(Solved): Problem 1 [30 Points] Consider the following mechanical system with F(t) as the sole input and the ...

Problem 1 [30 Points] Consider the following mechanical system with $$F(t)$$ as the sole input and the angular position of $$m_3$$ as the sole output: Assumptions - Coordinates are defined from static equilibrium. - Mass $$m_2$$ moves without friction. - Mass $$m_1$$ moves relative to $$m_2$$, without friction. - Mass $$m_1$$ will not fall off $$m_2$$. - Pulley $$m_3$$ rolls without slipping, and will not fall off the ledge. - The moment of inertia of mass $$m_3$$ about its center of mass is $$J$$. Problem Statement a) Determine the number of degrees of freedom the system has. Fully justify your answer. Clearly define the coordinates. Derive the applicable kinematic constraints, if any. b) Draw free-body diagrams and derive the governing differential equations of motion. c) Put the equations of motion derived in Part b) into state-space matrix form.