Mechanical Properties of Fluids Test

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Mechanical Properties of Fluids Test - 33

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Question 1
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Water flows through a frictionless duct with a crosssection varying as shown in fig. Pressure p at points along the axis is represented by

A

B

C

D

Solution

Answer is A.

When cross section of duct decreases the velocity of water increases and in accordance with Bernoulli's theorem the pressure decreases at that place.
Therefore, in this case, the pressure remains constant initially and then decreases as the area of cross section decreases along the neck of the tube and then remains constant along the mouth of the tube.
Hence, graph in option A is correct.
Question 2
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For a fluid which is flowing steadily, the level in the vertical tubes is best represented by

A

B

C

D

Solution

The tube is wider earlier and it becomes narrow as it progresses. Pressure in the thin tube will be more compared to that of the thick region of the tube. So, there would be enough pressure due to the thin region, so that liquid will move in that vertical part. But as the liquid is moving forward, its velocity will decrease. So, figure D is incorrect. And height in the second vertical part will be less compared to the height in the first vertical tube.

$$Answer:$$

Hence, option A is the correct answer.
Question 3
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A common hydrometer has a long uniform stem. When floating in pure water, 4.5 cm of its stem lies below the surface of water. In a liquid of specific gravity 2.0, 1.5 cm of the stem of the same hydrometer is immersed. Find the specific gravity of the liquid in which the hydrometer is immersed upto 0.5 cm.

A
2

B
3

C
4

D
5

Solution
Question 4
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The height of mercury which exerts the same pressure as 20 cm of water column is :( Given relative density of mercury=13.56)

A
1.47 cm

B
14.8 cm

C
148 cm

D
None of these

Solution

Let pressure $$= P,$$ density $$= \rho,$$ mass $$= m$$
$$P_{\text{mercury}} = h_m\rho_mg \rightarrow$$ Pressure exerted by mercury
$$P_{\text{water}} = h_w\rho_wg \rightarrow$$ Pressure exerted by water
Since given $$P_{\text{mercury}}$$ $$= P_{\text{water}}$$ at $$h_w = 20 cm$$
$$\Rightarrow h_m\rho_m = h_w\rho_w$$ where $$\rho_m = 13.56 \rho_w$$ relative density
$$\Rightarrow \dfrac{\rho_w}{\rho_m} = \dfrac{h_m}{h_w}$$
$$\Rightarrow \dfrac{1}{10.8}= \dfrac{h_m}{20}$$
$$\Rightarrow h_m = \dfrac{20}{10.8} = 1.47 cm$$
Question 5
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Two identical cylindrical vessels with their bases at the same level; contain liquid of density $$\rho$$. The area of both is $$S$$, but the height of liquid in one vessel is $${h}_{1}$$ and in other $${h}_{2}$$. The work done when both cylinders are connected, by gravity in equalising levels is:

A
$$\displaystyle\frac { 1 }{ 4 } g\rho S{ \left( { h }_{ 2 }-{ h }_{ 1 } \right) }^{ 2 }$$

B
$$g\rho S{ \left( { h }_{ 2 }-{ h }_{ 1 } \right) }^{ 2 }$$

C
$$g\rho S\left( { h }_{ 2 }-{ h }_{ 1 } \right) $$

D
$$\displaystyle\frac { 1 }{ 4 } g\rho S\left( { h }_{ 2 }-{ h }_{ 1 } \right) $$

Solution

The water level will rise to a height of $$\displaystyle\frac{1}{2}\left({h}_{2} - {h}_{1}\right)$$
$$\therefore$$ Increase in volume $$= \displaystyle\frac{1}{2}\left({h}_{2} - {h}_{1}\right)S$$
$$\therefore$$ Work done $$= \displaystyle\frac{1}{2}\left({h}_{2} - {h}_{1}\right)S \times \rho \times g\displaystyle\frac{1}{2}\left({h}_{2} - {h}_{1}\right)$$
$$= \displaystyle\frac{1}{4}g \rho S{ \left( { h }_{ 2 }-{ h }_{ 1 } \right) }^{ 2 }$$.
Question 6
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The pressure of gas in a metal cylinder is 4 atmospheres at $$27 C$$, then the pressure at $$54 C$$: (in atmosphere)

A
$$4.36$$

B
$$8$$

C
$$3$$

D
$${400}/{109}$$

Solution

According to the gas equation
$$\therefore \displaystyle\frac{{P}_{1}{V}_{1}}{{T}_{1}} = \displaystyle\frac{{P}_{2}{V}_{2}}{{T}_{2}}$$
Here $${P}_{1} = 4 atm V = V$$,
and $${T}_{1} = 27 + 273 = 300 K$$
$${P}_{2} = $$?, $${V}_{2} = V$$
and $${T}_{2} = 54 + 273 = 327 K$$
$$\therefore \displaystyle\frac{4 \times V}{300} = \displaystyle\frac{{P}_{2} \times V}{327}$$
or, $${P}_{2} = \displaystyle\frac{4 \times 327}{300}$$
$$ = 4.36 atm$$.
Question 7
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A container is filled with water to a height of 10 m. The pressure exerted by the water at the bottom of the container is _____ Pa.

A
$$9.8 * 10^4$$

B
980

C
$$980 * 10^5$$

D
$$19.6 * 10^4$$

Solution
Question 8
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The pressure exerted by a liquid column at the bottom of the container at a point inside a fluid

A
does not depend on the area of cross-selection of container.

B
dependent of the density of the fluid.

C
equal in all directions.

D
All the above are true

Solution
Question 9
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The height of mercury which exerts the same pressure as 20 cm of water column, is_____

A
1.47 cm

B
14.8 cm

C
148 cm

D
none of these

Solution

$$density\times area\times h=constant$$
$$Swater\times area\times hwater=Smercury\times area\times hmercury$$
$$1{ 9/cc }\times 20cm\quad =\quad 13.56\quad 9/cc\times h$$
$$h=\dfrac { 20 }{ 13.56 } cm=1.47cm$$
$$\boxed { h=1.47cm } $$
Question 10
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Two capillary tubes of diameters 3.0 mm and 6.0 mm are joined together to form a U-tube open at both ends. If the U-tube is filled with water, what is the difference in its levels in the two limbs of the tube? Surface tension of water at the temperature of the experiment is $$7.3 \times 10^{-2} N/m$$. Take the angle of contact to be zero and density of water to be $$10^3 kg/m^3(g = 9.8 m/s^2)$$

A
5 mm

B
10 mm

C
15 mm

D
20 mm

Solution

$$\varrho gh=\dfrac { 2T }{ R } $$
$${ h }_{ 1 }=\dfrac { 2\times 7.3\times { 10 }^{ -2 } }{ { 10 }^{ 3 }\times 9.8\times 1.5\times { 10 }^{ 3 } } $$
$${ h }_{ 2 }=\dfrac { 2\times 7.3\times { 10 }^{ -2 } }{ { 10 }^{ 3 }\times 9.8\times 3\times { 10 }^{ -3 } } $$
So $$\triangle h={ h }_{ 1 }-{ h }_{ 2 }$$
$$=\dfrac { 2\times 7.3\times { 10 }^{ -2 } }{ { 10 }^{ 3 }\times 9.8\times { 10 }^{ -3 } } \left( \dfrac { 2 }{ 3 } -\dfrac { 1 }{ 3 } \right) $$
$$=\dfrac { 2\times 7.3\times { 10 }^{ -2 } }{ { 10 }^{ 3 }\times 9.8\times 3\times { 10 }^{ -3 } } $$
$$\boxed { \triangle h=5mm } $$