find the gravitational force of attraction between a uniform sphere of mass M and a uniform rod of length L and mass m, placed such that r is the distance between the centre of the sphere and the near end of the road.

Options

A. `("GMm")/(r (r + l))`
B. `(GM)/(r^(2))`
C. `Mmr^(2) + l `
D. `(r^(2) + l)`mM

Answer

A

Explanation

The force of attraction between sphere and shaded position dF = GM `(((m)/(l) dx))/(x^(2))`
`F = int_(r)^(r+l) ("GMm")/(lx^(2)) dx = ("GMm")/(l) int_(r)^(r+l) (1)/(x^(2)) dx `
`= ("GMm")/(l) int_(r)^(r+l) x^(-2) dx = ("GMm")/(l) [ (x^(-2+1))/(- 2 + 1) ]_(r)^(r+l)`
= `("GMm")/(l) [ x^(-1)]_(r)^(r+l) = ("GMm")/(l) [(1)/(x)]_(r)^(r+l) = ("GMm")/(r(r+l))`
` (##DSH_NTA_JEE_MN_PHY_C07_E03_010_S01.png" width="80%">