1. The problem statement, all variables and given/known data
To relate E to the total energy of the expanding sphere, we need to integrate over
the sphere to determine its total energy. These integrals are most easily carried out
by dividing the sphere into shells of radius r, and thickness dr, so that each shell
has a volume dV = 4πr^2dr .
(b) (10 points) Show that the total kinetic energy K of the sphere is given by
K = ck * MRmax,i2{1/2*d2r/dt2}
where cK is a numerical constant, M is the total mass of the sphere, and Rmax,i
is the initial radius of the sphere. Evaluate the numerical constant cK.
(c) (10 points) Show that the total potential energy of the sphere can similarly be
written as U = cU*MR2max,i {-4∏/3 Gρi/a}
(Suggestion: calculate the total energy needed to assemble the sphere by bringing in one shell of mass at a time from infinity.) Show that cU = cK, so thatthe total energy of the sphere is given by
Etotal = cK*MR2max,iE .
2. Relevant equations
dV=4∏r^2dr
3. The attempt at a solution
Rather at a loss here, I tried starting out with their suggestion but given what I know E to be from the Freidmann eqs I don't see how that would work. These results are kind if intuitive, so I don't see how to shpw them per se.Like I've tried fiddling with integrals to little avail...something like k = 1/2mv^2 m = 4/3 ∏ri^3pi ??
To relate E to the total energy of the expanding sphere, we need to integrate over
the sphere to determine its total energy. These integrals are most easily carried out
by dividing the sphere into shells of radius r, and thickness dr, so that each shell
has a volume dV = 4πr^2dr .
(b) (10 points) Show that the total kinetic energy K of the sphere is given by
K = ck * MRmax,i2{1/2*d2r/dt2}
where cK is a numerical constant, M is the total mass of the sphere, and Rmax,i
is the initial radius of the sphere. Evaluate the numerical constant cK.
(c) (10 points) Show that the total potential energy of the sphere can similarly be
written as U = cU*MR2max,i {-4∏/3 Gρi/a}
(Suggestion: calculate the total energy needed to assemble the sphere by bringing in one shell of mass at a time from infinity.) Show that cU = cK, so thatthe total energy of the sphere is given by
Etotal = cK*MR2max,iE .
2. Relevant equations
dV=4∏r^2dr
3. The attempt at a solution
Rather at a loss here, I tried starting out with their suggestion but given what I know E to be from the Freidmann eqs I don't see how that would work. These results are kind if intuitive, so I don't see how to shpw them per se.Like I've tried fiddling with integrals to little avail...something like k = 1/2mv^2 m = 4/3 ∏ri^3pi ??
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