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In summary: Good luck!In summary, we discussed a non-uniform density planet with an object falling through it. Using the standard trick of writing dr/dt = v, we were able to simplify the differential equation to \frac{d^2 r}{dt^2} = v \frac{dv}{dr}. However, when trying to solve for r as a function of t, we encountered a difficult integral that may not have a closed form solution. It is possible to solve it numerically or with the use of elliptical functions.

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WraithM said:[tex]\frac{d^2 r}{dt^2}=\frac{4}{3}\pi a G(r^2 - r)[/tex]. I don't really know much beyond solving simple differential equations. Is this solvable? If there is a solution, how do you get it?

Hi WraithM!

Use the standard trick of writing dr/dt = v, and then

d

= dv/dt

= dv/dr dr/dt [chain rule]

= v dv/dr

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Does that actually work? Do you just separate variables from there? I see the logic of making [tex]\frac{d^2 r}{dt^2} = v \frac{dv}{dr}[/tex], but I don't want to believe that it's that easy.

For example, take [tex]\frac{d^2 r}{dt^2} = g[/tex] (constant acceleration). Separating variables using v dv/dr: v dv = g dr, that's: v^2/2 = gr

[tex]\frac{dr}{dt} = \sqrt{2gr}[/tex]

[tex]\frac{dr}{\sqrt{2gr}} = dt[/tex]

[tex]1/g \sqrt{2gr}=t[/tex]

[tex]r = \frac{gt^2}{2}[/tex]

Wow, so that actually works?

:/ I'm feeling stupid for not knowing this.

EDIT: I just did this method with my first example, and I get an integral that isn't doable... :/ Am I just stuck then?

For example, take [tex]\frac{d^2 r}{dt^2} = g[/tex] (constant acceleration). Separating variables using v dv/dr: v dv = g dr, that's: v^2/2 = gr

[tex]\frac{dr}{dt} = \sqrt{2gr}[/tex]

[tex]\frac{dr}{\sqrt{2gr}} = dt[/tex]

[tex]1/g \sqrt{2gr}=t[/tex]

[tex]r = \frac{gt^2}{2}[/tex]

Wow, so that actually works?

:/ I'm feeling stupid for not knowing this.

EDIT: I just did this method with my first example, and I get an integral that isn't doable... :/ Am I just stuck then?

Last edited:

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WraithM said:Wow, so that actually works?

Woohoo!

(btw, that's how 1/2 mv

EDIT: I just did this method with my first example, and I get an integral that isn't doable... :/ Am I just stuck then?

hmm … let's see …

you didn't get vdv = 4/3 πaG (r

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You integrate from ([tex]r_{0}[/tex] to [tex]r[/tex]) that to get [tex]v = \sqrt{\frac{8}{3}Ga(r^3/3 - r^2/2 - r_{0}^3/3 + r_{0}^2/2)}[/tex]. That's all good, but then if you try to go another step further, it blows up. Now I'm solving for r(t). You get: [tex]\int_{r_{0}}^{r(t)}\frac{dr}{\sqrt{\frac{8}{3}Ga(r^3/3 - r^2/2 - r_{0}^3/3 + r_{0}^2/2)}} = \int_{0}^{t} dt[/tex] I don't know how to do that integral. I don't think it's doable. Do you know how to do it? Am I perhaps doing it wrong?

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WraithM said:… You get: [tex]\int_{r_{0}}^{r(t)}\frac{dr}{\sqrt{\frac{8}{3}Ga(r^3/3 - r^2/2 - r_{0}^3/3 + r_{0}^2/2)}} = \int_{0}^{t} dt[/tex] I don't know how to do that integral. I don't think it's doable. Do you know how to do it? Am I perhaps doing it wrong?

ah, I didn't realize you wanted to go as far as solving for r as a function of t

Yes, that looks correct … unfortunately, not all integrals are "doable"

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If the density is not uniform, the force acting on the mass when is at distance r from the center will be

G*m*M(r)/r^2

where M(r) is the mass contained in the sphere of radius r.

This is not 4/3pi*r^3*density . Density is not constant. You need to integrate from 0 to r in order to find the mass.

M(r) will be 4*pi(a/3*r^3-r^4/4) - you should do it again do double check.

But before you do this, check again your density. Of course, if you are interested in a math problem only, it does not matter.

But for a physics problem...

density=a-r ?

So density is zero at r=a and then it becomes negative for r<a.

Not mentioning the units. You will have density equal to distance. (a and r are disatnces, I suppose) Not very physical...

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[tex]\int_{r_{0}}^{r(t)}\frac{dr}{\sqrt{\frac{2}{3} \pi G(r^3 - 2ar^2-r_{0}^3+2ar_{0}^2)}}=\int_{0}^{t}dt[/tex]

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The integral that you show in the last post can be solved numerically or maybe can be expressed in term of elliptical functions, I am not sure. Try Mathematica or Maple.

The nonuniform density of a planet refers to the variation in the distribution of mass throughout the planet. This means that different parts of the planet have different densities, which can affect its overall gravitational pull and structure.

There are a few factors that can contribute to nonuniform density in a planet. These include the planet's formation process, geological activity, and the presence of different materials such as dense iron in the core and lighter rock in the crust.

Scientists can use various methods to measure the density of a planet, including gravitational measurements, seismic waves, and remote sensing techniques. These methods allow scientists to map out the variation in density throughout the planet.

The nonuniform density of a planet can have significant implications for its overall structure and composition. It can affect the planet's rotation, magnetic field, and even the possibility of supporting life. Understanding a planet's density is essential for studying its evolution and potential habitability.

Yes, the density of a planet can change over time due to various factors such as tectonic activity, volcanic eruptions, and impacts from other celestial bodies. These changes can also have a ripple effect on other aspects of the planet, such as its climate and surface features.

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