Consider a spherical planet of uniform density . The distance from the planet\'s
ID: 1452266 • Letter: C
Question
Consider a spherical planet of uniform density . 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.)
a) Find an expression for the magnitude of the acceleration due to gravity, g(R), inside the planet.
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.
Explanation / Answer
a) & b
at position R, we have M(R) = (4/3) R^3 p, and
g(R) = G M(R)/R^2 = (4/3) G R^3 p / R^2 = (4/3) G p R.
g(R) = (4/3) Gp R
B. is a simple corollary: A. shows that g(R) is proportional to R. But it has to equal the surface value at the surface. Hence it must satisfy
g(R) = (R/Rp) Gp