(a) 120.8 m/s^2
The gravitational acceleration at a generic distance r from the centre of the planet is
[tex]g=\frac{GM'}{r^2}[/tex]
where
G is the gravitational constant
M' is the mass enclosed by the spherical surface of radius r
r is the distance from the centre
For this part of the problem,
[tex]r=R=1.17\cdot 10^6 m[/tex]
so the mass enclosed is just the mass of the core:
[tex]M'=M=2.48\cdot 10^{24}kg[/tex]
So the gravitational acceleration is
[tex]g=\frac{(6.67\cdot 10^{-11})(2.48\cdot 10^{24}kg)}{(1.17\cdot 10^6 m)^2}=120.8 m/s^2[/tex]
(b) 67.1 m/s^2
In this part of the problem,
[tex]r=3R=3(1.17\cdot 10^6 m)=3.51\cdot 10^6 m[/tex]
and the mass enclosed here is the sum of the mass of the core and the mass of the shell, so
[tex]M'=M+4M=5M=5(2.48\cdot 10^{24}kg)=1.24\cdot 10^{25}kg[/tex]
so the gravitational acceleration is
[tex]g=\frac{(6.67\cdot 10^{-11})(1.24\cdot 10^{25}kg)}{(3.51\cdot 10^6 m)^2}=67.1 m/s^2[/tex]
