Universal Law of Gravitation Video Lessons

Concept

# Problem: Gravitational Acceleration inside a PlanetConsider a spherical planet of uniform density ρ. The distance from the planet's center to its surface (i.e., the planet's radius) is Rp. An object is located a distance R from the center of the planet, where R&lt;Rp. (The object is located inside of the planet.)Part AFind 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 BRewrite your result for g(R) in terms of gp, the gravitational acceleration at the surface of the planet, times a function of R.Express your answer in terms of gp, R, and Rp.

###### FREE Expert Solution

Acceleration due to gravity is given by:

Part A

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

$\begin{array}{rcl}{\mathbf{g}}_{\mathbf{p}}& \mathbf{=}& \frac{\mathbf{G}\mathbf{\left(}\mathbf{4}}{\mathbf{3}}\mathbf{\right)}\mathbf{\pi }{{\mathbf{R}}_{\mathbf{p}}}^{\mathbf{3}}\mathbf{\rho }}{{{\mathbf{R}}_{\mathbf{p}}}^{\mathbf{2}}}\\ & \mathbf{=}& \frac{\mathbf{4}}{\mathbf{3}}\mathbf{\pi }\mathbf{\rho }\mathbf{G}{\mathbf{R}}_{\mathbf{p}}\end{array}$

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###### Problem Details

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 Rp. An object is located a distance R from the center of the planet, where R<Rp. (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 ρRπ, and G, the universal gravitational constant.

Part B

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