The surface mass density of the disk 0= ap2, so the gravitational field is 8=-2nGok=constant 3. The Earth is not, in fact, a sphere of uniform density. a high-density core is surrounded by a shell or mantle of lower-density material. Suppose we model a planet of radius R as indicated in Figure 5 A core of density 2p and radius 3R/4 is surrounded by a mantle of density p and thickness R/4. Let M be the mass of the core and m be the mass of the mantle. (This is not an accurate model of the interior structure of the earth but makes for an interesting and tractable problem. Mantle density p (a) Find the mass of the core (b) find the mass of the mantle (c)What is the total mass of the planet(core mantle)? (d) Find the magnitude of the gravitational field at the surface of Core density 2p the planet (e)What is the magnitude of the gravitational field at the interface between the core and the mantle Solution (a)The mass of the core is 丌(R)3=zRp (b) The mass of the mantle is m=p Mante=p l mp34 R]=0TR'p (c)The total mass of the planet is M+m=rRp+.Rp=zrR'p (d) The magnitude of the gravitational field at the surface of the planet is G(M+m)91 丌GRp=1.896xGR R 48 (e) The magnitude of the gravitational field at the interface between the core and the mantle GM 16 g =G·mR`p·=2zGRpThe surface mass density of the disk 2 R M π σ = , so the gravitational field is constant ˆ g = −2πGσ k = r 3. The Earth is not, in fact, a sphere of uniform density. A high-density core is surrounded by a shell or mantle of lower-density material. Suppose we model a planet of radius R as indicated in Figure 5. A core of density 2ρ and radius 3R/4 is surrounded by a mantle of density ρ and thickness R/4. Let M be the mass of the core and m be the mass of the mantle. (This is not an accurate model of the interior structure of the Earth but makes for an interesting and tractable problem.) (a) Find the mass of the core . (b) Find the mass of the mantle (c) What is the total mass of the planet (core + mantle) ? (d) Find the magnitude of the gravitational field at the surface of the planet. (e) What is the magnitude of the gravitational field at the interface between the core and the mantle ?. Solution: (a) The mass of the core is ρ ρ π π ρ 3 3 core 8 9 ) 4 3 ( 3 4 M = V ⇒ 2 ⋅ R = R (b) The mass of the mantle is ρ ρ π π π ρ 3 3 3 mantle 48 37 ) ] 4 3 ( 3 4 3 4 m = ⋅V = ⋅[ R − R = R (c) The total mass of the planet is π ρ π ρ π ρ 3 3 3 48 91 48 37 8 9 M + m = R + R = R (d) The magnitude of the gravitational field at the surface of the planet is πGRρ πGRρ R G M m g 1.896 48 ( ) 91 surface 2 = = + = (e) The magnitude of the gravitational field at the interface between the core and the mantle is π ρ πGRρ R G R R GM g erface 2 9 16 8 9 ) 4 3 ( 2 3 2 int = = ⋅ ⋅ = 3R/4 R/4 Core density 2ρ Mantle density ρ Fig.5