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.

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 )

Match List - $${\rm I}$$ with List - $${\rm II}$$

List - $${\rm I}$$
$$A.$$ Cetane number
$$B.$$ Approach and range
$$C.$$ $${\left( {{{\delta T} \over {\delta P}}} \right)_h} \ne 0$$
$$D.$$ $$dh = {c_p}\,\,dT,$$ even when pressure varies

List - $${\rm II}$$
$$1.$$ Ideal gas
$$2.$$ Vander Waals gas
$$3.$$ $$S.\,{\rm I}.$$ engine
$$4.$$ $$C.\,{\rm I}.$$ engine
$$5.$$ Cooling towers
$$6.$$ Heat exchangers