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 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

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

For the component loaded with a force $$F$$ as shown in the figure, the axial stress at the corner point $$P$$ is