Gravity Inside the Earth. Find the gravitational force that the earth exerts on a 10.0-kg mass if it is placed at the following locations. Consult Fig. 13.9, and assume a constant density through each of the interior regions (mantle, outer core, inner core), but not the same density in each of these regions. Use the graph to estimate the average density for each region: (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.
Solution Step 1 The gravitational force that earth exerts on a body is given by the equation . F= G mM ……….( 1 ) r2 M = mass of the earth = 5.98× 10 m= mass of the body = 10kg 6 r = radius of the earth = 6.38 × 10 11 G = gravitational force constant = 6.67× 10 Step 2 The density of an objec is given by = M …………..( 2 ) v Density of the earth = ME vE ME = Mass of the earth = 5.98× 104 vE = Volume of the earth = 1.09× 10 m 5.98×1024 = 1.09×10 m3 = 5500 kg/m 3 Step 3 From equation (2) we can make an equation for M M = v V is the volume , volume of sphere is =4 r 3 So M = 4 r 3 Put this value of M in equation ( 1 ) 4 G m 3r3 4 F = r2 = 3G r Step 4 a)First case the body is on the surface of earth r = 6.38 × 106 Gravitational force at the surface of earth 4 F = 3 G m r F = 4 ( 3.14×6.67× 10 11×5500×10 × 6.38 × 10 ) 6 3 F = 98.296 N