A wooden plank of length $1\mathrm{m}$ and uniform cross section is hinged at one end to the bottom of a tank as shown in the figure. The tank is filled with water up to a height of $0.5\mathrm{m}.$ The specific gravity of the plank is $0.5.$ If the angle $\mathrm{\theta }$ by the inclination of that the plank makes with the vertical in the equilibrium position (exclude the case $\mathrm{\theta }=0$).
Find the value of $\frac{1}{{\cos }^2\mathrm{\theta }}.$

Length of a horizontal arm of a U-tube is $L=1\mathrm{m}$ and ends of both the vertical arms are open to atmospheric pressure $P_0={10}^5\mathrm{N}/{\mathrm{m}}^2.$ A liquid of density $\rho ={10}^3\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ is poured in the tube such that liquid just fills the horizontal part of the tube as shown in figure. Now one end of the open ends is sealed and the tube is then rotated about a vertical axis passing through the other vertical arm with angular speed ${\omega }_0=\sqrt{\frac{40}{3}}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}.$ If length of each vertical arm is $a=1\mathrm{m}$ and in the sealed end liquid rises to a height $y=1/2\mathrm{m},$ find pressure in the sealed tube during rotation. (in $\times {10}^5\mathrm{N}/{\mathrm{m}}^2$)

When air of density $1.3\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ flows across the top of the tube shown in the accompanying figure, water rises in the tube to a height of $1.0\mathrm{c}\mathrm{m}$. What is the speed of the air (in $\mathrm{m}/\mathrm{s}$)?

The diagram shows venturi meter through which water is flowing. The speed of water at $X$ is $2\mathrm{c}\mathrm{m}/\mathrm{s}$. Find the speed of water at $Y$ in $\mathrm{c}\mathrm{m}/\mathrm{s}$. (taking $g=1000\mathrm{c}\mathrm{m}/{\mathrm{s}}^2$).

In the figure shown, the heavy cylinder (radius, $\sqrt{\frac{8}{3}}\mathrm{m}$) kept on a smooth surface separates two liquids of densities $2\rho$ and $3\rho$. The height $h$ (in meter) for the translational equilibrium of the cylinder must be

For the system shown in the figure, the cylinder on the left at $L$ has a mass of $600\mathrm{k}\mathrm{g}$ and a cross sectional area of $800\mathrm{c}{\mathrm{m}}^2$. The piston on the right, at $S$, has cross sectional area $25\mathrm{c}{\mathrm{m}}^2$ and negligible weight. If the apparatus is filled with oil $(\rho =0.75\mathrm{g}/\mathrm{c}{\mathrm{m}}^3)$ Find the force $F$ (in $\mathrm{N}$) required to hold the system in equilibrium. Take $g=10\mathrm{m}/{\mathrm{s}}^2$.

A siphon has a uniform circular base of diameter $\frac{8}{\sqrt{\pi }}\mathrm{c}\mathrm{m}$ with its crests $1.8\mathrm{m}$ above water level as shown in figure. Find absolute pressure at the crest level $A$ (in $kPa$). [Use $P_0={10}^5\mathrm{N}/{\mathrm{m}}^2$ and $g=10\mathrm{m}/{\mathrm{s}}^2$]