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AP® Physics C: Mechanics

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Mass of a Planet with Non-Constant Density

APPHMC-VYXHYI

A particular planet is spherically symmetric, but has a non-constant density given by $\rho \left( r \right) ={ \rho }_{ 0 }\left( 1-\frac { r }{ R } \right)$, where $R$ is the radius of the planet.

What is the total mass of the planet?

A

$\cfrac { 4}{ 3 } \pi { R }^{ 3 }$

B

$\cfrac { 4 }{ 3 } {\rho}_{0} \pi { R }^{ 3 }$

C

$\cfrac { 1 }{ 3 } {\rho}_{0} \pi { R }^{ 3 }$

D

$\cfrac { 1 }{ 4 } {\rho}_{0} \pi { R }^{ 3 }$

E

$\cfrac { 1 }{ 12} {\rho}_{0} \pi { R }^{ 3 }$