The Hinged Counterweight
Using torques to find the motion of this model is difficult at best. So, we will look at the moment when the projectile is released. Total energy is conserved, since there is no friction in the treb, and all external forces are conservative. The beam is uniform. Thus, KEproj=DPEcw+DPEb+DPEproj-KEcw-KEb. Where DPEb and DPEproj are negative. Since the beam has to be rotating, and both the beam and the projectile must move up from the initial position, only KEcw can be removed. This happens if the counterweight stalls when the projectile is released. The motion can be determined using Lagrange's method, which gives this treb about twice the efficiency of the see saw treb.
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