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**[SOLVED] Force on Particle in Dust Cloud**

The following problem is from Thorton & Marion's Classical Dynamics, Ch. 5 Problem 5-13 (p. 205 in the 5th edition of the text)

**1. Homework Statement**

A planet of density [tex]\rho_{1}[/tex] (spherical core, radius [tex]R_{1}[/tex]) with a thick spherical cloud of dust (density [tex]\rho_{2}[/tex], radius [tex]R_{2}[/tex]) is discovered. What is the force on a particle of mass [tex]m[/tex] placed within the dust cloud?

**2. Homework Equations**

[tex]

V_{sphere}=\frac{4}{3}\pi \ r^{3}

[/tex]

[tex]

F = \frac{-GmM}{r^{2}}

[/tex]

[tex]

\rho = \frac{m}{v}

[/tex]

**3. The Attempt at a Solution**

So my intuition for this one is to solve for big M and add the mass of the cloud with the mass of the planet.

[tex]

M_{1} = \frac{4}{3}\pi\rho_{1} \ {R_{1}}^{3}

[/tex]

for the mass of the planet, and:

[tex]

M_{2} = \frac{4}{3} \pi\rho_{2} {R_{2}}^{3}

[/tex]

substituting [tex]M[/tex] with [tex]M_{1} + M_{2}[/tex] and a bit of factoring, I get:

[tex]

F = \frac{4}{3} \frac{Gm \pi ({R_{1}}^{3}\rho_{1} + {R_{2}}^{3}\rho_{2})}{r^{2}}

[/tex]

Yet somehow this doesn't feel right...

The text provides answers for the even numbers only, so I don't know how to verify this. I feel like there's something else I should be doing and it might involve calculus...

Any suggestions? Thanks in advance for any help.