The diagram shows venturimeter through which water is flowing. The speed of water at $X$ is $2 \mathrm{cm} {\sec }^{-1}.$ Find the speed of water $\left(\mathrm{in} \mathrm{cm} {\sec }^{-1}\right)$ at $Y$  $\left(\mathrm{taking} g=1000 \mathrm{cm} {\sec }^{-2}\right).$

In a movable container shown in figure a liquid of density $\rho$ is filled up to a height $h$ $(1.8 \mathrm{m}).$ The upper $\&$ lower tube cross sectional areas are $A_2 \&{ A}_1$ respectively $\left(A_2>>A_1\right).$ If the liquid leaves out the container through the tube of cross-sectional area $A_1$ then find the velocity of liquid $\left(\mathrm{in} \mathrm{m} {\sec }^{-1}\right)$ coming out. $\left(\mathrm{Take} \mathrm{g}=10 \mathrm{m} {\sec }^{-2}\right)$

A large tank is filled with two liquids of specific gravities $2\mathrm{\sigma }$ and $\mathrm{\sigma }.$ Two holes are made on the wall of the tank as shown. Find the ratio of the distance from $O$ of the points on the ground where the jets from holes $A \text{\&} B$ strike. If the ratio is $\sqrt{\frac{a}{b}},$ find the value of $(a+b).$

Length of horizontal arm of a uniform cross-section U-tube is $\ell =21 \mathrm{cm}$ and ends of both of the vertical arms are open to surrounding of pressure $10500 \mathrm{N} {\mathrm{m}}^{-2}$. A liquid of density $\rho ={10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ is poured into the tube such that liquid just fills the horizontal part of the tube. Now one of the open ends is sealed, and the tube is then rotated about a vertical axis passing through the other vertical arm with angular velocity ${\omega }_0=10 \mathrm{rad} s^{-1}.$ If length of each vertical arm be $a=6 \mathrm{cm}.$ Calculate the length of air column in the sealed arm : $\left[g=10 \mathrm{m} {\mathrm{s}}^{-2}\right]$

A ball of density d is dropped onto a horizontal solid surface. It bounces elastically from the surface and returns to its original position in a time t1. Next, the ball is released and it falls through the same height before striking the surface of a liquid of density${\mathrm{d}}_{\mathrm{L}}.$

$\left(\mathrm{a}\right)$ If $\mathrm{d} < {\mathrm{d}}_{\mathrm{L}},$ obtain an expression $(\mathrm{in} \mathrm{terms} \mathrm{of} \mathrm{d} , {\mathrm{t}}_1 \mathrm{and} {\mathrm{d}}_{\mathrm{L}})$ for the time ${\mathrm{t}}_2$ the ball takes to come back to the position from which it was released.

$\left(\mathrm{b}\right)$ Is the motion of the ball simple harmonic?

$\left(\mathrm{c}\right)$ If $\mathrm{d} = {\mathrm{d}}_{\mathrm{L}},$ how does the speed of the ball depend on its depth inside the liquid ?
     Neglect all frictional and other dissipative forces. Assume the depth of the liquid to be large.

A container of large uniform cross-sectional area $A$ resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities $\mathrm{d}$ and $2\mathrm{d},$ each of height $\frac{H}{2}$ as shown in figure. The lower density liquid is open to the atmosphere having, pressure $P_{0^{\text{'}}}$

(i) A homogeneous solid cylinder of length $L\left(L<\frac{H}{2}\right),$ crosssectional area $\frac{A}{5}$ is immersed such that if floats with its axis vertical at the liquid-liquid interface with length $\frac{L}{4}$ in the denser liquid. Determine:
(a) the density $D$ of the solid,

(b) the total pressure at the bottom of the container.

(ii) The cylinder is removed and the original arrangement is restored. A tiny hole of area s $(s<<A)$ is, punched on the vertical side of the container at a height $h\left(h<\frac{H}{2}\right).$ Determine:
(a) the initial speed of efflux of the liquid at the hole

(b) the horizontal distance $\mathrm{x}$ travelled by the liquid initially, and

(c) the height $h_m$ at which the hole should be punched so that the liquid travels the maximum distance) $x_m$ initially. Also calculate ${\mathrm{x}}_{\mathrm{m}}.$ (Neglect the air resistance in these calculations)