Calculate the density of water near the bottom of the ocean where the pressure is about $501$ atm and density of water is $1025 \mathrm{kg}/{\mathrm{m}}^3$ approximately. Take bulk modulus of water (in ${\mathrm{kgm}}^{-3}$) as for water as $2.0\times {10}^9 \mathrm{N}/{\mathrm{m}}^2$ and $1$ atm is $1000 \mathrm{kg}/{\mathrm{m}}^3$

A very stiff bar $\left(AB\right)$ of negligible mass is suspended horizontally by two vertical rods as shown in figure. Length of the bar is $2.5L.$ The steel rod has length $L$ and cross sectional radius of $r$ and the brass rod has length $2L$ and cross sectional radius of $2r.$ A vertically downward force $F$ is applied to the bar at a distance $X$ from the steel rod and the bar remains horizontal. If the value of $\frac{X}{L}=\frac{n}{2}$ find $n$. If it is given that ratio of Young's modulus of steel and brass is $\frac{Y_S}{Y_B}=2.$

A sphere of radius $0.1 \mathrm{m}$ and mass $8\pi \mathrm{kg}$ is attached to the lower end of a steel wire of length $5 \mathrm{m}$ and diameter ${10}^{-3} \mathrm{m}.$ The wire is suspended from $5.22 \mathrm{m}$ high ceiling of a room. When the sphere is made to swing as a simple pendulum, it just grazes the floor at its lowest point. Young's modulus of steel is $2\times {10}^{11}{\mathrm{Nm}}^{-2}.$ Find the velocity of the sphere at the lowest position in $\mathrm{m}/\mathrm{s}.$ (Given: $g=10 \mathrm{m}/{\mathrm{s}}^2$)

(Take $\pi =3.14$)