One model for a certain planet has a core of radius r and mass m surrounded by an outer shell of inner radius r, outer radius 2r, and mass 4m. if m = 6.24 × 10^24 kg and r = 4.11 × 10^6 m, what is the gravitational acceleration of a particle at points (a) r and (b) 3r from the center of the planet?

(a)

At a distance r=R from the centre of the planet, there is no effect due to the outer shell: so, the gravitational field strength at r=R is only determined by the gravity produced by the core of the planet.

So, the strength of the gravitational field is given by

where

G is the gravitational constant

M = 6.24 × 10^24 kg is the mass of the core of the planet

R = 4.11 × 10^6 m is the radius of the core

Substituting into the equation, we find

(b)

at distance r=3R from the centre, the particle feels the effect of gravity due to both the core of the planet and the outer shell between R and 2R.

So, we have to consider the total mass that exerts the gravitational attraction at r=3R, which is the sum of the mass of the core (M) and the mass of the shell (4M):

M' = M + 4M = 5M

Therefore, the gravitational acceleration at r=3R will be

And susbstituting

g = 24.6 m/s^2

found in the previous part, we find

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