The earth does not have a uniform density; it is most dense at its center and least dense at its surface. An approximati
Posted: Thu May 05, 2022 5:41 pm
The earth does not have a uniform density; it is most dense at
its center and least dense at its surface. An approximation of its
density is p(r) = A - Br, where A = 12,700 kg/m? and B = 1.50 x
10-3 kg/ mt. Use R = 6.37 x 106 m for the radius of the earth
approximated as a sphere. (a) Geological evidence indicates that
the densities are 13,100 kg/m? and 2400 kg/m? at the earth's center
and surface, respectively. What values does the linear
approximation model give for the densities at these two locations?
(b) Imagine dividing the earth into concentric, spherical shells.
Each shell has a radius r, thickness dr, volume dV = 4712 dr, and
mass dm = p(r) dV. By integrating from r= 0 to r = R, show that the
mass of the earth in this model is M=‡mR3 (A-}BR). d) 3 TR? (A
{BR). (c) Show that the given values of A and B give the correct
mass of the earth to within 0.4%. (d) Because a spherical shell
gives no contribution to g inside it, show that g(r) = inGr (A -
2Br) inside the earth in this model. (e) Verify 3 that the
expression of part (d) gives g = O at the center of the earth and g
= 9.85 m/s? at the surface. (f) Show that, in this model, g does
not increase uniformly with depth but rather has a maximum of
47GA2/9B = 10.01 m/s2 at r= 2A /3B = 5640 km.
its center and least dense at its surface. An approximation of its
density is p(r) = A - Br, where A = 12,700 kg/m? and B = 1.50 x
10-3 kg/ mt. Use R = 6.37 x 106 m for the radius of the earth
approximated as a sphere. (a) Geological evidence indicates that
the densities are 13,100 kg/m? and 2400 kg/m? at the earth's center
and surface, respectively. What values does the linear
approximation model give for the densities at these two locations?
(b) Imagine dividing the earth into concentric, spherical shells.
Each shell has a radius r, thickness dr, volume dV = 4712 dr, and
mass dm = p(r) dV. By integrating from r= 0 to r = R, show that the
mass of the earth in this model is M=‡mR3 (A-}BR). d) 3 TR? (A
{BR). (c) Show that the given values of A and B give the correct
mass of the earth to within 0.4%. (d) Because a spherical shell
gives no contribution to g inside it, show that g(r) = inGr (A -
2Br) inside the earth in this model. (e) Verify 3 that the
expression of part (d) gives g = O at the center of the earth and g
= 9.85 m/s? at the surface. (f) Show that, in this model, g does
not increase uniformly with depth but rather has a maximum of
47GA2/9B = 10.01 m/s2 at r= 2A /3B = 5640 km.