My model of torque in a rod with mass

[Levis2]

I've always seen attempting to create my own models in physics as a fun and enriching pastime, even though my models should come out wrong :) This time i have attempted to model the torque in a non-massless rod affected by gravity.

This expression is with the force of gravity acting perpendicular on the rod, but it can be easily modified for another angle of attack. What i imagined was a infinitesimally small subdivisioning (dunno if that's a word, I'm scandinavian:) of the rod. I also assumed that the density of the rod was uniform all over. Then i looked at each of the pieces, in order to determine the torque that was created by gravity from each "piece of mass". All the masses of the pieces are obviously equivalent, but each one is at a farther distance from the pivot point and thus produce a higher torque under the influence of gravity, so i ended up using an infinite sum, in order to describe it.

I imagined cutting the rod into n pieces. Then each would posses the mass of m/n, where m is the entire mass of the rod. Then of course the torque each piece generates is given my gm(r_i)/n, where r_i is the length that the given piece is away from the pivot point, which is ofcoure r_i=r/n*i, where i runs through the integers as one progresses from the pivot point to the end. I know this is confusing, but i ended up with this;

(Attachment of a screenshot, i am no good with latex :()

It is probably gibberish, but can you make any sense out of my thoughts? :)

EDIT: i added a poor drawing of the situation, i hope it helps .. :)

 

[Levis2]

I have now studied the present material on this subject, and it seems that the torque at a given point in a rod created by gravity can be thought of as the torque from the entire gravitational force affecting the mass of the rod placed in it's center of gravity. This is not so obvious to me, and I'm not sure i understand why such a simplification can be justified??

In order to see if my model was correct, i did a calculation on a horizontal rod, with the axis of rotation in the left end, as my previous sketch. The rod has a mass of 1kg, and is 10m long. It is uniform, and has a center of gravity at 10m/2m=5m. So the torque created by gravity in the left most point of the rod is 1kg*9.82m/s^2*5=49.1 Nm.

I then used my model to see if i could achieve the same using my model, and the results from maple are attached as a screenshot;

As you can see, the results are exactly identical.. I would appreciate if someone is able to explain to exactly why it is, that such a simplification is possible? I don't see why my model can be simplified to that :)