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

A pump handling a liquid raises its pressure from $$1$$ bar to $$30$$ $$bar.$$ Take the density of the liquid as $$990$$ $$kg/{m^3}$$. The isentropic specific work done by the pump in $$kJ/kg$$ is

The contents of a well-insulated tank are heated by a resistor of $$23\Omega $$ in which $$10A$$ current is flowing. Consider the tank along with its contents as a thermodynamic system. The work done by the system and the heat transfer to the system are positive. The rates of heat $$(Q),$$ work $$(W)$$ and change in internal energy $$\left( {\Delta U} \right)$$ during the process in $$kW$$ are

In an experimental set-up, air flows between two stations $$P$$ and $$Q$$ adiabatically. The direction of flow depends on the pressure and temperature conditions maintained at $$P$$ and $$Q$$. The conditions at station $$P$$ are $$150$$ $$kPa$$ and $$350$$ $$K.$$ The temperature at station $$Q$$ is $$300$$ $$K.$$ The following are the properties and relations pertaining to air:

Specific heat at constant pressure, $$${C_p} = 1.005\,\,kJ/kg\,\,K;$$$
Specific heat at constant volume, $$${C_v} = 0.718\,\,kJ/kg\,\,K;$$$
Characteristic gas constant, $$R=0.287$$ $$kJ/kgK.$$
Enthalpy $$h = {C_p}T,$$
Internal energy, $$u = {C_v}T.$$

If the air has to flow from station $$P$$ to station $$Q,$$ the maximum possible value of pressure in $$kPa$$ at station $$Q$$ is close to