Question 1: Gravity
10 points
The structure of the Earth
Treat the Earth as consisting of three distinct regions: the inner core (solid), the outer core (liquid), and the mantle (mostly solid). We are ignoring the thin crust at the surface for this problem. Say that the Earth has a total mass M and radius R, and the boundary of the inner and outer core is at r, and that of the outer core and mantle is at r. Further assume that each region has a uniform density P_inner core, P_outer core, and P_mantle.
Newton derived two results that are essential for this problem. The first effect concerns the gravitational field exerted on a test mass (m) inside a spherical shell of uniform density and total mass M. And the second concerns the effect of the total mass M.
2 points
Describe the two effects (no need to derive them here).
Find the magnitude of the force of gravity and the gravitational potential on a particle of mass m at the following locations:
b
1 point
10 meters away from the center of the Earth,
c
1 point
at r,
d
1 point
at r,
ze
1 point
at R.
Just use P_inner core, P_outer core, P_mantle, and R in your answers and not the actual numerical values from the figure.
4 points
The total mass of the Earth is 5.97x10^24 kg and the radius of the Earth is 6.37 x 10^6 m. What should be the velocity of an object ejected from the surface of the Earth radially outwards that will reach an altitude of 200 km before falling back to Earth?
