Question

In a certan region of space the gravitaional field is given by -k/r where r is the distance and k is a constant. if the gravitational potential at r= a be v then what is the expression for gravitational potential ?

Answer 1

• `U(r) = k ln(r/a) + V`

Explanation:

If the gravitational field is inverse (as opposed to inverse-square), then:

• `bbg = - k/r \ bb hat r`

The increase in potential `Delta U` of a unit mass that is forced to move from `rho = a` to `rho = r`, where `r > a`, will equal the work done on that unit mass:

`delta W = bb F * bb x`

`Delta U = W = int_C bb F * d bb x`

`Delta U(r) = int_(rho = a)^r \ k/rho \ d rho = (k ln rho)_a^r = k ln(r/a)`

Because `U(a) = V`:

`U(r) = Delta U(r) + V`

• `U(r) = k ln(r/a) + V`