Question

Consider a spherical planet of uniform density rho. 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
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 rho, R, π, 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.

Express your answer in terms of gp, R, and Rp.

Answer 1

Answer:

a) $g(r) = 4\pi \cdot G \cdot \rho\cdot r$, b) $g = 4\pi \cdot G \cdot \rho\cdot r_{P}$

Explanation:

a) The acceleration due to gravity inside the planet is:

$dg = G\cdot \frac{\rho \cdot dV}{r^{2}}$

$dg = G\cdot \frac{\rho \cdot dV}{r^{2}}$

$dg = G\cdot \frac{4\pi\cdot \rho \cdot r^{2}\,dr}{r^{2}}$

$dg = 4\pi\cdot G\cdot \rho \,dr$

$g(r) = 4\pi \cdot G \cdot \rho\cdot r$

b) The acceleration at the surface of the planet is:

$g = 4\pi \cdot G \cdot \rho\cdot r_{P}$