Question

What is the gravitational field g inside the following, spherical mass distributions as a function of the distance r from the center of the mass distribution? Let k = a constant and ρ(r) = density of mass distribution as a function of r.

a. ρ(r) = k
b. ρ(r) = k/r
c. ρ(r) = k/r2
d. In terms of k, r and G, what would be the velocity of an object in a circular orbit inside* the mass distribution about the center of it for the mass distrbutions specified by
i. ρ(r) = k
ii. ρ(r) = k/r
iii. ρ(r) = k/r2

Answer 1

For the first part of the question about the gravitational field for the given mass distributions:

• ρ(r) = k: The density is constant, so the gravitational field is also constant and is given by g = (4/3)πGk.
• ρ(r) = k/r: The density varies inversely with the distance, so the gravitational field decreases as r increases and is given by g = (4/3)πGk/r.
• ρ(r) = k/r^2: The density varies inversely with the square of the distance, so the gravitational field decreases even faster as r increases and is given by g = (4/3)πGk/r^2.

For the second part of the question about the velocity of an object in a circular orbit inside the mass distributions:

• For ρ(r) = k: The velocity of the object in a circular orbit is given by v = √(GMr).
• For ρ(r) = k/r: The velocity of the object in a circular orbit is given by v = √(GM/r).
• For ρ(r) = k/r^2: The velocity of the object in a circular orbit is given by v = √(GMr^3).

Step-by-step explanation:

The gravitational field g inside a spherical mass distribution is given by the equation g = (4/3)πGρ(r)r where G is the gravitational constant, ρ(r) is the density of the mass distribution as a function of r, and r is the distance from the center of the mass distribution.

To find the gravitational field for the given mass distributions:

1. ρ(r) = k: The density is constant, so the gravitational field is also constant and is given by g = (4/3)πGk.
2. ρ(r) = k/r: The density varies inversely with the distance, so the gravitational field decreases as r increases and is given by g = (4/3)πGk/r.
3. ρ(r) = k/r^2: The density varies inversely with the square of the distance, so the gravitational field decreases even faster as r increases and is given by g = (4/3)πGk/r^2.

For the second part of the question about the velocity of an object in a circular orbit inside the mass distributions:

1. For ρ(r) = k: The velocity of the object in a circular orbit is given by v = √(GMr).
2. For ρ(r) = k/r: The velocity of the object in a circular orbit is given by v = √(GM/r).
3. For ρ(r) = k/r^2: The velocity of the object in a circular orbit is given by v = √(GMr^3).