Estimate flattening of the Earth

[Yoonique]

Homework Statement

The flattening of the Earth is defined as
f_⊕ ≡ 1 - (R_p - R_e)
where R_e is the radius of the equator and R_p is half of the distance between the North Pole and the South Pole. Estimate the flattening of the Earth.

Homework Equations

F_g = Gm₁m₂/r²

The Attempt at a Solution

a_pole = GM_E/R_p²
a_equator = GM_E/R_E² = ω²R_E

How do I approach this problem, I'm so clueless. How do I carry on from here?

 

[haruspex]

1 - (Rp - Re)

Maybe 1 - (Rp / Re)?
Treat the Earth as a rotating ball of water. Since the water at the surface does not run towards the poles, despite the spin, what can you say about the surface - i.e. what is it that is constant over the surface?

 

[Yoonique]

Maybe 1 - (Rp / Re)?
Treat the Earth as a rotating ball of water. Since the water at the surface does not run towards the poles, despite the spin, what can you say about the surface - i.e. what is it that is constant over the surface?

Is a typo, suppose to be 1 - (Rp / Re). Ok, after reading about some article is it regarding about hydrostatic equilibrium? Because I haven't learn about it yet. So I think there must be a constant radial force all over the surface of the sphere for it to be in hydrostatic equilibrium? At the poles, the force is just GM_EM_W/R_p². At the equator, the force is GM_EM_W/R_e² + ω²R_e. For both of the forces to be balanced, R_p must decrease and R_e must increase thus resulting in the flattening of the Earth? Am I right? I don't know anything about hydrostatic equilibrium so my concept would be wrong.

 

[haruspex]

Is a typo, suppose to be 1 - (Rp / Re). Ok, after reading about some article is it regarding about hydrostatic equilibrium? Because I haven't learn about it yet. So I think there must be a constant radial force all over the surface of the sphere for it to be in hydrostatic equilibrium? At the poles, the force is just GM_EM_W/R_p². At the equator, the force is GM_EM_W/R_e² + ω²R_e. For both of the forces to be balanced, R_p must decrease and R_e must increase thus resulting in the flattening of the Earth? Am I right? I don't know anything about hydrostatic equilibrium so my concept would be wrong.

No, it's not a constant (magnitude) force. If you were in a boat on the surface, anywhere, which way would gravity act? What does that tell you about potential energy levels?

 

[Yoonique]

The gravity would act towards the center of the Earth. All points on the surface are equipotential when the sphere of water rotates so the surface water does not move around surface area of the sphere, it remains at 'rest'? So I assume this is the reason for hydrostatic equilibrium? The surface water at the poles must have the same potential as the surface water at the equator.

 

[haruspex]

The gravity would act towards the center of the Earth. All points on the surface are equipotential when the sphere of water rotates so the surface water does not move around surface area of the sphere, it remains at 'rest'? So I assume this is the reason for hydrostatic equilibrium? The surface water at the poles must have the same potential as the surface water at the equator.

Right, so can you figure out what the potential is at a given latitude and distance from Earth's centre? (For the purposes of calculating the gravitational potential, pretend the Earth is spherical, and you are considering a point outside that sphere.)

 

[Yoonique]

Right, so can you figure out what the potential is at a given latitude and distance from Earth's centre? (For the purposes of calculating the gravitational potential, pretend the Earth is spherical, and you are considering a point outside that sphere.)

I'm not very sure how does the centripetal force contribute to the potential. I assume potential is potential energy per unit mass and potential energy is ∫-F(r) dr
Potential the poles: -GM_e/R_p
Potential the equator: -1/M_w ∫F(R) dR= -1/M_w ∫[GM_eM_w/R_e² + M_wω²R_e dR = GM_e/R_e - ω²R_e²/2

 

[haruspex]

I'm not very sure how does the centripetal force contribute to the potential. I assume potential is potential energy per unit mass and potential energy is ∫-F(r) dr
Potential the poles: -GM_e/R_p
Potential the equator: -1/M_w ∫F(R) dR= -1/M_w ∫[GM_eM_w/R_e² + M_wω²R_e dR = GM_e/R_e - ω²R_e²/2

Sorry to do this to you but I'm going to change tack. I believe consideration of potentials will work, but I was finding it confusing and I wish to avoid confusing you.
Instead, consider the net force at some point on the surface at latitude theta. This will not point at the Earth's centre. You can get an expression for the approximate angular deviation from the radial line. You can use that angle to find out how rapidly radius of the surface changes as theta changes. Solving the ODE gives the profile.

 

[Yoonique]

Sorry to do this to you but I'm going to change tack. I believe consideration of potentials will work, but I was finding it confusing and I wish to avoid confusing you.
Instead, consider the net force at some point on the surface at latitude theta. This will not point at the Earth's centre. You can get an expression for the approximate angular deviation from the radial line. You can use that angle to find out how rapidly radius of the surface changes as theta changes. Solving the ODE gives the profile.

(GM_e/r²)cosθ = ω²rcosθ

How do I continue from here?

 

[haruspex]

(GM_e/r²)cosθ = ω²rcosθ

How do I continue from here?

That doesn't look like what I had in mind.
Consider a point at distance r from Earth's centre, at latitude theta. There is a gravitational force towards Earth's centre. Working in terms of pseudo forces, there is a centrifugal force. The resultant of these two forces makes an angle, alpha, to the radial line. Find alpha as a function of r, theta, etc.
Next, the local horizontal (i.e. the surface tangent) must be at right angles to this. Consider a small change $d\theta$ to the latitude. Find dr as a function of that and alpha.
Combining these gives you a differential equation.

 

[mfb]

Instead, consider the net force at some point on the surface at latitude theta. This will not point at the Earth's centre. You can get an expression for the approximate angular deviation from the radial line. You can use that angle to find out how rapidly radius of the surface changes as theta changes. Solving the ODE gives the profile.

Is that necessary? The first approach looked fine and just two steps away from the solution if you plug in some parameters at the right places.
It is an approximation, of course, but it will get the first order right and the second order is much smaller.

 

[haruspex]

Is that necessary? The first approach looked fine and just two steps away from the solution if you plug in some parameters at the right places.
It is an approximation, of course, but it will get the first order right and the second order is much smaller.

Yes, as I posted, I believe my original approach works, but it looked like it was going to be hard explaining why the signs are as they are. Please take it further.

 

[Yoonique]

Is that necessary? The first approach looked fine and just two steps away from the solution if you plug in some parameters at the right places.
It is an approximation, of course, but it will get the first order right and the second order is much smaller.

The answer should be 0.003 but I got 0.002 using the potential method. So this method isn't accurate enough?

 

[Yoonique]

That doesn't look like what I had in mind.
Consider a point at distance r from Earth's centre, at latitude theta. There is a gravitational force towards Earth's centre. Working in terms of pseudo forces, there is a centrifugal force. The resultant of these two forces makes an angle, alpha, to the radial line. Find alpha as a function of r, theta, etc.
Next, the local horizontal (i.e. the surface tangent) must be at right angles to this. Consider a small change $d\theta$ to the latitude. Find dr as a function of that and alpha.
Combining these gives you a differential equation.

I can't really visualize or understand this as I'm not used to using a rotating frame of reference. So centrifugal force is in the opposite direction of centripetal force right? So if I'm in the same rotating frame of reference, I'm seeing the point being push outwards? The resultant force is the apparent weight?
Because in an inertial frame of reference, the resultant of gravitational force and normal force provide the centripetal force. The apparent weight is the normal force. I'm little confused in the case of a rotating frame of reference.

 

[haruspex]

So alpha + theta is 90°?

No.

 

[Yoonique]

No.
View attachment 82226

Is the resultant force the apparent gravitational force felt?

 

[haruspex]

Is the resultant force the apparent gravitational force felt?

Yes.

 

[Yoonique]

Yes.

Do I need to approximate anything to find a relationship between α and θ? I can't find a direct relationship ;/

 

[haruspex]

Do I need to approximate anything to find a relationship between α and θ? I can't find a direct relationship ;/

An approximation valid for small alpha would certainly be reasonable.

 

[Yoonique]

An approximation valid for small alpha would certainly be reasonable.

Why does the potential method yield 1.7x10^-4 while the given answer is 3.4x10^-4?

 

[mfb]

I guess you mean 10^-3. I get 10.9 km difference in radius which is 1.7*10^-3. The true value is ~21.4 km.
The factor 2 suggests some stupid mistake, but I don't find one now. Check the approach via the local angle of net force to see if that works.

1/(1/(6370km)+(2 pi/day)^2(6370km)^2/(2*gravitational constant * mass of earth))