Simple Harmonic Motion Inside earth

In summary, the conversation discusses the motion of a particle dropped into a hole drilled through the center of the Earth. By assuming a constant density and using the 1/r^2 force law, it can be shown that the particle's motion is simple harmonic with a period of approximately 84 minutes. The conversation also touches on the method of solving the problem using a surface integral and the concept of net force only being exerted by the matter inside the current radius.
  • #1
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Homework Statement


A particle is dropped into a hole drilled straight through the center of the Earth. Neglecting rotational effects, show that the particle's motion is simple harmonic if you assume Earth has uniform density. Show that the period of the oscillation is about 84 min.


Homework Equations


[tex]F = -G m \int_V \frac{\rho(r') e_r}{r^2}dv'[/tex]


The Attempt at a Solution



I was going to use Newton's second law to show that
[tex] m \frac{d^2 r}{dt^2} = -G m \int_V \frac{\rho(r') e_r}{r^2}dv'[/tex]

Where the volume integral should produce some function of r. So, I started off the integration by choosing an arbitrary point in the sphere a distance r' away, and using R as the distance from the origin to the point mass, r as the distance between the arbitrary distance and the distance of the point mass, theta as the asimuthal angle, and phi as the rotational angle I got.

[tex]m \frac{d^2 r}{dt^2} = -G m \int_0^{R} \int_0^\pi \int_0^{2\pi} \frac{\rho r'^2 sin\theta}{r^2}dr' d\theta d\phi[/tex]

integrating with respect to phi brings in a factor of 2π, and now I used law of cosines

[tex]r^2 = r'^2 + R^2 - 2r'Rcos\theta[/tex]

[tex] 2r dr = 2r'Rsin\theta d\theta[/tex]

[tex]\frac{sin\theta}{r}d\theta = \frac{dr}{r'R}[/tex]

so the substitute the law of cosines stuff into the force equation

[tex]F = \frac{-2 \pi G m \rho}{R} \int_0^R r'^2 dr \int_{r'-R}^{r'+R} \frac{1}{r}dr[/tex]

doing the r' integral I can see that this ultimately won't give a linear function of R

[tex]\frac{2 \pi}{3} \frac{G m \rho}{R} R^3 \int_{r'-R}^{r'+R} \frac{1}{r}dr [/tex]

Can someone help me out, point out what I did wrong, put me on the right track?

I think that once I find the right equation for force, which I actually know from experience should be [tex]F(r) = -\frac{4 \pi}{3}G m r \rho[/tex], that I can do the differential equation stuff.
 
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  • #2
You're making it too complicated. Assume that the density of the Earth is a constant function, and realize that because of the character of the 1/r^2 force law, you can turn this into a surface integral. Once you do that, you're golden.
 
  • #3
At any point in the fall, the only mass exerting a nett force on the test body is that contained inside the current radius because the shell of matter outside the radius has no effect. So the force at point x is

[tex]F(x) = -\frac{4}{3}\pi Gm\rho x^3/x^2[/tex]

which gives

[tex]F(x) = -Kx[/tex]

QED I think. Assuming uniform density.
 
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1. What is Simple Harmonic Motion (SHM) inside the Earth?

Simple Harmonic Motion, also known as oscillatory motion, is a type of periodic motion in which an object moves back and forth in a predictable pattern. Inside the Earth, SHM refers to the movement of tectonic plates, which cause earthquakes and other seismic activity.

2. What factors affect Simple Harmonic Motion inside the Earth?

The main factors that affect SHM inside the Earth are the composition and density of the Earth's layers, as well as the movement of tectonic plates. Additionally, the presence of fluids, such as molten rock and water, can also affect the motion of the Earth's crust.

3. How is Simple Harmonic Motion inside the Earth measured?

SHM inside the Earth is measured using seismometers, which are instruments that detect and record the vibrations caused by earthquakes and other seismic activity. Seismologists analyze these recordings to better understand the patterns and behavior of SHM inside the Earth.

4. How does Simple Harmonic Motion inside the Earth impact our daily lives?

Simple Harmonic Motion inside the Earth can have a significant impact on our daily lives through the occurrence of earthquakes and other seismic events. These events can cause damage to buildings and infrastructure, as well as pose a threat to human safety. Understanding SHM inside the Earth is crucial for predicting and preparing for these natural disasters.

5. Can Simple Harmonic Motion inside the Earth be controlled or manipulated?

No, Simple Harmonic Motion inside the Earth cannot be controlled or manipulated by humans. It is a natural phenomenon that is influenced by various factors and cannot be altered by human intervention. However, scientists continue to study SHM inside the Earth in order to better understand and predict its behavior.

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