Question

Mountain pull. A large mountain can slightly affect the direction of "down" as determined by a plumb line. Assume that we can model a mountain as a sphere of radius R -3.00 km and density (mass per unit volume) 2.6 x 103 kg/m3. Assume also that we hang a 0.500m plumb line at a distance of 3R from the sphere's center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere? Number Units the tolerance is +12%

Answer 1

Let:

M be the mass of the large sphere,

m be the mass of the plumb bob,

G be the gravitational constant,

g be the acceleration due to gravity,

r be the density of the material of the large sphere,

T be the tension in the string,

a be the inclination to vertical of the plumb line,

s be the required distance

L be the length of the plumb line.

Then:

M = (4pi/3) R^3 r

= (4pi/3)(3 * 10^3)^3 (2.6 * 10^3)

= 2.94 * 10^14 kg.

Resolving horizontally and vertically for the plumb line:

T cos(a) = mg ...(1)

T sin(a) = GMm / (3R)^2 ...(2)

s = L sin(a)

As a is very small, approximation is justified, and it is reasonable to replace sin(a) by tan(a):

s = L tan(a) ...(3)

Dividing (2) by (1):

tan(a) = GM / 9R^2 g

From (3):

s = LGM / 9R^2g

= (0.5)(6.67 * 10^(-11))(2.94 * 10^14) / 9(3 * 10^3)^2 (9.81)

= 1.23 * 10^-5 m