As shown in the figure, a plane mirror is fixed at a height of $$50 \mathrm{~cm}$$ from the bottom of tank containing water $$\left(\mu=\frac{4}{3}\right)$$. The height of water in the tank is $$8 \mathrm{~cm}$$. A small bulb is placed at the bottom of the water tank. The distance of image of the bulb formed by mirror from the bottom of the tank is ___________ $$\mathrm{cm}$$.

A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity $$\omega$$. A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If E be the initial Kinetic energy of the system, then final Kinetic energy will be $$\frac{E}{x}$$. The value of $$x$$ is

The surface tension of soap solution is $$3.5 \times 10^{-2} \mathrm{~Nm}^{-1}$$. The amount of work done required to increase the radius of soap bubble from $$10 \mathrm{~cm}$$ to $$20 \mathrm{~cm}$$ is _________ $$\times ~10^{-4} \mathrm{~J}$$.

$$(\operatorname{take} \pi=22 / 7)$$

A wire of density $$8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$$ is stretched between two clamps $$0.5 \mathrm{~m}$$ apart. The extension developed in the wire is $$3.2 \times 10^{-4} \mathrm{~m}$$. If $$Y=8 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$$, the fundamental frequency of vibration in the wire will be ___________ $$\mathrm{Hz}$$.