Recently I have been trying to solve a problem relating to trebuchet mechanics for physics, but I am stuck on the mathematics.
The problem goes like this;
For a trebuchet with arm ratio $d_1\over d_2$, sling length $l$, counterweight drop height $h$, projectile mass $m$ counterweight mass $M$ and sling angle to arm $\theta$ (Trebuchet Variables Picture) calculate the final kinetic energy of the projectile (on release) to determine the effect of changing the mass of the projectile each launch.
My solution is as follows (it is shortened):
$E_k={1\over 2} mv^2,$ $K_r={1\over 2} I_t ω^2,$ $E_g=mgh$
$ω=\sqrt{2K_r\over I_t}$
$\therefore v=r\sqrt{2K_r\over I_t}$
$\therefore E_k={1\over 2} mr\sqrt{2K_r\over I_t}^2$
$Assuming$ $100%$ $energy$ $transfer:$
$E_g=K_r$
$I_t=Σmr^2$
$∴I_t=m(d_1+{l\over \sinh(θ)} )^2 +Md_2^2$
$∴E_k={1\over 2} mr \sqrt{2Mgh\over m(d_1+{l\over \sinh(θ)})^2 +Md_2^2}^2$
$Since$ $r=d_1+d_2+{l\over \sinh(θ)}$
$∴E_k={m(d_1+d_2+{l\over \sinh(θ)})^2 Mgh\over (m(d_1+{l\over \sinh(θ)})^2 +Md_2^2}$
This solution assumes a lot, (In Picture Form for Space) can anyone present a more accurate solution than mine which still fulfills the initial requirements?
Also, any ideas of how to calculate the efficiency of a system like this? If you have any ideas please contribute :)