Two coaxial discs, having moments of inertia I₁ and I₁/2, are rotating with respective angular velocities $$\omega $$₁ and $$\omega $$₁/2 , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If E_f and E_i are the final and initial total energies, then (E_f - E_i) is:

A thin disc of mass M and radius R has mass per unit area $$\sigma $$(r) = kr² where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is :

A cylinder with fixed capacity of 67.2 lit contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by 20°C is : [Given that R = 8.31 J mol^–1 K^–1]

One plano-convex and one plano-concave lens of same radius of curvature 'R' but of different materials are joined side by side as shown in the figure. If the refractive index of the material of 1 is $$\mu $$₁ and that of 2 is $$\mu $$₂, then the focal length of the combination is :