help virial theorem

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help virial theorem

Post by betelgeuse1 » Tue Aug 18, 2009 2:43 am

Can anyone give me a simple explanation for the virial theorem and an example. I understand it's something like <T>=-(1/2)U where T=total kinetic energy and U=total potential energy. Now I saw a solution on for 0177 problem 3 where somebody says U=3/2T. I may understand 3 as being something like the number of degrees of freedom for translational motion but I don't get how the 2 comes down instead of up.
Thanks a lot

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Re: help virial theorem

Post by blackcat007 » Tue Aug 18, 2009 8:45 am

why do you need virial thm to answer this question.. simply equate the centrifugal with the gravitational force (non inertial frame)? in there are many redundant NEC's

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Re: help virial theorem

Post by physics_auth » Tue Aug 18, 2009 9:46 am

I will explain briefly, not providing a full proof. Consider a collection of particles whose position vectors r_a and momenta p_a are both bounded (i.e. they remain finite for all values of the time). The virial theorem is -in a more general case- an expression about the average kinetic energy of the system of the forementioned particles. Specifically it is:
<T> = -(1/2) * <Σ F_a*r_a> (1)
T = total kinetic energy
Σ = summation over a = the number of particles of the system
F_a = the force on the a-th particle
r_a = the position of the a-th particle
where the right hand side was named by Clausius "virial". Notice also the factor -1/2 on the right hand side (RHS). If the forces on each particle are derivable from some potential U_a, then (1) transforms to the following form:
<T> = (1/2) <Σ r_a * grad(U_a)> (2), since in this case F_a = - grad(U_a).

Of particular interest is the case of 2 particles that interact via some central force-> F analogous to (r^n) , where n = some exponent.
In this case the potential energy of interaction takes the form U = k*r^(n+1), where k = proportionality constant. Thus, for this particular case it is (r = relative position of the 2 particles, U = potential energy of interaction), in spherical coordinates (see the RHS side of (2)):
r*grad(U) = r*dU/dr = k(n+1)*r^(n+1) = (n+1)*U (3), since U = k*r^(n+1).
Combining (3) and (2) (for the case of 2 particles always) we find that
<T> = (n+1)/2 * <U> (4).

If the mutual interaction is gravitational then n = -2 (see the force dependence on r above, how it was defined) and from (4) it comes out that
<T> = -(1/2) * <U> .
This is a useful relation for calculations concerning the planetary motion.

In the alternative answer of GR0177 #3, the argumentation is correct but the virial theorem does not seem to be employed correctly. In general, think carefully before reading the solutions given on this site. I have met several errors in the quoted answers. Think before you "digest" the quoted answer.

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Re: help virial theorem

Post by betelgeuse1 » Tue Aug 18, 2009 10:34 am

Thank you physics_auth. I was scared that all I know suddenly became wrong... :)

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