Four_FUN

Solved the first one mathematically, although it turned out to actually be easier than I had thought... For a uniform ring, you only need to consider the x-direction of the force. So for the contribution of a small unit mass dM, the force dF_x on the mass m can be given by df_x = dF*cos(\theta). They way I am picturing the ring, \theta is the angle between the direction the force dF is acting and the x-axis. Using basic trigonometry, cos(\theta) = x / (A^2 + x^2)^1/2. Also we know dF = GmdM / (A^2 + x^2). therefore you get : dF_x = (Gmx / ((A^2 + x^2)^3/2)) dM. I f you integrate the LHS you get F_x, and the right hand side is an integral in terms of dM. Since everything else in the RHS is a constant, this is basically a very easy integral. Note that this solution is much more mathematically rigorous than the previous solution. Although I generally support strict mathematical practice, as long as you understand the concepts behind the procedure you will be fine. Hope I was helpfull and good luck with the rest of your homework.

For your second problem, all the gravitational force is acting in the x-direction (positive or negative depends on how you define the axes). Since the rod has uniform mass, it has a linear mass density of \lamda = (M/L). Each small segment of mass dm = \lamda*dl where dl is the length of the small mass segment. They way I understood the problem is that dF=(-GmdM) / (x+n*dl), where lim (n-> infinity) [ n*dl = L]. In order to solve this you take the integral from 0 to infinity of (Gm*(M/L)*dl) / (x+n*dl) in terms of n. You can take Gm*M/L outside the integral and you are left to calculate the integral |0infinity [ dl / (x+n*dl)] dn. All the mathematical manipulations are not entirely correct here because I arrived at this result using the idea of first principles. Also it is important to consider dl and x as constants and only n as the variable.... I feel that you can prceed from here by yourself. If you solve the integral and multiply with the terms out side (G, m , etc). you should get the right answer. I have not worked on your other question yet. If I solve it I will inform you. Good luck with the rest of your homework