A circular disc of radius $$R$$ is removed from a bigger circular disc of radius $$2R$$ such that the circumferences of the discs coincide. The center of mass of the new disc is $$\alpha R$$ form the center of the bigger disc. The value of $$\alpha $$ is

A round uniform body of radius $$R,$$ mass $$M$$ and moment of inertia $$I$$ rolls down (without slipping) an inclined plane making an angle $$\theta $$ with the horizontal. Then its acceleration is

Angular momentum of the particle rotating with a central force is constant due to

A Carnot engine, having an efficiency of $$\eta = 1/10$$ as heat engine, is used as a refrigerator . If the work done on the system is $$10$$ $$J$$, the amount of energy absorbed from the reservoir at lower temperature is