Consider a steel (Young’s modulus $$E = 200$$ $$GP$$a) column hinged on both sides. Its height is $$1.0$$ $$m$$ and cross-section is $$10 mm$$ $$ \times $$ $$20 mm$$. The lowest Euler critical buckling load (in $$N$$) is __________ .

A wheel of radius r rolls without slipping on a horizontal surface shown below. If the velocity of point P is 10m/s in the horizontal direction, the magnitude of velocity of point Q (in m/s) is ______

A pinion with radius $${r_1}$$, and inertia $${{\rm I}_1}$$ is driving a gear with radius $${r_2}$$ and inertia $${{\rm I}_2}$$ . Torque $${\tau _1}$$ is applied on pinion. The following are free body diagrams of pinion and gear showing important forces ($${F_1}$$and $${F_2}$$) of interaction. Which of the following relations hold true?

Consider a slider crank mechanism with nonzero masses and inertia. A constant torque $$\tau $$ is applied on the crank as shown in the figure. Which of the following plots best resembles variation of crank angle, $$\theta $$ versus time