Acceleration due to gravity is given by:

**Part A**

$\overline{){{\mathbf{g}}}_{{\mathbf{p}}}{\mathbf{=}}\frac{\mathbf{G}\mathbf{M}}{{{\mathbf{R}}_{\mathbf{p}}}^{\mathbf{2}}}}$

$\begin{array}{rcl}{\mathbf{g}}_{\mathbf{p}}& \mathbf{=}& \frac{\mathbf{G}\mathbf{\left(}{\displaystyle \raisebox{1ex}{$\mathbf{4}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}\mathbf{\right)}\mathbf{\pi}{{\mathbf{R}}_{\mathbf{p}}}^{\mathbf{3}}\mathbf{\rho}}{{{\mathbf{R}}_{\mathbf{p}}}^{\mathbf{2}}}\\ & \mathbf{=}& \frac{\mathbf{4}}{\mathbf{3}}\mathbf{\pi}\mathbf{\rho}\mathbf{G}{\mathbf{R}}_{\mathbf{p}}\end{array}$

Gravitational Acceleration inside a Planet

Consider a spherical planet of uniform density *ρ*. The distance from the planet's center to its surface (i.e., the planet's radius) is *R*p. An object is located a distance *R* from the center of the planet, where *R*<*R*p. (The object is located inside of the planet.)

**Part A**

Find an expression for the magnitude of the acceleration due to gravity, *g*(*R*), inside the planet.

Express the acceleration due to gravity in terms of *ρ*, *R*, *π*, and *G*, the universal gravitational constant.

**Part B**

Rewrite your result for *g*(*R*) in terms of *g*p, the gravitational acceleration at the surface of the planet, times a function of R.

Express your answer in terms of *g*p, *R*, and *R*p.

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