A thin cylinder of $$100$$mm internal diameter and $$5$$mm thickness is subjected to an internal pressure of $$10$$ MPa and a torque of $$2000$$ Nm. Calculate the magnitudes of the principal stresses.

Below Fig show a planetary gear train. Gears 2, 4 and 5 have 24, 40 and 144 teeth respectively. Gear 5 is fixed. Gear 2 is rotating clockwise at 700 rpm. What will be the rpm of the arm and gear 4?

A rod of length $$1$$ $$m$$ is sliding in a corner as shown in Fig.$$2.7.$$ At an instant when the rod makes an angle of $$60$$ degrees with the horizontal plane, the velocity of point $$A$$ on the rod is $$1$$ $$m/s$$. The angular velocity of the rod at this instant is

A cylinder of mass 1 kg and radius 1 m is connected by two identical springs at a height of $$0.5m$$ above the center as shown in the Fig. The cylinder rolls without slipping. If the spring constant is $$30$$ $$kN/m$$ for each spring, find the natural frequency of the system for small oscillations.