If you think of this as a simple pendulum, increasing the weight on the end of the rod should not affect the speed of rotation of the rod, and therefore not affect the distance that the ball bearing travels.
But the rotational velocity of a conventional pendulum weight
does change, twice per period of oscillation. I think it would graph out as some sort of damped sine wave (similar to the middle of the grey illustration here - http://www.oz.net/~coilgun/theory/dampedoscillator.htm ) but move all the negative peaks up to the X-axis. (If you don't care about the direction of the motion, then just leave it centred on the X-axis.) The positive and negative peaks represent the velocities as it passes through bottom dead centre.
What
doesn't change in a pendulum, is the period of the oscillation, which is just a time interval.
Before the rod has even completed one cycle of oscillation, the masses change. This sounds more like a trebuchet-style siege engine problem.
In case everybody is not familiar with trebuchets, here's how they work:
There is a beam (B) supported by an off-centre axle, with a large counterweight (CW) mass on the short end (length "CA" or Counterweight Arm) and a sling with a pouch on the longer end (length "TA" or Throwing Arm). CA+TA = B. The CW may or may not be suspended by a short linkage that is hinged at the tip of the CWA. We'll ignore that for this equation to follow. The projectile (P) is some fraction of CW.
There is no "stop bar" to fling the projectile out of the pouch of the sling; rather, one of the two cords of the sling is allowed to slip off the end of a pin on the tip of the throwing arm, at exactly the right time for a 45 degree launch angle.
The Beam Ratio (BR) is the ratio of the length TA over CA.
The Mass Ratio (MR) is the ratio of mass CW over P.
It is assumed that a launch angle of 45 degrees is used for maximum range.
The next problem, when sketching out a preliminary design for a trebuchet, is to determine the optimum values of both BR and MR to obtain the maximum range. If you increase BR, you would expect greater velocity of the tip of the TA, but then it has less mechanical advantage, so it will go slower.
There a thing called "phsstpok's Rule" among trebuchet hobbyists, which is a rule of thumb for calculating the initial values of BR and MR. phsstpok is the user name of a guy that frequents the "Catapult Message Board", which has now migrated to
http://www.thehurl.org/tiki-view_articles.php
MR = 20 x BR
Typically, you start with one or more given values of B, CW, or P. You might wish to throw an egg, and you have a hockey stick to use for B. Or you might have a 10 kg dumbell and want to build a treb that will fit in the trunk of a car. whatever. For most competitions, there is an overall size limit and a given object for a projectile.
Given a P of 2 kg, a CW of 200 kg, and a 4 meter Beam:
MR = 200/2 = 100
Solving for BR:
BR = MR/20 = 100/20 = 5
length TA becomes BR/(BR+1) x B = 5/6 x 4.00 m = 3.34 m
and CA = 4.00 m - 3.34 m = 0.66 m
Badda-boom badda-bing. The rest is just common-sense carpentry, raising the axle high enough that the CW doesn't bottom out on the launch trough when it's fired and so on. This does not give you any estimates of the range of the hurled projectile, only some design parameters to obtain the
best practical results.
Now, you ask, why not just load up on the CW mass and thereby increase the range?
In a real-world trebuchet, there are stresses that can damage the trebuchet itself, such as "dry-firing" - firing it off with no projectile can bend the axle or snap the beam, with tragic results. An extremely high mass ratio is similar to dry-friring. You could make the beam thicker, but then it increases the inertia of the beam. If you're making a very large trebuchet (to throw a Mini-Cooper car, or a cow), you don't want to add more CW than necessary, because it'll be difficult and dangerous enough just to cock it into the firing position.
What I discovered with one of my small trebuchets, with a 1 foot long beam -
http://northernelectric.ca/medieval/acme/acme.htm - is that when I increased my CW from 393 grams to 792 grams (nearly double) to hurl the same wooden bead (unknown mass, maybe 3-5 grams? let's say 4 gm), my range only increased from about 7 meters to 7.65 meters, something like 9% further.
I expected better than that. Then I looked at the numbers. My mass ratio was initially about 100:1, but my beam ratio was only 3:1, where it should have been 5:1. The newer, bigger CW cast from lead gave it a mass ratio of close to 200:1, which calls for a beam ratio of 10:1.
In a blinding flash, it occurred to me that I was operating it in nearly a dry-fire state. If you ignore the very small mass (and inertia) of the beam itself,
as you increase the mass ratio the CW starts approaching free-fall conditions 9.8 meters per second squared. Even when you include the beam inertia, it eventually approaches free-fall. The only way I was going to get it to drop any faster was to move to a bigger planet, or to operate it in an elevator during its brief acceleration stage.
Now, there is still one advantage to increasing the CW mass - you can throw
more massive projectiles to nearly the same distance as the lighter projectiles, but that's changing more than one parameter.
There is software available that will allow you to consider parameters such as counterweight hanger length, sling length, release pin angle (critical for timing), arm lengths and mass, and so on, and they'll even give you a flight path, projectile velocity, and range, but for seat-of-the-pants trebuchet design, it's hard to do much better than phsstpok's Rule.
More details here:
http://northernelectric.ca/medieval/siege/phsstpoks_rule.htm
More of my siege engines here:
http://northernelectric.ca/medieval/index.htm
For what it's worth, after a trebuchet has released the projectile, the arm and counterweight continue to oscillate a couple of times before coming to rest, much like a pendulum does. If it's a hinged hanging counterweight model, the oscillations are just a bit more complex.