A number of water droplets each of radius r coalesce to form a drop of radius R. Are consuming the whole of the energy liberated due to  coalescence goes into hitting the drop, the rise and temperature $\mathrm{d\theta }$, is

1. $\frac{2\mathrm{T}}{\mathrm{r} \mathrm{J}}$

2. $\frac{3\mathrm{T}}{\mathrm{r} \mathrm{J}}$

3. $\frac{3\mathrm{T}}{\mathrm{J}}\left(\frac{1}{\mathrm{r}}-\frac{1}{\mathrm{R}}\right)$

4. $\frac{3\mathrm{T}}{\mathrm{J}}\left(\frac{1}{\mathrm{r}}+\frac{1}{\mathrm{R}}\right)$

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If ${\rho }_c$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

1. More than filled if ${\rho }_c$ is less than 0.5

2. More than half filled if ${\rho }_c$ is less than 1.0

3. Half filled if ${\rho }_c$ is less than 0.5

4. Less than half filled if ${\rho }_c$ is less than 0.5

A large tank is filled with water (density= ${10}^3 \mathrm{kg}/{\mathrm{m}}^3$). A small hole is made at a depth 10m below water surface. The range of water issuing out of  the hole is R on ground. What extra p[ressure must be applied on the water surface so that the range become 2R (take 1 atm = ${10}^5$ Pa and g= 10 m/${\mathrm{s}}^2$)

1. a atm

2. 2 atm

3. 4 atm

4. 3 atm

When a block of iron floats in mercury at 0$°$C, a fraction $k_1$ of its volume is submerged, while at 60$°$C, a fraction $k_2$ is seen to be submerged, If the coefficient of volume expansion of iron is ${\mathrm{\gamma }}_{\mathrm{Fe}}$ and that of mercury is ${\gamma }_{Hg}$, then the ratio of $k_1$/$k_2$ can be expressed as

1. $\frac{1+{{60}_{\gamma }}_{Fe}}{1+{{60}_{\gamma }}_{He}}$

2. $\frac{1-{{60}_{\gamma }}_{Fe}}{1+{{60}_{\gamma }}_{He}}$

3. $\frac{1+{{60}_{\gamma }}_{Fe}}{1-{{60}_{\gamma }}_{He}}$

4. $\frac{1+{{60}_{\gamma }}_{Hg}}{1+{{60}_{\gamma }}_{Fe}}$

The diameter of a big drop of water is 2.8 mm. It is broken into 125 equal drops. If the surface tension of water is 75 dyne/cm, then the work done is

1. zero

2. 19 erg

3. 46 erg

4. 74 erg

If pressure at half the depth of a lake is equal to 2/3 pressure at the botom of the lake then what is the depth of the lake

1. 10 m

2. 20 m

3. 60 m

4. 30 m

Acubical block if floating in a liquid with half of its volume immersed in the liquid. When the whole system accelerates upwards with acceleration of $\frac{\mathrm{g}}{3}$, the fraction of volume immersed in the liquid will be

1. 1/2

2. 3/8

3. 2/3

4. 3/4

A pump is designed as a horizontal cylinder with a piston area A and an outlet orifice arranged near the axis of the cylinder. Find the velocity of outflow of liquid from pump, if the piston movos with a constant velocity under the action of a constant force F. The density of liquid is $\rho$

1. $\sqrt{\frac{F}{A\rho }}$

2. $\sqrt{\frac{2F}{A\rho }}$

3. $\sqrt{\frac{A\rho }{F}}$

4. $\sqrt{\frac{A\rho }{2F}}$

A liquid is flowing in a horizontal uniform capillary tube under a constant pressure difference P. The value of pressure for which the rate of flow of the liquid is doubled when the radius and length both are doubled is 

(1) P

(2) $\frac{3P}{4}$

(3) $\frac{P}{2}$

(4) $\frac{P}{4}$

A metallic Square Plate ABCD is suspended vertically with a pair of sides horizontal by a light in elastic string as shown in figure. A beaker of water is brought below the plate and raised till the plate is completely immersed and the level of water is well above the plate.

If the point of support is slowly raised vertically at a constant velocity, the graph of tension T in the string against the displacement s of the point of support is best represented by by

1.

2.

3.

4.