A rigid uniform rod $$AB$$ of length $$L$$ and mass $$m$$ is hinged at $$C$$ such that $$AC = L/3, CB = 2L/3.$$ Ends $$A$$ and $$B$$ are supported by springs of spring constant $$k.$$ The natural frequency of the system is given by

A single degree of freedom system having mass $$1$$ $$kg$$ and stiffness $$10$$ $$kN/m$$ initially at rest is subjected to an impulse force of magnitude $$5$$ $$kN$$ for $${10^{ - 4}}$$ seconds. The amplitude in $$mm$$ of the resulting free vibration is

A concentrated mass $$m$$ is attached at the centre of a rod of length $$2L$$ as shown in the figure. The rod is kept in a horizontal equilibrium position by a spring of stiffness $$k.$$ For very small amplitude of vibration, neglecting the weights of the rod and spring, the un damped natural frequency of the system is

A disc of mass m is attached to a spring of stiffness $$k$$ as shown in the figure. The disc rolls without slipping on a horizontal surface. The natural frequency of vibration of the system is