Hole though the Earth and SHM: why is uniform density assumed?

[Vitani11]

Homework Statement

A particle of mass m is dropped into a hole drilled straight through the center of the Earth. neglecting rotational effects and friction, show that the particle’s motion is a simple harmonic if it is assumed that the Earth has a uniform mass density. Obtain an expression for the period of oscillation.

Homework Equations

The answer is that K = (4/3)Gmπρ and so T = sqrt(3π/Gρ), right?
K = force constant
G = gravitational constant
ρ = density of earth
m = mass of object dropped
T = period

The Attempt at a Solution

I want to know why the constant K is not actually GmM/r³ and the force is not GmMx/r³ where x is the displacement of the object from equilibrium at any point within the Earth (I defined equilibrium to be at the center of earth). I think my issue is not understanding what it means to solve a problem using uniform density. I thought that uniform density meant that the mass per unit volume was the same everywhere in the earth, so the constant M is okay to leave in the equation since it's density is not changing. I want to understand this because there is a problem in my homework set which asks to find the electrostatic potential energy of a sphere of uniform density with charge Q and radius R. My mind tells me this is simply Q²/4πεR... but that can't be correct because it's too easy and that is the same as a point charge. What am I not getting?

 

[haruspex]

why the constant K is not actually GmM/r³

Depending on exactly how you are defining M and r, that is the same as (4/3)Gmπρ.
For this question it is better to work in terms of the density because you have to consider different radii, and hence different masses.

 

[ehild]

Homework Statement

A particle of mass m is dropped into a hole drilled straight through the center of the Earth. neglecting rotational effects and friction, show that the particle’s motion is a simple harmonic if it is assumed that the Earth has a uniform mass density. Obtain an expression for the period of oscillation.

Homework Equations

The answer is that K = (4/3)Gmπρ and so T = sqrt(3π/Gρ), right?
K = force constant
G = gravitational constant
ρ = density of earth
m = mass of object dropped
T = period

The Attempt at a Solution

I want to know why the constant K is not actually GmM/r³ and the force is not GmMx/r³ where x is the displacement of the object from equilibrium at any point within the Earth (I defined equilibrium to be at the center of earth). I think my issue is not understanding what it means to solve a problem using uniform density. I thought that uniform density meant that the mass per unit volume was the same everywhere in the earth, so the constant M is okay to leave in the equation since it's density is not changing. I want to understand this because there is a problem in my homework set which asks to find the electrostatic potential energy of a sphere of uniform density with charge Q and radius R. My mind tells me this is simply Q²/4πεR... but that can't be correct because it's too easy and that is the same as a point charge. What am I not getting?

Use the density ρ in the formulas. How is M related to the density and radius of the Earth ?

 

[Vitani11]

Last question - this problem never gave me any variables other than m. However in the final answer there is ρ. Why is it convention to allow this in the final answer when it wasn't given in the problem? I understand why G, π, etc. are in the final answer. Is it because it's a constant? So whenever something is constant, I can use it in my final answer?

 

[ehild]

Last question - this problem never gave me any variables other than m. However in the final answer there is ρ. Why is it convention to allow this in the final answer when it wasn't given in the problem? I understand why G, π, etc. are in the final answer. Is it because it's a constant? So whenever something is constant, I can use it in my final answer?

The problem said "A particle of mass m is dropped...", that means the mass of the particle was given, although not numerically. It was also said to assume that the mass density of the Earth was constant.
You need to give the answer in terms of the given and known data. They are m and ρ.

 

[Vitani11]

I know. My problem is not with the mass m, it is obviously given. I said ρ

 

[JoePhysics]

Last question - this problem never gave me any variables other than m. However in the final answer there is ρ. Why is it convention to allow this in the final answer when it wasn't given in the problem? I understand why G, π, etc. are in the final answer. Is it because it's a constant? So whenever something is constant, I can use it in my final answer?

The problem statement says that the planet is assumed to have a uniform mass density, which it then simply calls ρ.

 

[haruspex]

in the final answer there is ρ

Yes, that should have been specified in the problem statement. Looks like an omission.