A uniform rigid slender bar of mass $$10$$ $$kg.$$ is hinged at the left end is suspended with the help of spring and damper arrangement as shown in the figure where $$K = 2kN/m,$$ $$C =500Ns/m$$ and the stiffness of the torsional spring $${K_\theta }$$ is 1 kN/m/rad. Ignore the hinge dimensions.

The un-damped natural frequency of oscillations of the bar about the hinge point is

A uniform rigid slender bar of mass $$10$$ $$kg.$$ is hinged at the left end is suspended with the help of spring and damper arrangement as shown in the figure where $$K = 2kN/m,$$ $$C =500Ns/m$$ and the stiffness of the torsional spring $${K_\theta }$$ is 1 kN/m/rad. Ignore the hinge dimensions.

The damping coefficient in the vibration equation is given by

Nitrogen gas (molecular weight $$28$$) is enclosed in a cylinder by a piston, at the initial condition of $$2$$ bar, $$298$$ $$K$$ and $$1\,\,{m^3}$$. In a particular process, the gas slowly expands under isothermal condition, until the volume becomes $$2{m^3}.$$ Heat exchange occurs with the atmosphere at $$298$$ $$K$$ during this process.

The entropy change for the system during the process in $$kJ/K$$ is

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