An initially stress-free massless elastic beam of length $$L$$ and circular cross-section with diameter $$d (d < < L)$$ is held fixed between two walls as shown. The beam material has Young's modulus $$E$$ and coefficient of thermal expansion $$\alpha .$$

If the beam is slowly and uniformly heated, the temperature rise required to cause the beam to buckle is proportional to

For an inline slider-crank mechanism, the lengths of the crank and connecting rod are 3 m and 4 m respectively. At the instant when the connecting rod is perpendicular to the crank, if the velocity of the slider is 1 m/s, the magnitude of angular velocity (upto 3 decimal points accuracy) of the crank is _________ radian/s.

In an epicyclic gear train, shown in the figure, the outer ring gear is fixed, while the sun gear rotates counterclockwise at 100 rpm. Let the number of teeth on the sun, planet and outer gears to be 50, 25 and 100, respectively. The ratio of magnitudes of angular velocity of the planet gear to the angular velocity of the carrier arm is _________.

The damping ratio for a viscously damped spring mass system, governed by the relationship $$\,m{{{d^2}x} \over {d{t^2}}} + c{{dx} \over {dt}} + kx = f\left( t \right),\,\,\,$$ is given by