17.31  (Rigid Body Work Energy Equation)  (Sorry for the tilt!)

Comments/Hints:
1.  Disk A rolls down the right hand rope;  thus disk A stores kinetic energy in both translation (T = 1/2·m_A·v_A² ) and rotation (T = 1/2·I_A·w_A² ).
2.  Disk B rotates about a fixed pin, so it stores energy in rotation only ((T = 1/2·I_B·w_B² ).
3.  The only work done is the mass center of disk A falling 3 ft.
4.  Use the work-energy equation:  ST₁ + SU_1-2 = ST₂
5.  You need kinematics relationships:  w_B = 2w_A, and v_A = 5/12·w_A.
6.  Answers:   w_A = 17.83 rad/sec, v_A = 7.43 fps, and w_B = 35.66 rad/sec.