Question

I found the value for A which proved to be very simple, however the ones involving density are giving me headaches.

Thanks in advance....

Find the gravitational force that the earth exerts on a 10.0 kg mass if it is placed at the following locations. (a) at the surface of the earth; (b) at the outer surface of the molten outer core; (c) at the surface of the solid inner core; (d) at the center of the earth.

Consult the figure 12.9 in the textbook, and assume a constant density through each of the interior regions (mantle, outer core, inner core), butnot the same density in each of these regions. Use the values given below for average density for each region.

Assume the inner core has outer radius 1.2 × 106m, inner radius zero and density 1.3 × 104kg/m3, the outer core has inner radius 1.2 × 106m, outer radius 3.6 × 106 m and density 1.1 ×104 kg/m3, the total mass of the earth is mE = 5.97 × 1024kg and its radius is RE = 6.38 ×106m.

Answer 1

Here ,

as the gravitational force that hollow sphere inside the sphere is zero

a)

mass of sphere , m = volume * density

at the outer surface ,

mass of earth , M = 5.97 *10^24 Kg

force on the mass = G * M * m/R^2

force on the mass = 6.673 *10^-11 * 5.98 *10^24 * 10/(6.38 *10^6)^2

force on the mass at the surface= 98 N

the force on the mass at the surface is 98 N
b)

at the outer surface of molten outer core

mass of volume inside the outer core

m = (4/3 )* pi * 1.3 *10^4 * (1.2 *10^6)^3 + (4/3) * pi * 1.1 *10^4 *((3.6 *10^6)^3 - (1.2 *10^6)^3)

m = 2.164 *10^24 Kg

force on the mass = G * M * m/d^2

force on the mass = 6.673 *10^-11 * 2.164 *10^24 * 10/(3.6 *10^6)^2

force on the mass = 111.4 N

the force on the mass at the outer surface of the molten outer core is 111.4 N

part c)

at the surface of solid core

mass , m = (4/3 )* pi * 1.3 *10^4 * (1.2 *10^6)^3

m = 9.41 *10^22 Kg

force = 6.673 *10^-11 * 9.41 *10^22 * 10/(1.2 *10^6)^2

force = 43.6 N

the force on the mass is 43.6 N

part d)

at the centre as the mass inside the sphere is zero

force on the mass is zero