Figure shows a single degree of freedom system. The system consists of a mass less rigid bar $$OP$$ hinged at $$O$$ and a mass $$m$$ at end $$P.$$ The natural frequency of vibration of the system is

A rigid container of volume $$0.5$$ $${m^3}$$ contains $$1.0$$ $$kg$$ of water at $${120^ \circ }C$$
$$\left( {{V_f} = 0.00106\,{m^3}/kg,\,\,{v_g} = 0.8908\,\,{m^3}/kg} \right).$$ The state of water is

Steam enters a turbine at $$30bar,$$ $${300^ \circ }C$$ $$\left( {u = 2750\,\,kJ/kg,\,\,h = 2993\,\,kJ/kg} \right)$$ and exits the turbine as saturated liquid at $$15$$ $$kPa$$ $$\left( {u = 225\,\,kJ/kg,\,\,h = 226\,\,kJ/kg} \right)$$.
Heat loss to the surrounding is $$50$$ $$kJ/kg$$ of steam flowing through the turbine. Neglecting changes in kinetic energy and potential energy, the work output of the turbine (in $$kJ/kg$$ of steam) is _____________

A mixture of ideal gases has the following composition by mass:

If the universal gas constant is $$8314$$ $$J/kmol$$-$$K,$$ the characteristic gas constant of the mixture (in $$J/kg$$-$$K$$) is ____________.