### Part C : Dynamics of Beam

Here we finally pull everything together to devœlope a differetial equation for $$\ddot{\alpha}$$ which contains only constants reflecting the physical characteristics of the beam plus all independent integration variables which are :

$$\alpha \; \dot{\alpha} \; \beta \; \dot{\beta} \; \gamma \dot{\gamma}$$

In Fig.(C.1) the three forces, $$P_w$$ from the counter-weight, $$P_p$$ from the pumpkin and $$M_b g$$ the weight of the beam act to rotate the beam about the pivot.

\begin{equation*} I_b \ddot{\alpha} = a P_w + b P_p - M_b\: g\: c\: \sin\alpha \tag{C.1} \end{equation*}

Here :

Mb = mass of the beam
c  = location of center of mass of beam, can have negative or positive value
Ib = mass moment of inertia of beam w.r.t. pivot
Pw = force of counter-weight onto beam, Eq.(A.7)
Pp = force of pumpkin rope onto beam, Eq.(B.5) or (B.7), respectively.


\begin{equation*} I_b \ddot{\alpha} = a^2 M_w ( w_1 - w_2 \ddot{\alpha} ) + b a M_p (p_1 - p_2 \ddot{\alpha} ) - M_b\: g\: c\: \sin\alpha \end{equation*}

In order to relate this equation more easily to what we use in the program, see file diffeqs.js, we divide through by Mb a2 and we utilize the abbreviations $$w_1, w_2, p_1, p_2$$ as indicated in Eqs. (A.7a), (B.5) and (B.7), respectively .

\begin{equation*} \left( \frac{I_b}{a^2 M_b} + \frac{M_w}{M_b} w_2 + \frac{b}{a} \frac{M_p}{M_b} p_2 \right)\;\ddot{\alpha} = \frac{M_w}{M_b} w_1 + \frac{b}{a} \frac{M_p}{M_b} p_1 -\frac{c}{a} \frac{g}{a} \sin\alpha \tag{C.2} \end{equation*}

Zig Herzog; hgnherzog@yahoo.com
Last revised: xx