux of gravitational field is zero => the force (which is the product of gravitational field strength and the mass of the object) will be zero.
To understand why the gravitational force on a mass located at the exact center of the Moon is zero using Gauss' law, let's break it down step by step.
Gauss' law for gravity states that the gravitational flux through a closed surface is proportional to the mass enclosed within that surface. Mathematically, it can be expressed as:
Φ = -4πGM
Where Φ is the gravitational flux, G is the gravitational constant, and M is the mass enclosed within the closed surface.
Now, let's consider a spherical Gaussian surface centered at the exact center of the Moon. Since the Moon is spherically symmetric and has uniform density, the mass enclosed within any sphere concentric with the Moon is the same. Let's call this mass M_enclosed.
Now, when the Gaussian surface is at the exact center of the Moon, the gravitational force exerted by the mass M_enclosed on a mass m located at the center would be directed radially inwards toward the center of the Moon due to the symmetry.
However, due to the spherical symmetry and uniform density of the Moon, the gravitational force exerted by every point on the Moon cancels out by another point of equal magnitude and opposite direction. Therefore, the net gravitational force on the mass m at the center is zero.
In other words, since the gravitational flux through the Gaussian surface is zero (Φ = 0), using Gauss' law, we can conclude that the gravitational force at the exact center of the Moon is zero.