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AP® Physics C: Mechanics

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Gravitational Field Outside a Planet of Non-Constant Density

APPHMC-41HHY$

A particular planet is spherically symmetric, but has a non-constant density given by $\rho \left( r \right) ={ \rho }_{ 0 }\left( 1-\frac {{ r}^{2} }{ {R}^{2} } \right)$, where $R$ is the radius of the planet.

What is the magnitude of the gravitational field at a distance of $3R$ from the center of the planet?

A

$\cfrac { 8G{ \pi { \rho }_{ 0 }R } }{ 135 }$

B

$\cfrac { 4G{ \pi { \rho }_{ 0 }R } }{ 27 }$

C

$\cfrac { 8G{ \pi { \rho }_{ 0 }R } }{ 9 }$

D

$\cfrac { 4G \pi { \rho }_{ 0 } }{ 27{R}^{2} }$

E

$\cfrac { 8G \pi { \rho }_{ 0 } }{ 9{R}^{2} }$