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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 nonmassless 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 thats 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 .. :)
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 thats 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 .. :)
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