Two large masses with mass moments of inertia i1, and i2, are fixed at either end of a uniform shaft with a length l and polar area moment of inertia j. find the characteristic equation for the torsional vibration of the system

A2-m rigid tank initially contains saturated water vapor at 100 kpa. the tank is connected to a supply line through a valve. steam is flowing in the supply line at 600 kpa and 300 c. the valve is opened, and steam is allowed to enter the tank until the pressure in the tank reaches the line pressure, at which point the valve is closed. a thermometer placed in the tank indicates that the temperature at the final state is 200°c. determine (a) the mass of steam that has entered the tank (b) the amount of heat transfer.