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 = 2.00 km and density (mass per unit volume) 2.6 × 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?

Answer 1

we first find the mass of the sphere

M = density * volume

M = 2.6e3 * ( 4 / 3) * 2000³

M = 8.71e13 kg

If plumb lines makes an angle

then

In X - direction

T sin - F ---------- (1)

where F is gravitational force = GMm / 9R²

so,

In y - direction

Tcos = mg --------- (2)

divide (1) by (2) , we have

tan = GM / 9gR²

from figure

a = Ltan

so,

a = GML / 9gR²

a = 6.67e-11 * 8.71e13 * 0.5 / 9 * 9.8 * 2000²

a = 8.236e-6 m