The thin disk of radius r and mass 3m is attached to a slend

The thin disk of radius r and mass 3m is attached to a slender rod of length L and mass m which is attached to a pin joint at A. The disk spins about the rod at a constant rate of psi. A torque T is applied to the vertical shaft to cause it to rotate with an angular velocity of phi at the same time the vertical shaft moves with a constant velocity of u upward. Derive the equations of motion for the system using Lagrange\'s Equations.

Solution

Kinetic energy of the system.

Spinning disk: 1/2 I (shi\')⁾² + 1/2(Lsin theta)m( phi\')² + 1/m u²  = ke of disk about own axis L + KE of mass of disk rotating about vertical axis with radius L sin theta + ke of mass moving up with vel u

PE = mgLsintheta

Apply Lagrange eqns and equate to torque GAMMA.

Best