# Math Help - Schwartschild Radius Problem (With Limits, etc.)

1. ## Schwartschild Radius Problem (With Limits, etc.)

Background formula for this problem:

$F = \frac{mMG}{r^2}$ where F = force of gravity.

For a given mass, the radius of the sphere to which it must be compressed to be a black whole is its Schwartzschild Radius.

The work energy principle (work done on a particle by a gravitiational force is equal to the change in the kinetic energy of the particle) means that:

$
Work = \int_R^r F*dr = [\frac{1}{2}mv^2 - \frac{1}{2}mv_{0}^2]
$

R is radius of object in question.
r is a random distance from the center of gravity
v = velocity of particle as it goes from r to the dist. R (and of course initial velocity $v_{0}$ is the initial velocity at distance r).

------------------------(Now the Problem)------------------------

*To find the escape velocity, 2 quantities are equated: the work done in moving a particle of mass m from the distance R out to infinity, and the kinetic energy gained by the particle in letting it fall from infinity with initial velocity = 0. The final velocity is called the escape velocity.

To deduce $R_{s}$ of a mass M object .....

$
\frac{1}{2}mv^2 = \lim_{x\to\infty} \int_R^x F*dr = \lim_{x\to\infty} \int_R^x mMG(r)^{-2} * dr
$

^Solve the above for R and we can have v=c (speed of light), and the m will cancel out. R = Schwartzschild Radius of the object M.

PART A

Show that $R_{s} = \frac{2MG}{c^2}$

PART B

Calculate $R_{s}$ for both the sun/earth.

Sun = Mass $2(10)^{30} kg$

Earth = Mass $6(10)^{24} kg$

$
G = 6.7(10)^{-11}
$

$
C = 3 * 10^8 \frac{m}{s}
$

--------------------

I'm not sure how to start Part A. How do I start?

$
\frac{1}{2}mv^2 = \lim_{x\to\infty} \int_R^x mMG(r)^{-2} * dr
$

Could I just do something like this?

$
\lim_{x\to\infty} \frac{F^2}{2}
$

$
\frac{mMG}{r^2} - \frac{mMG}{r^2}
$

(As x goes to infinity, the r in the denominator makes the first fraction above zero?) But this doesn't get me close to the part a proof.

Sorry if it seems confusing, it's a long problem, ask if you need help reading instructions, etc.

2. Can anyone help?

It's just part A, part B i can figure out even without looking at the rest of the question.

3. Never mind, problem solved. I figured it out.