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

A large tank of cross-section area A is filled with water to a height H. A small hole of area 'a' is made at the base of the tank. It takes time ${\mathrm{T}}_1$ to decrease the height of water to $\frac{\mathrm{H}}{\mathrm{\eta }}(\mathrm{\eta }>1)$ ; and it takes ${\mathrm{T}}_2$ time to take out the rest of water. If ${\mathrm{T}}_1={\mathrm{T}}_2$, then the value of $\mathrm{\eta }$ is
(a) 2                                    (b) 3
(c) 4                                    (d) $2\sqrt{2}$

 As the temperature of water increases, its viscosity

(1) Remains unchanged

(2) Decreases

(3) Increases

(4) Increases or decreases depending on the external pressure

A small drop of water falls from rest through a large height h in air; the final velocity is

(1) $\propto \sqrt{\mathrm{h}}$

(2) $\propto \mathrm{h}$

(3) $\propto (1/\mathrm{h})$

(4) Almost independent of h

The rate of flow of liquid in a tube of radius r, length l, whose ends are maintained at a pressure difference P is $\mathrm{V}=\frac{{\mathrm{\pi QPr}}^4}{\mathrm{\eta l}}$ where $\mathrm{\eta }$ is coefficient of the viscosity and Q is-

(1) 8                     

(2) $\frac{1}{8}$

(3) 16                     

(4) $\frac{1}{16}$

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 a radius \(2r\) at the point \(A\) and a radius \(r\) at the point \(B\) further along the flow direction. If the velocity at the point \(A\) is \(v,\) its velocity at the 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 figure. 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) $\frac{d}{5}$