A tank is filled with water up to a height \(H.\) The water is allowed to come out of a hole \(P\) in one of the walls at a depth \(D\) below the surface of the water. The horizontal distance \({x}\) in terms of \(H\) and \({D}\) is:
    
1. \(x = \sqrt{D\left(H-D\right)}\)
2. \(x = \sqrt{\frac{D \left(H - D \right)}{2}}\)
3. \(x = 2 \sqrt{D \left(H-D\right)}\)
4. \(x = 4 \sqrt{D \left(H-D\right)}\)

A streamlined body falls through air from a height h on the surface of a liquid. If d and D(D > d) represents the densities of the material of the body and liquid respectively, then the time after which the body will be instantaneously at rest, is

1. $\sqrt{\frac{2\mathrm{h}}{\mathrm{g}}}$                         

2. $\sqrt{\frac{2\mathrm{h}}{\mathrm{g}}·\frac{D}{d}}$

3. $\sqrt{\frac{2\mathrm{h}}{\mathrm{g}}·\frac{d}{D}}$                   

4. $\sqrt{\frac{2\mathrm{h}}{\mathrm{g}}}\left(\frac{d}{D-d}\right)$

 As the temperature of water increases, its viscosity

1. Remains unchanged

2. Decreases

3. Increases

4. Increases or decreases depending on the external pressure

If a small drop of water falls from rest through a large height h in air, then the final velocity is:

Water flows in a streamlined manner through a capillary tube of radius a, the pressure difference being P and the rate of flow Q. If the radius is reduced to a/2 and the pressure increased to 2P, the rate of flow becomes

1. $4\mathrm{Q}$                          

2. $\mathrm{Q}$

3. $\frac{\mathrm{Q}}{4}$                         

4. $\frac{\mathrm{Q}}{8}$

Water is flowing in a pipe of diameter 4 cm with a velocity 3 m/s. The water then enters into a tube of diameter 2 cm. The velocity of water in the other pipe is

1. 3 m/s                                       

2. 6 m/s

3. 12 m/s                                       

4. 8 m/s

What is the velocity v of a metallic ball of radius r falling in a tank of liquid at the instant when its acceleration is one-half that of a freely falling body ? (The densities of metal and of liquid are $\mathrm{\rho }$ and $\mathrm{\sigma }$ respectively, and the viscosity of the liquid is $\mathrm{\eta }$).

1. $\frac{{\mathrm{r}}^2\mathrm{g}}{9\mathrm{\eta }}\left(\mathrm{\rho }-2\mathrm{\sigma }\right)$                     

2. $\frac{{\mathrm{r}}^2\mathrm{g}}{9\mathrm{\eta }}\left(2\mathrm{\rho }-\mathrm{\sigma }\right)$

3. $\frac{{\mathrm{r}}^2\mathrm{g}}{9\mathrm{\eta }}\left(\mathrm{\rho }-\mathrm{\sigma }\right)$                       

4. $\frac{2{\mathrm{r}}^2\mathrm{g}}{9\mathrm{\eta }}\left(\mathrm{\rho }-\mathrm{\sigma }\right)$

An incompressible fluid flows steadily through a cylindrical pipe which has radius 2r at point A and radius r at B further along the flow direction. If the velocity at point A is v, its velocity at point B is 

1. 2v                               

2. v

3. v/2                               

4. 4v

A homogeneous solid cylinder of length L(L<H/2) . Cross-sectional area A/5 is immersed such that it floats with its axis vertical at the liquid-liquid interface with length L/4 in the denser liquid as shown in the fig. The lower density liquid is open to atmosphere having pressure ${\mathrm{P}}_0$. Then density D of solid is given by

1. $\frac{5}{4}\mathrm{d}$                     

2. $\frac{4}{5}\mathrm{d}$

3. $\mathrm{d}$                         

4. $\frac{d}{5}$

A block of ice floats on a liquid of density 1.2 $g/c{m}^3$ in a beaker. The level of liquid when ice completely melts-

1. Remains same                         

2. Rises

3. Lowers                                   

4. (1), (2) or (3)