A planet of mass M has its center fixed at the origin. The gravitational potential energy of a mass m at a point r = (x, y, z) is V = -GMm/r, where r =sqrt(x^2 + y^2 + z^2) and where G is a positive constant. The force on m is given by F = -grad(V).
Calculate F and show that F(dot)dr is exact.
Hence calculate the work done by F on a particle which travels from infinity to r.
Show how to get the answer please =)

Question

A planet of mass M has its center fixed at the origin. The gravitational potential energy of a mass m at a point r = (x, y, z) is V = -GMm/r, where r =sqrt(x^2 + y^2 + z^2) and where G is a positive constant. The force on m is given by F = -grad(V).
Calculate F and show that F(dot)dr is exact.
Hence calculate the work done by F on a particle which travels from infinity to r.
Show how to get the answer please =)

anonymous · Answer

I got GMm/r not sure if its correct though

anonymous · Answer

since there exists a function (V in this case) that means F is exact, so we can use fundamental theorem of Calculus to evaluate F from r to infinity, I dont know how to evaluate it at infinity so I thought ill make it zero and I get GMm/r. Who is good with Line integrals Help Please

dominusscholae · Answer

Since F(dot)r is a gradient of another function, it is by definition exact. You are on the right track. You can evaluate it at infinity using the limit. Since $$\lim 1/r$$$$=$$ 0 as it goes to infinity, so your upper bound for GMm/r is zero. That leaves your lower bound, R, intact. Thus, your integral is 0-GMm/r, or GMm/r.

dominusscholae · Answer

correction: 0-GMm/R = -GMm/R