# Drilled hole through earth - Diff EQ w/ Gauss' Law

## Homework Statement

Suppose that a hole has been drilled through the center of the earth, and that an object is droppped into this hole. Write a first order linear differential equaiton for the object's velocity, v as a function of the distance r from the earth's center (i.e, and equation involving dv/dr), and solve it to determine the speed the object achieves as it reaches the center of the earth. Check this speed with the result you get from simple conservation of energy considerations. Consider the Earth's mass density to be uniform throughout.

Hint : recall Gauss' law as it applies to the gravitational field of a spherically symmetric mass distribution

## The Attempt at a Solution

I am having trouble relating Gauss' law to this problem , and from there, I don't know how to use the equation hopefully involving velocity and radius to use it in my differential equation. I'm very lost on how to start this problem. I don't need help doing the math, just the concept is not clicking.

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gabbagabbahey
Homework Helper
Gold Member

## Homework Statement

Suppose that a hole has been drilled through the center of the earth, and that an object is droppped into this hole. Write a first order linear differential equaiton for the object's velocity, v as a function of the distance r from the earth's center (i.e, and equation involving dv/dr), and solve it to determine the speed the object achieves as it reaches the center of the earth. Check this speed with the result you get from simple conservation of energy considerations. Consider the Earth's mass density to be uniform throughout.

Hint : recall Gauss' law as it applies to the gravitational field of a spherically symmetric mass distribution

## The Attempt at a Solution

I am having trouble relating Gauss' law to this problem , and from there, I don't know how to use the equation hopefully involving velocity and radius to use it in my differential equation. I'm very lost on how to start this problem. I don't need help doing the math, just the concept is not clicking.

What does Gauss' law (for gravity) tell you the gravitational field is a distance $r \leq R_{\text{Earth}}$ from the center of the Earth?

You might be better off trying to find the position, [;r;], as a function of time and then differentiating. To do this, use Gauss's Law to find the force acting on the particle as a function of [;r;] and then use Newton's Law: [F=ma=\frac{d^2r}{dt^2}] to get an ordinary differential equation. Solve said ODE.