# The density within of sphere so that gravity is constant

• richard7893
In summary, the question asks how the density of a sphere varies as a function of the distance from the center, given that the gravitational field inside the sphere is constant. The solution involves using Gauss's law for gravity and setting the integral of g over the surface area of the sphere equal to -4*pi*GM.
richard7893

## Homework Statement

If the gravitational field vector g inside a sphere is independent of the
distance from the center of the sphere r, how does the density ρ(r) of the
sphere vary as a function of r?

## Homework Equations

gauss' law for gravity: integrate g*da=4*pi*G integrate ρ(r) dv

## The Attempt at a Solution

So far i have g=contant= (a/r^2) integrate 0 to r r^2 dr I am not sure what to do next.

If the field vector g inside is constant with respect to radii, then you can pull it outside the integral for Gauss's law as long as your gaussian surface is a sphere. So you get $$g\oint_{dS}dA = -4\pi GM$$ The integral is just the surface area of the sphere you have, so $$g(4\pi r^2)=-4\pi GM$$ Can you figure it out from there?

## 1. What is the density within a sphere?

The density within a sphere is the amount of mass per unit volume of the sphere. It is typically measured in units of grams per cubic centimeter (g/cm3).

## 2. How is density related to gravity within a sphere?

Density and gravity are directly related within a sphere. The higher the density of the sphere, the stronger its gravitational pull will be. This is because the more mass an object has, the stronger its gravitational force.

## 3. How does the density within a sphere affect the strength of gravity?

The density within a sphere directly affects the strength of gravity. The higher the density, the stronger the gravitational pull will be. This is why objects with higher densities, such as planets, have stronger gravitational forces than objects with lower densities, such as asteroids.

## 4. Can the density within a sphere be changed to alter gravity?

Yes, the density within a sphere can be changed to alter gravity. By increasing or decreasing the density, the strength of the gravitational force can be adjusted. This is why scientists can predict the gravity on different planets, as they know the densities of those planets.

## 5. How is the density within a sphere calculated?

The density within a sphere can be calculated by dividing the mass of the sphere by its volume. The mass can be determined by weighing the sphere, and the volume can be calculated using the formula V = (4/3)πr3, where r is the radius of the sphere. The resulting value will be the density in units of mass per volume.

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