Torque produced by balls in Fantastic Contraption

The fun part about exploring the physics of [Fantastic Contraption](http://fantasticcontraption.com/) is coming up with new setups to test ideas. Torque is not too difficult to set up. Here is what I did:


In this setup, I have a "turning ball" with a wood stick attached to the side. I increased the length of the stick until the ball does not turn. At this point, the torque from the gravitational force on the stick is equal to the torque from the ball. I can use [Tracker Video Analysis](http://www.cabrillo.edu/~dbrown/tracker/) to find the lengths of the two wood sticks. The torque from each stick will be its gravitational weight times the perpendicular distance to the center of the turning ball.


In order to calculate the gravitational force, I need the mass of each "stick". [From my previous post](http://scienceblogs.com/dotphysics/2008/10/physics-of-fantastic-contrap…), I found that the mass density per length for sticks was


where m_b is the mass of a ball and U is the diameter of a ball. I also need to find the horizontal distance from the center of the stick to the center of the ball. I will call the top stick 1 and the bottom 2. This gives:


Notice that stick 2 is connected at the same x-value as the ball, so I did not need to add the radius of the ball to its r value. Now I can calculate the total torque:


Although I do have an ok value for U in meters, I do not have a value for the mass of the ball, so no point in multiplying in the constant g. Anyway, let me test this. If this is true, how many balls could I hang right off the circle and lift? In that case, r would be 0.5 U (U is the diameter). So if the torque is around 3, I should be able to lift 6 balls (depending on the mass of string used). Let me try it.


I love it when a plan comes together. Actually, this was a little more than the weight of 6 balls, it also had the short length of water-sticks. But also, according to my calculation, this should not be able to lift 7 balls. Again, success.