A solid disc of radius $$r$$ rolls without slipping on a horizontal floor with angular velocity $$\omega $$ and angular acceleration $$\alpha $$. The magnitude of the acceleration of the point of contact on the disc 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 circular solid disc of uniform thickness $$20mm,$$ radius $$200mm$$ and $$mass$$ $$20kg,$$ is used as a fly wheel. If it rotates at $$600$$ $$rpm,$$ the kinetic energy of the flywheel, in joules is

An ideal gas of mass $$m$$ and temperature $${T_1}$$ undergoes a reversible isothermal process from an initial pressure $${P_1}$$ to final pressure $$P{}_2.$$ The heat loss during the process is $$Q$$. The entropy changes $$\Delta S$$ of the gas is