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.

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

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 mass of $$1$$ kg is suspended by means of $$3$$ springs as shown in Figure. The spring constants $$\,{K_1},\,{K_2}\,\,$$ and $$\,{K_3},$$ are respectively $$1$$ $$kN/m,$$ $$3$$ $$kN/m$$ and $$2$$ $$kN/m.$$ The natural frequency of the system is approximately )