The gravitational force of attraction between a uniform sphere of mass M and a uniform rod of length l and mass in as shown in figure is

Options

A. `(GMm)/(r (r + 1))`
B. `(GM)/(r^2)`
C. `Mm^2`
D. `(r^2 + 1) mM`

Answer

A

Explanation

Mass per unit length of the rod = m/l
Mass of element of length , `dx = m/l dx`

Let dF be the gravitational force of attraction between this element and sphere . Then , `dF = GM ((m/l dx) )/(x^2)`
` therefore F = int_r^(r +l) (GMm)/(lx^2) dx = (GMm)/(l)int_(r)^(r +1) (1)/(x^2) dx`
`= (GMm)/(l) int_(r )^(r + l) x^(-2) dx = (GMm)/(l) [ (x^(-2 + 1) )/(-2 + 1) ]_r^(r + 1) = (GMm)/(r (r +l))`
If `r gt gt l, ` then ` F = (GmM)/(r^2)`