A cantilever type gate hinged at $$Q$$ is shown in the figure. $$P$$ and $$R$$ are the centers of gravity of the cantilever part and the counterweight respectively. The mass of the cantilever part is $$75$$ kg. The mass of the counterweight, for static balance, is

The rod $$PQ$$ of length $$L$$ and with flexural rigidity $$EI$$ is hinged at both ends. For what minimum force $$F$$ is expected to buckle?

A cylindrical container of radius $$R=1$$m, wall thickness $$1$$mm is filled with water upto a depth of $$2$$m and suspended along with its upper rim. The density of water is $$1000$$kg/m³ and acceleration due to gravity is $$10$$ m/s². The self weight of the cylinder is negligible. The formula for hoop stress in a thin walled cylinder can be used at all points along the height of the cylindrical container.

If the Young's modulus and Poisson's ratio of the container material are $$100$$GPa and $$0.3$$, respectively. The axial strain in the cylinder wall at mid height is

A compression spring is made of music wire of $$20$$ mm diameter having a shear strength and shear modulus of $$800$$ MPa and $$80$$ GPa respectively. The mean coil diameter is $$20$$ mm, free length is $$40$$ mm and the number of active coils is $$10.$$ If the mean coil diameter is reduced to $$10$$ mm, the stiffness of the spring is approximately