A simply supported beam of length $$2L$$ is subjected to a moment $$M$$ at the mid-point $$x = 0$$ as shown in the figure. The deflection in the domain $$0 \le x \le L$$ is given by $$$W = {{ - Mx} \over {12EIL}}\left( {L - x} \right)\left( {x + c} \right)$$$

Where $$E$$ is the Young’s modulus, $$I$$ is the area moment of inertia and $$c$$ is a constant (to be determined).

The slope at the center $$X=0$$ is

A thin cylindrical pressure vessel with closed-ends is subjected to internal pressure. The ratio of circumferential (hoop) stress to the longitudinal stress is

A single degree of freedom mass-spring - viscous damper system with mass $$m,$$ spring constant $$k$$ and viscous damping coefficient $$q$$ is critically damped. The correct relation among $$m, k$$ and $$q$$ is

The rod AB, of length 1 m, shown in the figure is connected to two sliders at each end through pins. The sliders can slide along QP and QR. If the velocity V_A of the slider at A is 2 m/s, the velocity of the midpoint of the rod at this instant is ___________ m/s.