A cantilever beam of length $$L$$ is subjected to a moment $$M$$ at the free end. The moment of inertia of the beam cross section about the neutral axis is $$I$$ and the Young’s modulus is $$E$$. The magnitude of the maximum deflection is

For a long slender column of uniform cross section, the ratio of critical buckling load for the case with both ends clamped to the case with both ends hinged is

A thin walled spherical shell is subjected to an internal pressure. If the radius of the shell is increased by 1% and the thickness is reduced by 1%, with the internal pressure remaining the same, the percentage change in the circumferential (hoop) stress is

A solid disc of radius $$r$$ rolls without slipping on a horizontal floor with angular velocity $$\omega $$ and angular acceleration $$\alpha $$. The magnitude of the acceleration of the point of contact on the disc is