Question

8. Bonus: Consider the gravitational force of attraction F (r) on a mass m located at a point r R3 which is exerted by a mass M at the origin r 0, GMm r. (a) From your results in 7(b), find the scalar-valued function V(r) such that F(r)VV(r) with the condition V(r) -0 as lrl->oo. The function V(r) is the potential energy function associated with the gravitational force F(r). The existence of a scalar-valued potential energy function V R-Rimplies that the vector field F is conservative. (This concept will be discussed later in the course.)

Answer 1

F(r) = -GM_m/r^3 r vector (or) = -GM_m/r^2 r^cap F(r) = -nabla vector V (r) (or) v(r) = -integral F vector (r) middot dr^bar = -integral c - GM_m/r^2 dr = G M_m integral dr/r^2 + C v(r) = -GM_m/r + C so, as r rightarrow, v(r) = 0 So 0 = 0 + c = > c = 0 So v(r) = -GM_m/r Scalar valued function