A solid disc with radius a is connected to a spring at a point $$d$$ above the center of the disc. The other end of the spring is fixed to the vertical wall. The disc is free to roll without slipping on the ground. The mass of the disc is $$M$$ and the spring constant is $$K$$. The polar moment of inertia for the disc about its centre is $$J = {{M{a^2}} \over 2}$$

The natural frequency of this system in rad/s is given by

In a steam power plant operating on an ideal Rankine cycle, super-heated steam enters the turbine at $$3$$ $$MPa$$ and $${350^ \circ }C.$$ The condenser pressure is $$75$$ $$kPa.$$ Thermal efficiency of the cycle is ___________ percent.

Given data:

For saturated liquid, at $$P=75$$ $$kPa,$$
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,{h_f} = 384.39\,\,kJ/kg, \cr & \,\,\,\,\,\,\,\,\,\,{v_f} = 0.001037\,\,{m^3}/kg, \cr & \,\,\,\,\,\,\,\,\,\,{s_f} = 1.213\,\,kJ/kg K \cr} $$

At $$75$$ $$kPa,$$ $${h_{fg}} = 2278.6\,\,kJ/kg,$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,{s_{fg}} = 6.2434\,\,kJ/kg$$-$$K$$

At $$P=3$$ $$MPa$$ and
$$\,\,\,\,\,\,\,\,\,\,T = {350^ \circ }C\,\,\,$$ (Superheated steam),
$$\,\,\,\,\,\,\,\,\,\,h = 3115.3\,\,kJ/kg,$$
$$\,\,\,\,\,\,\,\,\,\,s = 6.7428\,\,kJ/kg$$-$$K$$

The INCORRECT statement about regeneration in vapor power cycle is that

Which of the following statements are TRUE with respect to heat and work?
$$(i)$$ They are boundary phenomena
$$(ii)$$ They are exact differentials
$$(iii)$$ They are path functions