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pirland

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A space probe of mass m is dropped into a previously unexplored

spherical cloud of gas and dust, and accelerates toward

the center of the cloud under the influence of the cloud’s gravity.

Measurements of its velocity allow its potential energy, [tex]U[/tex], to be

determined as a function of the distance [tex]r[/tex] from the cloud’s center.

The mass in the cloud is distributed in a spherically symmetric way,

so its density, [tex]\rho(r)[/tex], depends only on [tex]r[/tex] and not on the angular coordinates.

Show that by finding [tex]U(r)[/tex], one can infer [tex]\rho(r)[/tex] as follows:

[tex]\rho(r)=\frac{1}{4 \pi Gmr^2} \frac{d}{dr}(r^2 \frac{dU}{dr}) [/tex]

The first problem I am running into is finding the Equation for U. I assume it as a lot to do with the equation for work=[tex]GMm(\frac{1}{r_{2}}-\frac{1}{r_{1}}) [/tex] but I am getting tripped up over the value of M, or the mass of the cloud itself. I believe that this would change as r decreased, and less and less of the cloud would be exerting force on the probe. I assume I need to find a value for M in terms of r, but I am seemingly unable to come up with anything that works with later parts of the problem. If anyone could give me a push in the right direction it would be appreciated.