- #1
mbcrute
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Homework Statement
A sphere with radius R = 0.200 m has density ρ that decreases with distance r from the center of the sphere according to [tex]\rho = 3.00 \times 10^3 \frac{kg}{m^3} - (9.00 \times 10^3 \frac{kg}{m^4})r[/tex]
a) Calculate the total mass of the sphere.
b) Calculate the moment of inertia of the sphere for an axis along the diameter.
Homework Equations
Moment of inertia for a solid sphere: [tex]\frac{2}{5}MR^2[/tex]
Density: [tex]\rho = \frac{M}{V}[/tex]
Volume of a sphere: [tex]V = \frac{4}{3} \pi R^3[/tex]
The Attempt at a Solution
[/B]
I managed to figure out part a). I started with [itex]\rho = \frac{dM}{dV}[/itex] and [itex]V = \frac{4}{3} \pi r^3[/itex]. Solving for dM gets you [itex]dM = \rho dV[/itex] and taking the derivative of V yields [itex]dV = 4 \pi r^2 dr[/itex]. Substituting that back into the expression for dM yields [itex]dM = 4 \rho \pi r^2 dr[/itex]. I then substituted the expression for ρ given in the problem statement and integrated from 0 to R to find the mass of the sphere, 55.3 kg. According to the back of the book this is correct.
For part b) I know there's going to be another integral involved but I can't seem to get it right. I know the moment of inertia for a solid sphere (stated above under the relevant equations) so I thought I could plug my expression for dm back into it and integrate from 0 to R but that didn't get me a correct answer. I'm not quite sure where to go next. Any help would be greatly appreciated!