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## Homework Statement

(a) A particle is dropped into a hole drilled straight through the center of Earth.

Neglecting rotational eects, show that the particle's motion is simple harmonic if

you assume Earth has uniform density. Find the period of the oscillation in terms

of Earth mass M and radius R (and G). Find its numerical value in minutes.

**SOLVED BY MYSELF**

This is needed to describe part (b), which is the problem I am stuck on.

**Problem of Concern**

(b) Consider the same problem for the case when the hole is drilled obliquely

through Earth (not going through the center). Show that the particle's motion is

simple harmonic with the same period as in (a). Ignore eects of rotation, friction,

etc. Express period through g and REarth.

## Homework Equations

[tex] F = m_{p} \ddot{r} = -G \frac{m_{p} M}{r^2}[/tex]

[tex] T = 2 \pi \sqrt{\frac{R_{E}^{3}}{G M_E}} = 84 min[/tex]

## The Attempt at a Solution

To solve the first problem was not that difficult. All I did was use the first of the relevant equations where [tex] m_{p} [/tex] will cancel out. I then say that the Mass from 0 to r will be [tex] M = \frac{4}{3} \pi r^3 \rho [/tex]. I ignore the R-r mass assuming that it can be neglected. This results in the wonderful ODE:

[tex] \ddot{r} + (\frac{4}{3} G \pi \rho) r = 0 [/tex]

with

[tex] \omega^2 = \frac{4}{3} G \pi \rho [/tex]

and since [tex] \omega = \frac{2 \pi}{T} [/tex]

Plugging everything in and saying that [tex] \rho = \frac{M_E}{\frac{4}{3} \pi R_{E}^3} [/tex]

we get the second relevant equation.

Now, the second question, part (b), says that this particle is not going to go through the center, that it will have an oblique path.

So, I though that there will be then a minimal r which will be the closest it can be to the center and a maximum r of [tex] R_E [/tex].

But I really don't know where to go with this. Any help would be greatly appreciated especially before 1:00 p.m. Tuesday, November 9th PDT GMT -0800.