Example of gravitation
Gravitation Example. A solid sphere of lead has a mass M and radius R. A spherical cavity is made in it (fig.1) , the boundary passing through the centre O and also touching the boundary of the solid sphere, Find the gravitational force on a mass m placed at point P, distance r from O, along the line of ceneres.
Solution: Let M be the mass and ρ be the density of the entire solid sphere. Let C be the centre of the spherical cavity. For an external point, the massive sphere behaves as if its entire mass is concentrated at its centre O. The gravitational attraction on a body of mass M at P due to complete solid sphere with centre at O is:
F1=GMm/r2 ; along PO
The diameter of smaller sphere at place of hollow spherical cavity is R . Its radius is R/2. Therefore mass of this sphere is :
M’ =(4/3)π(R/2)3ρ=(1/8){(4/3)πR3ρ}=M/8
The gravitational force on m due to this small sphere with centre at C is
F2 = GM’m/(r-R/2)2 = {G(m/8)m}/(r-R/2)2 along PO.
As both the forces F1 and F2 are acting in the same direction, the gravitational pull on m due to the lead sphere with spherical cavity is :
F= F1 - F2 =(GMm/r2 ) – GMm/{8(r-R/2)2}=GMm/ r2[1-1/8(r-R/2)2]