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

  1. #1
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    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:

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

    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 .....

    <br />
\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<br />

    ^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

     <br />
G = 6.7(10)^{-11}<br />

    <br />
C = 3 * 10^8 \frac{m}{s}<br />

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

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

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

    Could I just do something like this?

     <br />
\lim_{x\to\infty} \frac{F^2}{2}<br />

    <br />
\frac{mMG}{r^2} - \frac{mMG}{r^2}<br />

    (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.
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  2. #2
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    Can anyone help?

    It's just part A, part B i can figure out even without looking at the rest of the question.
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  3. #3
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    Never mind, problem solved. I figured it out.
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