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) $\frac{d}{5}$

 A wooden block with a coin placed on its top, floats in water as shown in fig. The distance \(l\) and \(h\) are shown there. After some time the coin falls into the water. Then:
                      

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

(1) Remains same                         

(2) Rises

(3) Lowers                                   

(4) (a), (b) or (c)

A vessel of area of cross-section A has liquid to a height H. There is a hole at the bottom of vessel having area of cross-section a. The time taken to decrease the level from ${\mathrm{H}}_1$ to ${\mathrm{H}}_2$ will be

(1)$\frac{\mathrm{A}}{\mathrm{a}}\sqrt{\frac{2}{\mathrm{g}}}\left[\sqrt{{\mathrm{H}}_1}-\sqrt{{\mathrm{H}}_2}\right]$             

(2) $\sqrt{2\mathrm{gh}}$

(3) $\sqrt{2\mathrm{gh}({\mathrm{H}}_1-{\mathrm{H}}_2)}$                     

(4) $\frac{\mathrm{A}}{\mathrm{a}}\sqrt{\frac{g}{2}}\left[\sqrt{{\mathrm{H}}_1}-\sqrt{{\mathrm{H}}_2}\right]$

A lead shot of 1mm diameter falls through a long column of glycerine. The variation of its velocity v with distance covered is represented by

Water flows through a frictionless duct with a cross-section varying as shown in fig. Pressure p at points along the axis is represented by

From amongst the following curves, which one shows the variation of the velocity v with time t for a small sized spherical body falling vertically in a long column of a viscous liquid -

A liquid does not wet the sides of a solid if the angle of contact is

(1) Zero                                     

(2) Obtuse (More than 90°)

(3) Acute (Less than 90°)           

(4) 90°