(Solved): Question 1 Motion of a Rigid Rod (25 Points) As shown in Figure 1, a thin uniform rod of mass m an ...

Question 1 Motion of a Rigid Rod (25 Points) As shown in Figure 1, a thin uniform rod of mass $$m$$ and length $$2?$$ is pin-jointed at $$O$$. A force $$\mathbf{P}$$ of constant magnitude $$P$$ is transmitted by a cable to point $$A$$ which is located at the end of the rod. During the ensuing motion, a vertical gravitational force $$mg{\mathbf{E}}_x$$ also acts on the rigid body. The position vectors of the center of mass $$C$$ of the rigid body, the point $$A$$ relative to the point $$O$$, and the point $$B$$ relative to $$O$$, and the angular momentum of the rigid body relative to $$C$$ have the representations $$\overline{\mathbf{x}}=?{\mathbf{e}}_x,{\mathbf{x}}_A=2?{\mathbf{e}}_x,{\mathbf{x}}_B=2?{\mathbf{E}}_y,\mathbf{H}=\left(I_{zz}=\frac{m{?}^2}{3}\right)\stackrel{?}{?}{\mathbf{E}}_z.$$ We also note that $$?{\mathbf{x}}_B?{\mathbf{x}}_A?=2?\sqrt{2(1?\sin (?))}.$$ Figure 1: A rigid body of mass $$m$$ is free to rotate about $$O$$. The rigid body is subject to a gravitational force and a force $$\mathbf{P}$$ which has a constant magnitude. (a) (5 Points) Establish expressions for the angular momentum $${\mathbf{H}}_O$$ and kinetic energy $$T$$ of the rigid body when it is rotating about $$O$$. (b) (5 Points) Draw a free-body diagram of the rigid body when it is rotating about $$O$$. For credit, give clear representations for the forces and moments in this diagram. (c) (5 Points) Show that the following differential equation governs $$?$$ when the body is rotating about $$O$$ : $$\frac{4m{?}^2}{3}\ddot{?}=?mg?\sin (?)+\frac{2P?\cos (?)}{\sqrt{2(1?\sin (?))}}.$$ (d) (5 Points) Show that the force $$\mathbf{P}$$ has a potential energy function $$U_P=P?{\mathbf{x}}_B?{\mathbf{x}}_A?.$$ (e) (5 Points) Starting from the work-energy theorem (5), prove that the total energy $$E$$ of the rigid body is conserved. For credit, supply an expression for the total energy $$E$$.