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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]