A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency $$\omega$$ and amplitude $$a$$. If at a height $$y^{\prime}$$ from the mean position, the body gets detached from the spring, calculate the value of $$y^{\prime}$$ so that the height $$\mathrm{H}$$ attained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment $$\left(a \omega^{2}>g\right)$$

In the given circuit, the switch S is closed at time $$t=0$$. The charge Q on the capacitor at any instant t is given by $$Q(t)=Q_0(1-e^{-\alpha t})$$. Find the value of Q$$_0$$ and $$\alpha$$ in terms of given parameters shown in the circuit.

Two identical prisms of refractive index $$\sqrt{3}$$ are kept as shown in the figure. A light ray strikes the first prism at face AB. Find

(A) the angle of incidence so that the emergent ray from the first prism has minimum deviation;

(B) through what angle, the prism DCE should be rotated about C so that the final emergent ray also has minimum deviation?

A cylinder of mass $$1 \mathrm{~kg}$$ is given heat of $$20000 \mathrm{~J}$$ at atmospheric pressure. If initially temperature of cylinder is $$20^{\circ} \mathrm{C}$$, find

(A) The final temperature of the cylinder;

(B) The work done by the cylinder;

(C) The change in internal energy of the cylinder.

Given :

The specific heat of cylinder

$$=400 \mathrm{~J} \mathrm{~kg}^{-1 \circ} \mathrm{C}^{-1}$$

Coefficient of volume expansion

$$=9 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1} \text {; }$$

Atmospheric pressure $$=10^{5} \mathrm{~N} / \mathrm{m}^{2}$$ Density of cylinder $$=9000 \mathrm{~kg} / \mathrm{m}^{3}$$ )