Mechanical Properties of Fluids Test

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Mechanical Properties of Fluids Test - 45

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Question 1
1 / -0

A tank with a small hole at the bottom has been filled with water and kerosene (specific gravity $$0.8$$). The height of water is $$3\ m$$ and that of kerosene $$2m$$. When the hole opened the velocity of fluid coming out from it is nearly: (take $$g=10\ {ms}^{-2}$$ and density of water $${10}^{3}\ kg\ {m}^{-3}$$)

A
$$9.6\ {ms}^{-1}$$

B
$$8.5\ {ms}^{-1}$$

C
$$7.6\ {ms}^{-1}$$

D
$$10.7\ {ms}^{-1}$$

Solution

Applying Bernoulli's law between the kerosene - water interface and the hole :
$$P_{0}+(\rho _{w}gh_{1}+\rho _{k}gh_{2})+\dfrac{1}{2}\rho _{w}v_{1}^{2}=P_{0}+0+\dfrac{1}{2}\rho _{w}v_{2}^{2}$$ (hole is at base so $$h=0$$)
$$\Rightarrow 3\times 10^{3}g+2\times 0.8\times 10^{3}g=\dfrac{1}{2}\times 10^{3}\times v_{2}^{2}$$ ($$v_{1}\approx 0$$ as flow is very slow at interface)
$$\Rightarrow v_{2}^{2}=9.2g$$
$$\Rightarrow v_{2}=\sqrt{92}$$ (given : $$g=10 m/s^{2}$$)
$$\Rightarrow v_{2}=9.6 m/s$$
Question 2
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A container is filled with water, accelerating with acceleration $$10 m/s^2$$, long $$+ve$$ X-axis on a smooth horizontal surface. The velocity of efflux of water at a point $$P$$ at the bottom of the tank and near its left most corner is

A
$$4.43 m/s$$

B
$$5.48 m/s$$

C
$$4 m/s$$

D
$$3 m/s$$

Solution

Due to acceleration liquid profile gets tilted as shown
$$tan\theta =\dfrac { a }{ g } =1$$
$$\theta ={ 45 }^{ 0 }$$
So,
$$x=\dfrac { 1 }{ 2 } m$$
Now, height of the liquid in left most section $$=\left( 1+\dfrac { 1 }{ 2 } \right) m=\dfrac { 3 }{ 2 } m$$
So, $$V=\sqrt { 2gh } $$
$$V=\sqrt { 2\times 10\times \dfrac { 3 }{ 2 } } $$
$$V=\sqrt { 30 } $$ m/s
$$V=5.48$$ m/s
Question 3
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Water flow through a vertical tube of variable cross-section. The area of cross-section at $$A$$ and $$B$$ are $$6$$ and $$3\ mm^{2}$$ respectively. If $$12\ cc$$ of water enters per second through $$A$$, find the pressure difference $$\left|P_{A}-P_{B} \right|$$ $$(g=10\ m/s^{2})$$. The separation between the cross-sections at $$A$$ and $$B$$ is $$100\ cm$$.

A
$$0.4\ \times 10^{5}\ dyne/cm^{2}$$

B
$$2.29\ \times 10^{5}\ dyne/cm^{2}$$

C
$$5.9\ \times 10^{5}\ dyne/cm^{2}$$

D
$$3.9\ \times 10^{5}\ dyne/cm^{2}$$

Solution
Question 4
1 / -0

Water if flowing steadily through a horizontal pipe of non-uniform cross-section. If the velocity of water at a point where cross-section is $$0.02 m^3$$ is $$2m/s$$, what is the velocity of water at another point where the cross-section is $$0.01 m^3$$ ?

A
$$1 m/s$$

B
$$2 m/s$$

C
$$3 m/s$$

D
$$4 m/s$$

Solution

According to torricellis theorem.
$${ A }_{ 1 }{ V }_{ 1 }={ A }_{ 2 }{ V }_{ 2 }$$
where
$$A-$$ Area of cross section
$$V-$$ Velocity of liquid
So,
$$0.02\left( 2 \right) ={ V }_{ 2 }\left( 0.01 \right) $$
$$\Rightarrow \quad { V }_{ 2 }=4m/s$$
Question 5
1 / -0

A solid sphere falls with a terminal velocity of $$10 cm/sec$$ in air. If its is allowed to fall in vacuum, the terminal velocity will be

A
equal to $$10 cm/s$$

B
less than $$10 m/s$$

C
more that $$10 cm/s$$

D
never be attained

Solution

As there is no opposing medium in vacuum. So velocity will never become constant.
Question 6
1 / -0

In the diagram shown, the difference In the two tubes of the manometer is $$5\ cm$$, the cross section of the tube at $$A$$ and $$B$$ is $$6\ {mm}^{2}$$ and $$10\ {mm}^{2}$$ respectively. The rate at which liquid flows though the tube is $$(g=10\ {m/s}^{2})$$

A
$$7.5\ cc/s$$

B
$$8.0\ cc/s$$

C
$$10.0\ cc/s$$

D
$$12.5\ cc/s$$

Solution
Question 7
1 / -0

An open U-tube contains mercury. When $$13.6 cm$$ of water is pourd into one of the arms of the tube then the mercury rise in the other arm from its initial level is:

A
$$1 cm$$

B
$$0.5 cm$$

C
$$10 cm$$

D
$$5 cm$$

Solution

$$(2x+h)P_m g=p_m h\ g+P_w\ g\ 13.6$$
$$2P_m\ g\ x=P/w\ g\ 13.6 \left(\dfrac {P_m}{P_w}=13.6\right)$$
$$\boxed {x=1/2}cm$$
Question 8
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When an air bubble of radius $$r$$ rises from the bottom to the surface of a lake, the radius becomes $$\dfrac {5r}{4}$$. Taking the atmospheric pressure to be equal to $$10\ m$$ height of water column, the depth of the lake would approximately be (ignore the surface tension and the effect of temperature):

A
$$11.2\ m$$

B
$$8.7\ m$$

C
$$9.5\ m$$

D
$$10.5\ m$$

Solution

$$\quad Given\quad that\quad temperature\quad and\quad surface\quad tension\quad \\ \quad \quad does\quad not\quad play\quad any\quad role\quad in\quad the\quad process\\ \quad \quad Let\quad the\quad temperature\quad remain\quad constant\quad throughout\quad the\quad process\\ \quad \quad so\quad { p }_{ 1 }{ v }_{ 1 }={ p }_{ 2 }{ v }_{ 2 }\quad since\quad T=c\\ \quad \quad { p }_{ 1 }=\quad { p }_{ atm }+\rho gh\quad \quad \quad \quad ;\quad h=\quad depth\quad of\quad the\quad lake\\ \quad \quad { p }_{ 2 }=\quad { p }_{ atm }=10\rho g\\ { \quad v }_{ 1 }=\frac { 4 }{ 3 } \pi { r }^{ 3 }\quad ;\quad { v }_{ 2 }=\frac { 4 }{ 3 } \pi { \frac { 5r }{ 4 } }^{ 3 }\quad putting\quad the\quad value\quad in\quad { p }_{ 1 }{ v }_{ 1 }={ p }_{ 2 }{ v }_{ 2 }\quad we\quad get\\ \quad h=\quad 9.53m\quad $$
Question 9
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A hydraulic lift is used to lift a car of mass $$2000\ kg$$. The cross-sectional area of the larger piston on which the car is supported is $$10^{-1} m^{2}$$. The smaller pistion is $$5m$$ below the bigger piston. The cross-sectional area of the smaller piston is $$10^{-2} m^{2}$$. What weight placed on smaller piston will be sufficient to keep the car in equilibrium?

A
$$250\ kg$$

B
$$170\ kg$$

C
$$50\ kg$$

D
$$350\ kg$$

Solution
Question 10
1 / -0

An open U-tube contain mercury when $$13.6cm$$ of water is poured into one of the arms of the mercury rise in the other its initial level is

A
$$1cm$$

B
$$0.5cm$$

C
$$10cm$$

D
$$5cm$$

Solution