A point mass is placed at a distance a from one end of a rod of mass M and length l on the line which passes through the central axis of the rod. The potential energy of the system.

Options

A. `U =- ( GM m )/( a) log _(e) ((a + 1 )/(a))`
B. ` U =- G M m ((1)/(a) - (1)/(a +1))`
C. `U=- ( GMm )/(1) log _(e) ((a + 1)/( a))`
D. `U =- ( GMm )/(a)`

Answer

C

Explanation

Mass per unit length of the rod `=( M)/(1)`
Consider a small element of width dx of the rod at a distance x from m.
P.E. of the element `dU =- ( G m ((M )/(1)) dx)/( x)`
`therefore U = - (GM m )/(1) int _(0) ^( a+1) (dx)/(x)`
`=- ( GM m )/(1) [ log _(e) (x) ] _(a) ^(a +1)`
`U = - ( Gm m )/(1) log _(e) ((a + l )/( a ))`