Glycerin of density $$1.25 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$$ is flowing through the conical section of pipe The area of cross-section of the pipe at its ends are $$10 \mathrm{~cm}^{2}$$ and $$5 \mathrm{~cm}^{2}$$ and pressure drop across its length is $$3 ~\mathrm{Nm}^{-2}$$. The rate of flow of glycerin through the pipe is $$x \times 10^{-5} \mathrm{~m}^{3} \mathrm{~s}^{-1}$$. 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 ___________ $$\mathrm{Hz}$$.

The length of a wire becomes $$l_{1}$$ and $$l_{2}$$ when $$100 \mathrm{~N}$$ and $$120 \mathrm{~N}$$ tensions are applied respectively. If $$10 ~l_{2}=11~ l_{1}$$, the natural length of wire will be $$\frac{1}{x} ~l_{1}$$. Here the value of $$x$$ is _____________.