Mechanical Properties of Fluids Test

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Mechanical Properties of Fluids Test - 71

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Question 1
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A cylindrical vessel of height $$500$$ mm has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height $$H$$. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes from the orifice and the water level in the vessel becomes steady with height of water column being $$200$$ mm. Find the fall in height (in mm) of water level due to opening of the orifice.

A
6mm

B
12mm

C
16mm

D
18mm

Solution
Question 2
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A drop of liquid of density $$\rho$$ is floating half-immersed in a liquid of density $$d$$. If $$\rho$$ is the surface tension the diameter of the drop of the liquid is

A
$$\sqrt {\dfrac {\sigma}{g(2\rho - d)}}$$

B
$$\sqrt {\dfrac {3\sigma}{g(2\rho - d)}}$$

C
$$\sqrt {\dfrac {6\sigma}{g(2\rho - d)}}$$

D
$$\sqrt {\dfrac {12\sigma}{g(2\rho - d)}}$$

Solution

The equation for the surface tension of the drop is:
$$2\pi r\sigma + \dfrac {1}{2} \times \dfrac {4}{3} \pi r^{3} dt = \dfrac {4}{3} \pi r^{3}\rho g$$

or $$2\pi r\sigma = \dfrac {\pi r^{3} g}{3} [4\rho - 2d]$$

or $$r^{2} = \dfrac {3\times 2\pi \sigma}{\pi g(2\rho - 2d)}$$

or $$r^{2} = \dfrac {3\sigma}{g(2\rho - d)}$$

or $$r = \sqrt {\dfrac {3\sigma}{g(2\rho - d)}}$$

$$Diameter = 2r = \sqrt {\dfrac {12\sigma}{g(2\rho - d)}}$$.
Question 3
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A small spherical body of radius $$r$$ is falling under gravity in a viscous medium and due to friction the medium gets heated. When body attains termional velocity then rate of heating is proportional to

A
$$r$$

B
$$r^{3/2}$$

C
$$r^5$$

D
$$r^{1/2}$$

Solution
Question 4
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A test tube filled with water is being spun around in an ultracentrifuge with angular velocity $$\omega$$. The test tube is lying along a radius and the free surface of the water is at radius $$r_0$$ (Figure). The length of the test tube is $$l$$ density of the water. Ignore gravity and ignore atmospheric pressure. Choose the correct statement.

A
The pressure at the mid point of test tube is given by $$\dfrac {1}{8}\rho l\ \omega^2(l+4r_0)$$

B
The pressure at the mid point of test tube is given by $$\dfrac {1}{2}\rho l\ \omega^2 \left (\dfrac {l}{2}+r_0 \right)$$

C
The pressure at the mid point of test tube is given by $$\dfrac {1}{4}\rho l\ \omega^2 \left (\dfrac {l}{2}+r_0 \right)$$

D
The pressure remains constant through out the tube
Question 5
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Figure shows a large open tank. Water emerging out from holes $$O_1$$ and $$O_3$$ strikes the ground at same point but from $$O_2$$ has maximum range. The height of $$O_2$$ from ground

A
$$\frac{{{h_1} - {h_2}}}{2}$$

B
$$\frac{{{h_2} + {h_1}}}{2}$$

C
$$\frac{{\sqrt {{h_1}{h_2}} }}{2}$$

D
$$2 {{\sqrt {{h_1}{h_2}} }}$$
Question 6
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A thin square plate of side $$5\ cm$$ is suspended vertically from a balance so that lower side just dips into water with side to surface. When the plate is clean $$(\theta = 0^{\circ})$$, it appears to weigh $$0.044\ N$$. But when the plate is greasy $$(\theta = 180^{\circ})$$, it appears to weigh $$0.03\ N$$. The surface tension of water

A
$$3.5 \times 10^{- 2}N/m$$

B
$$7.0 \times 10^{- 2}N/m$$

C
$$14.0 \times 10^{- 2}N/m$$

D
$$1.08 N/m$$

Solution

Difference in apparent weights is due to differences in forces of surface tension. Due to $$180^{\circ}$$, the force surface tension in one case is opposite to the force of surface tension in the other case.
$$\therefore 2\times \sigma_{w}\times \dfrac {10}{100} = 0.004 - 0.03$$
or $$\sigma_{w}\times \dfrac {0.014}{1}\times 5 N/m = 0.07\ N/m$$.
Question 7
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A small spherical body of radius r is falling under gravity in a viscous medium. Due to friction the medium gets heated. How does the rate of heating depends on radius of body when it attains terminal velocity?

A
Rate of heat produced $$\propto r^4$$

B
Rate of heat produced $$\propto r^3$$

C
Rate of heat produced $$\propto r^5$$

D
None

Solution
Question 8
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A constant of $$20 cm$$ maintained in the contains water of $$1 kg$$ as shown in the figure. A small orifice area $${ 10 }^{ -2 }{ m }^{ 2 }$$ is made at the bottom of the vertical wall of the container the ejected water is directed as shown in the figure. Assuming the mass of container is negligible, the net force on the container is

A
$$Zero$$

B
$$2\,N$$

C
$$11\,N$$

D
$$18\,N$$

Solution
Question 9
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Water is flowing continuously from a tap of area $$10^{-4} m^2$$. The water velocity as it leaves the top is $$1 m/s$$.
Find out area of the water stream at a distance $$0.15 m$$ below the top.

A
$$0.5 \times 10^{-4} m^2$$

B
$$1 \times 10^{-4} m^2$$

C
$$2 \times 10^{-4} m^2$$

D
$$0.25 \times 10^{-4} m^2$$

Solution

$$A_1 V_1 = A_2 V_2$$
$$V_2 = \sqrt{V_1^2 + 2 gh}$$
$$V_2 = \sqrt{1^2 + 2 \times 10 \times 0.15}$$
$$V_2 = 2 \, m/s$$
$$10^{-4} \times 1 = A_2 \times 2$$
$$A_2 = 0.5 \times 10^{-4} m^2$$.
Question 10
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An ideal liquid of density $$\rho $$ is filled in a horizontally fixed syringe fitted with piston. There is no friction between the piston and the inner surface of the syringe. Cross-section area of the syringe is A An orifice is made at one end of the syringe. When the piston is pushed into the syringe, the liquid comes out of the orifice and then following a parabolic path falls on the ground.
With what velocity does the liquid strike the ground?

A
$$\sqrt { \dfrac { F+\rho ghA }{ \rho A } } $$

B
$$\sqrt { \dfrac { 2F}{ \rho A } } $$

C
$$\sqrt { \dfrac { 2F+\rho ghA }{ \rho A } } $$

D
$$\sqrt { \dfrac { 2(F+\rho ghA) }{ \rho A } } $$

Solution