Particle falling inside a sphere due to gravity

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Homework Help Overview

The discussion revolves around a theoretical scenario where a test particle is dropped through a hole drilled from the surface of the Earth to the other side, exploring how the time taken for the particle to return depends on the Earth's radius and density. The subject area includes gravitational physics and oscillatory motion.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the gravitational field inside a sphere and its relation to the motion of the particle. There are attempts to derive the relationship between time, radius, and density using differential equations. Questions arise about the dependence of the period on these variables and whether they can be combined in a single equation.

Discussion Status

Some participants have provided insights into the oscillatory nature of the motion and suggested forms of equations to describe it. There is acknowledgment that the period does not explicitly depend on the radius, but further exploration of how density might relate to radius is suggested. The conversation reflects a mix of interpretations and ongoing inquiry.

Contextual Notes

Participants note the challenge of integrating both density and radius into a single equation, indicating potential constraints in the problem setup. The discussion also touches on the theoretical aspects of gravitational motion and oscillation.

Robin04
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Homework Statement


We drill a hole from the surface of the Earth to the other side through the center, drop a test particle and measure the time it takes to come back. How does this time depend on the radius and density of the Earth?

Homework Equations

The Attempt at a Solution


The gravitation field inside a sphere is ##g=\frac{4\pi G\rho}{3}r##, where ##G## is the gravitational constant, ##\rho## is the density, ##r## is the distance from the center.
Now I have to find the function ##r(t)##. I already found ##v(r)=r\sqrt{4\pi G \rho}## (the initial velocity is zero). How can I bring time into the equation?
 
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Robin04 said:

Homework Statement


We drill a hole from the surface of the Earth to the other side through the center, drop a test particle and measure the time it takes to come back. How does this time depend on the radius and density of the Earth?

Homework Equations

The Attempt at a Solution


The gravitation field inside a sphere is ##g=\frac{4\pi G\rho}{3}r##, where ##G## is the gravitational constant, ##\rho## is the density, ##r## is the distance from the center.
Now I have to find the function ##r(t)##. I already found ##v(r)=r\sqrt{4\pi G \rho}## (the initial velocity is zero). How can I bring time into the equation?

Since the motion is oscillatory motion, you can think a form of Second order DE, such that

$$d^2r/dt^2+ω^2r=0$$ where ##ω## will be the anguler frequency or ##w=2π/T##
 
Arman777 said:
Since the motion is oscillatory motion, you can think a form of Second order DE, such that

$$d^2r/dt^2+ω^2r=0$$ where ##ω## will be the anguler frequency or ##w=2π/T##
So I could say ##g+\omega^2r=0##, from which ##T=\sqrt{\frac{3\pi}{G\rho}}## Is this correct? It doesn't depend on the radius of the sphere.
 
Robin04 said:
So I could say ##g+\omega^2r=0##, from which ##T=\sqrt{\frac{3\pi}{G\rho}}## Is this correct? It doesn't depend on the radius of the sphere.
It's true yes. I don't think we can get the density and the radius in the same equation. For the radius dependence put the ##ρ## value in the equation that you find and you ll see that it depends also on radius in some other aspect.
 
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Arman777 said:
It's true yes. I don't think we can get the density and the radius in the same equation. For the radius dependence put the ##ρ## value in the equation that you find and you ll see that it depends also on radius in some other aspect.
Thank you! :)
 
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A bit of trivia: the period for your particle is the same as for a (theoretical) satellite orbiting the Earth at its surface.
 
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