## H C VERMA PHYSICS BOOK SOLUTIONS FLUID MECHANICS

__Solved Examples__

1. Water is filled in a flask upto a height of 20 cm. The bottom of the flask is circular with radius 10 cm. If the atmospheric pressure is 1.01 × 10^{5} Pa, find the force exerted by the water on the bottom. Take g = 10 m/s^{2} and density of water = 1000 kg/m^{3}.

2. A 700 g solid cube having an edge of length 10 cm floats in water. How much volume of the cube is outside the water ? Density of water = 1000 kg/m^{3}.

3. Figure shows a liquid being pushed out of a tube by pressing a piston. The area of cross-section of the piston is 1.0 cm^{2} and that of the tube at the outlet is 20 mm^{2}. If the piston is pushed at a speed of 2 cm/s, what is the speed of the outgoing liquid ?

4. Figure shows a liquid of density 1200 kg/m^{3} flowing steadily in a tube of varying cross-section. The cross-section at a point A is 1.0 cm^{2} and that at B is 20 mm^{2}, the points A and B are in the same horizontal plane. The speed of the liquid at A is 10 cm/s. Calculate the difference in pressures at A and B.

5. A water tank is constructed on the top of a building. With what speed will the water come out of a tap 6.0 m below the water level in the tank ? Assume steay flow and that the pressure above the water level is equal to the atmospheric pressure.

__Questions for Short answer__

1. It is always true that the molecules of a dense liquid are heavier than the molecules of a lighter liquid ?

2. If someone presses a pointed needle against your skin, you are hurt. But if someone presses a rod against your skin with the same force, you easily tolerate. Explain.

5. The free surface of a liquid resting in an inertial frame is horizontal. Does the normal to the free surface pass through the centre of the earth ? Think separately if the liquid is (a) at the equator (b) at a pole (c) some where else.

6. A barometer tube reads 76 cm of mercury. If the tube is gradually inclined keeping the open end immersed in the mercury reservoir, will the length of mercury column be 76 cm, more than 76 cm or less than 76 cm?

7. A one meter long glass tube is open at both ends. One end of the tube is dipped into a mercury cup, the tube is kept vertical and the air is pumped out of the tube by connecting the upper end to a suction pump. Can mercury be pulled up into the pump by this process?

8. A satellite revolves round the earth. Air pressure inside the satellite is maintained at 76 cm of mercury. What will be the height of mercury column in a barometer tube 1 m long placed in the satellite ?

9. Consider the barometer shown in figure. If a small hole is made at a point P in the barometer tube, will the mercury come out from this hole ?

10.

11. Why is it easier to swim in sea water than in fresh water ?

12. A glass of water has an ice cube floating in water. The water level just touches the rim of the glass. Will the water overflow when the ice melts?

13. A ferry boat loaded with rocks has to pass under a bridge. The maximum height of the rocks is slightly more than the height of the bridge so that the boat just fails to pass under the bridge. Should some of the rocks be removed or some more rocks be added ?

14. Water is slowly coming out from a vertical pipe. As the water descends after coming out, it area of cross-section reduces. Explain this on the basis of the equation of continuity.

15. While watering a distant plant, a gardener partially closes the exit hole of the pipe by putting his finger on it. Explain why this results in the water stream going to a larger distance.

16. A Gipsy car has a canvass top. When the car runs at high speed, the top bulges out. Explain.

__Objective - I__

1. A liquid can easily change its shape but a solid can not because

(A) the density of a liquid is smaller than that of a solid

(B*) the forces between the molecules is stronger in solid than in liquids

(C) the atoms combine to form bigger molecules in a solid

(D) the average separation between the molecules is larger in solids.

2. Consider the equations

In a elevator accelerating upward

(A) both the equations are valid

(B*) the first is valid but not the second

(C) the second is valid but not the first

(D) both are invalid

3. Three vessels shown in figure have same base area. Equal volumes of a liquid are poured in the three vessels. The force on the base will be

5. Figure shows a siphon. The liquid shown is water. The pressure difference P_{B} – P_{A} between the points A and B is

(A) 400 N/m^{2} (B) 3000 N/m^{2} (C) 1000 N/m^{2} (D*) zero

6. A breaker containing a liquid is kept inside a big closed jar. If the air inside the jar is continuously pumped out, the pressure in the liquid near the bottom of the liquid will

(A) increase (B*) decrease

(C) remain constant (D) first decrease and then increase

7. The pressure in a liquid at two points in the same horizontal plane are equal. Consider an elevator accelerating upward and a car accelerating on a horizontal road. The above statement is correct in

(A) the car only (B*) the elevator only (C) both of them (D) neither of them

8. Suppose the pressure at the surface of mercury in a barometer tube is P_{1} and the pressure at the surface of mercury in the cup is P_{2}.

(A*) P_{1} = 0, P_{2} = atmospheric pressure (B) P_{1} = atmospheric pressure, P_{2} = 0

(C) P_{1} = P_{2 }= atmospheric pressure (D) P_{1} = P_{2} = 0. k

9. A barometer kept in an elevator reads 76 cm when it is at rest. If the elevator goes up with increasing speed, the reading will be

(A) zero (B) 76 cm (C*) < 76 cm (D) > 76 cm

10. A barometer kept in an elevator accelerating upward reads 76 cm. The air pressure in the elevator is

(A) 76 cm (B) < 76 cm (C*) > 76 cm (D) Zero

11. To construct a barometer, a tube of length 1 m is filled completely with mercury and is inverted in a mercury cup. The barometer reading on a particular day is 76 cm. Suppose a 1 m tube is filled with mercury up to 76 cm. It is inverted in a mercury cup. The height of mercury column in the tube over the surface in the cup will be

(A) zero (B) 76 cm (C) > 76 cm (D*) < 76 cm

12. A 20 N metal block is suspended by a spring balance. A beaker containing some water is placed on a weighing machine which reads 40 N. The spring balance is now lowered so that the block gets immersed in the water. The spring balance now reads 16 N. The reading of the weighing machine will be

(A) 36 N (B) 60 N (C*) 44 N (D) 56 N

13. A piece of wood is floating in water kept in a bottle. The bottle is connected to an air pump. Neglect the compressibility of water. When more air is pushed into the bottle from the pump, the piece of wood will float with HCV_Ch-13_Obj I_13

(A) larger part in the water (B) lesser part in the water

(C*) same part in the water (D) it will sink.

14. A metal cube is placed in an empty vessel. When water is filled in the vessel so that the cube is completely immersed in the water, the force on the bottom of the vessel in contact with the cube

(A) will increase (B) will decrease

(C*) will remain the same (D) will become zero

15. A wooden object floats in water kept in a beaker. The object is near a side of the beaker. Let P_{1}, P_{2}, P_{3} be the pressures at the three points A, B and C of the bottom as shown in the figure.

(A*) P_{1} = P_{2} = P_{3} (B) P_{1} < P_{2} < P_{3} (C) P_{1} > P_{2} > P_{3} (D) P_{2} = P_{3} not equal to P_{1}

16. A closed cubical box is completely filled with water and is accelerated horizontally towards right with an acceleration a. The resultant normal force by the water on the top of the box

(A) passes through the centre of the top

(B) passes through a point to the right of the centre

(C*) passes through a point to the left of the centre

(D) becomes zero

17. Consider the situation of the previous problem. Let the water push the left wall by a force F_{1} and the right wall by a force F_{2}.

(A) F_{1} = F_{2} (B*) F_{1} > F_{2} (C) F_{1} < F_{2}

(D) The information is insufficient to know the relation between F_{1} and F_{2}

18. Water enters through end A with a speed v_{1} and leaves through end B with a speed v_{2} of a cylindrical tube AB. The tube is always completely filled with water. In case I the tube is horizontal, in case II it vertical with the end A upward and in case III it is vertical with the end B upward. We have v_{1} = v_{2} for

HCV_Ch-13_Obj I_18

(A) case I (B) case II (C) case III (D*) each case

19. Bernoulli’s theorem is based on conservation of

(A) momentum (B) mass (C*) energy (D) angular momentum

20. Water is flowing through a long horizontal tube. Let PA and PB be the pressures at two points A and B of the tube

(A) P_{A} must be equal to P_{B}

(B) P_{A} must be greater than P_{B}

(C) P_{A} must be smaller than P_{B}

(D*) P_{A} = P_{B} only if the cross-sectional area at A and B are equal

22. A larger cylindrical tank has a hole of area A at its bottom. Water is poured in the tank by a tube of equal cross-sectional area A ejecting water at the speed v.

(A) The water level in the tank will keep on rising.

(B) No water can be stored in the tank

(C*) The water level will rise to a height v^{2}/2 g and then stop

(D) The water level will oscillate.

__Objective - II__

1. A solid floats in a liquid in a partially dipped position.

(A*) the solid exerts a force equal to its weight on the liquid

(B*) the liquid exerts a force of buoyancy on the solid which is equal to the weight of the solid

(C*) the weight of the displaced liquid equals the weight of the solid

(D) the weight of the dipped part of the solid is equal to the weight of the displaced liquid

2. The weight of an empty balloon on a spring balance is W_{1}. The weight becomes W_{2} when the balloon is filled with air. Let the weight of the air itself be w. Neglect the thickness of the balloon when it is filled with air. Also neglect the difference in the densities of air inside and outside the balloon.

(A*) W_{2} = W_{1} (B) W_{2} = W_{1} + w (C*) W_{2} < W_{1} + w (D) W_{2} > W_{1}

3. A solid is completely immersed in a liquid. The force exerted by the liquid on the solid will

(A) increase if it is pushed deeper inside the liquid

(B) change if its orientation is changed

(C*) decrease if it is taken partially out of the liquid

(D*) be in the vertically upward direction

4. A closed vessel is half filled with water. There is a hole near the top of the vessel and air is pumped out from this hole

(A) the water level will rise up in the vessel

(B*) the pressure at the surface of the water will decrease

(C*) the force by the water on the bottom of the vessel will decrease

(D) the density of the liquid will decrease

5. In a streamline flow,

(A) the speed of a particle always remains same

(B) the velocity of a particle always remains same

(C*) the kinetic energies of all the particles arriving at a given point are the same

(D*) the momenta of all the particles arriving at a given point are the same

6. Water flows through two identical tubes A and B. A volume V_{0} of water passes through the tube A and 2V_{0} through B in a given time. Which of the following may be correct ?

(A*) flow in both the tubes are steady (B*) flow in both the tubes are turbulent

(C*) flow in steady in A but turbulent in B (D) flow is steady in B but turbulent in A

7. Water is flowing in streamline motion through a tube with its axis horizontal. Consider two points A and B in the tube at the same horizontal level.

(A) the pressures at A and B are equal for any shape of the tube

(B) the pressures are never equal

(C*) the pressure are equal if the tube has a uniform cross-section

(D*) the pressures may be equal even if the tube has a nonuniform cross-section.

8. There is a small hole near the bottom of an open tank filled with a liquid. The speed of the water ejected does not depend on

(A*) area of the hole (B*) density of the liquid

(C) height of the liquid from the hole (D) acceleration due to gravity

__Worked Out Examples__

1. A beaker of circular cross-section of radius 4 cm is filled with mercury upto a height of 10 cm. Find the force exerted by the mercury on the bottom of the beaker. The atmospheric pressure = 10^{5} N/m^{2}. Density of mercury = 13600 kg/m^{3} . Take g = 10 m/s^{2}.

(g = 10 m/s^{2} ysaA)

Sol. The pressure at the surface = atmospheric pressure

= 10^{5} N/m^{2}.

The pressure at the bottom = 10^{5} N/m^{2} + h(density)g

= 10^{5} N/m^{2} + h(density)g

= 10^{5} N/m^{2} + (0.1 m) (13600)(10)

= 10^{5} N/m^{2} + 13600 N/m^{2}

= 1.136 × 10^{5} N/m2.

The force exerted by the mercury on the bottom

= (1.136 × 10^{5} N/m^{2}) × (3.14 × 0.04 m × 0.04 m)

= 571 N.

2. The density of air near earth’s surface is 1.3 kg/m^{3} and the atmospheric pressure is 1.0 × 10^{5} N/m^{2}. If the atmosphere had uniform density, same as that observed at the surface of the earth, what would be the height of the atmosphere to exert the same pressure ?

Even Mount Everest (8848 m) would have been outside the atmosphere.

3. The liquid shown in figure in the two arms are mercury (specific gravity = 13.6) and water. If the difference of heights of the mercury columns is 2 cm, find the height h of the water column.

4. A cylindrical vessel containing a liquid is closed by a smooth piston of mass m as shown in the figure. The area of cross-section of the piston is A. If the atmospheric pressure is P_{0}, find the pressure of the liquid just below the piston.

Sol. Let the pressure of the liquid just below the piston be P. The forces acting on the piston are

(a) its weight, mg (downward)

(b) force due to the air above it, P_{0} A (doward)

(c) force due to the liquid below it, PA (upward).

If the piston is in equilibrium

5. The area of cross-section of the two arms of a hydraulic press are 1 cm^{2} and 10 cm^{2} respectively (figure). A force of 5 N is applied on the water in the thinner arm. What force should be applied on the water in the thicker arms so that the water may remain in equilibrium?

6. A copper piece of mass 10 g is suspended by a vertical spring. The spring elongates 1 cm over its natural length to keep the piece in equilibrium. A beaker containing water is now placed below the piece so as to immerse the piece completely in water. Find the elongation of the spring. Density of copper = 9000 kg/m^{3}. Take g = 10 m/s^{2}.

7. A cubical block of wood of edge 3 cm floats in water. The lower surface of the cube just touches the free end of a vertical spring fixed at the bottom of the pot. Find the maximum weight that can be put on the block without wetting it. Density of wood = 800 kg/m^{3} and spring constant of the spring = 50 N/m. Take g = 10 m/s^{2}.

8. A wooden plank of length 1 m and uniform cross-section is hinged at one end to the bottom of a tank as shown in figure. The tank is filled with water up to a height of 0.5 m. The specific gravity of the plank is 0.5. Find the angle theta that the plank makes with the vertical in the equilibrium position.

(Exclude the case theta = 0).

9. A cylindrical block of wood of mass M is floating in water with its axis vertical. It is depressed a little and then released. Show that the motion of the block is simple harmonic and find its frequency.

10. Water flows in a horizontal tube as shown in figure. The pressure of water changes by 600 N/m^{2} between A and B where the areas of cross-section are 30 cm2 and 15 cm^{2} respectively. Find the rate of flow of water through the tube. (Water density=1000kg/m^{3})

11. The area of cross-section of a large tank is 0.5 m^{2}. It has an opening near the bottom having area of cross-section 1 cm^{2}. A load of 20 kg is applied on the water at the top. Find the velocity of the water coming out of the opening at the time when the height of water level is 50 cm above the bottom. Take g = 10 m/s^{2}.

__Exercise__

__Exercise__

1. The surface of water in a water tank on the top of a house is 4 m above the top level. Find the pressure of water at the tap when the tap is closed. It is necessary to specify that the tap is closed? Take g = 10 m/s^{2}.

1. 40000 N/m^{2}, Yes

2. The heights of mercury surfaces in the two arms of the manometer shown in figure are 2 cm and 8 cm. Atmospheric pressure = 1.01 × 10^{5} N/m^{2}. Find (a) the pressure of the gas in the cylinder and (b) the pressure of mercury at the bottom of the U-tube.

2. (a) 1.09 × 10^{5} N/m^{2} (b) 1.12 × 10^{5} N/m^{2}

3. The area of cross-section of the wider tube shown in figure is 900 cm^{2}. If the boy standing on the piston weighs 45 kg, find the difference in the levels of water in the two tubes.

3. 50 cm

4. A glass full of water has a bottom of area 20 cm^{2}, top of area 20 cm^{2}, height 20 cm and volume half a litre.

HCV_Ch-13_Ex._4

(a) find the force exerted by the water on the bottom.

(b) Considering the equilibrium of the water, find the resultant force exerted by the sides of the glass on the water. Atmospheric pressure = 1.0 × 10^{5} N/m^{2}. Density of water 1000 kg/m^{3} and g = 10 m/s^{2}. Take all numbers to be exact.

Ans. (a) 204 N (b) 1 N upward

5. Suppose the glass of the previous problem is convered by a jar and the air inside the jar is completely pumped out. (a) what will be the answers to the problem? (b) Show that the answers do not change if a glass of different shape is used provided the height, the bottom area and the volume are unchanged.

5. 4 N, 1 N upward

6. If water be used to construct a barometer, what would be the height of water column at standard atmospheric pressure (76 cm of mercury)?

6. 1033.6cm

7. Find the force exerted by the water on a 2 m^{2} plane surface of a large stone placed at the bottom of a sea 500 m deep. Does the force depend on the orientation of the surface?

Sol. P = 1000 × 10 × 500 = 5 × 10^{6}

F = PA = (5 × 10^{6}) (2) = 10^{7} N.

No the force on stone is independent of its orientation.

Ans. 10^{7 }N, No

9. An ornanment weighing 36 g in air, weighs only 34 g in water. Assuming that some copper is mixed with gold to prepare the ornament, find the amount of copper in it. Specific gravity of gold is 19.3 and that the copper is 8.9.

9. 2.2 g

10. Refer to the previous problem. Suppose, the goldsmith argues that he has not mixed copper or any other material with gold, rather some cavities might have been left inside the ornament. Calculate the volume of the cavities left that will allow the weights given in that problem.

10. 0.112 cm^{3 }

11. A metal piece of mass 160 g lies in equilibrium inside a glass of water. The piece touches the bottom of the glass at a small number of points. If the density of the metal is 8000 kg/m^{3}, find the normal force exerted by the bottom of the glass on the metal piece.

11. 1.4 N

12. A ferry boat has internal volume 1 m3 and weight 50 kg. (a) Neglecting the thickness of the wood, find the fraction of the volume of the boat immersed in water. (b) If a leak develops in the bottom and water starts coming in, what fraction of the boat’s volume will be filled with water before water starts coming in from the sides ?

12. (a) 1/20 (b) 19/20

13. A cubical block of ice floating in water has to support a metal piece weighing 0.5 kg on top of it. What can be the minimum edge of the block so that it does not sink in water? Specific gravity of ice = 0.9.

Ans. (10)5^{1/3} cm = 17 cm

14. A cube of ice floats partly in water and partly in K.oil. Find the ratio of the volume of ice immersed in water to that in K.oil. Specify gravity of K.oil is 0.8 and that of ice is 0.9.

14. 1 : 1

15. A cubical box is to be constructed with iron sheets 1 mm in thickness. What can be the minimum value of the external edge so that the cube does not sink in water? Density of iron = 8000 kg/m^{3} and density of water = 1000 kg/m^{3}.

Ans. 4.8 cm

16. A cubical block of wood weighing 200 g has a lead piece fastened underneath. Find the mass of the lead piece which will just allow the block to float in water. Specific gravity of wood is 0.8 and that of lead is 11.3.

Ans. 54.8 g

17. Solve the previous problem if the lead piece is fastened on the top surface of the block and the block is to float with its upper surface just dipping into water.

17. 50 g

18. A cubical metal block of edge 12 cm floats in mercury with one fifth of the height insde the mercury. Water is poured till the surface of the block is just immersed in it. Find the height of the water column to be poured. Specific gravity of mercury = 13.6.

18. 10.4 cm

19. A hollow spherical body of inner and outer radii 6 cm and 8 cm respectively floats half submerged in water. Find the density of the material of the sphere.

19. 865 kg/m^{3}

20. A solid sphere of radius 5 cm floats in water. If a maximum load of 0.1 kg can be put on it without wetting the load, find the specific gravity of the material of the sphere.

20. 0.8

21. Find the ratio of the weights, as measured by a spring balance, of a 1 kg block of iron and a 1 kg block of wood. Density of iron = 7800 kg/m^{3}, density of wood = 800 kg/m^{3} and density of air = 1.293 km/m^{3}.

21. 1.0015

22. A cylindrical object of out diameter 20 cm and mass 2 kg floats in water with its axis vertical. If it is slightly depressed and then released, find the time period of the resulting simple harmonic motion of the object.

22. 0.5 s

23. A cylindrical object of outer diameter 10 cm, height 20 cm and density 8000 kg/m^{3} is supported by a vertical spring and is half dipped in water as shown in figure. (a) Find the elongation of the spring in equilibrium condition. (b) If the object is slightly depressed and released, find the time period of resulting oscillations of the object. The spring constant = 500 N/m.

24. A wooden block of mass 0.5 kg and density 800 kg/m^{3} is fastened to the free end of a vertical spring of spring constant 50 N/m fixed at the bottom. If the entire system is completely immersed in water, find (a) the elongation (or compression) of the spring in equilibrium and (b) the time period of vertical oscillations of the block when it is slightly depressed and released.

24. (a) 2.5 cm (b) 3.14/5s

25. A cube of ice of edge 4 cm is placed in an empty cylindrical glass of inner diameter 6 cm. Assume that the ice melts uniformly from each side so that it always retains its cubical shape. Remembering that ice is lighter than water, find the length of the edge of the ice cube at the instant it just leaves contact with the bottom of the glass.

25. 2.26 cm

26. A U-tube containing a liquid is accelerated horizontally with a constant acceleration a_{0}. If the separation between the vertical limbs is l, find the difference in the heights of the liquid in the two arms.

26. a0l/g

27. At Deoprayag (Garhwal, UP) river Alaknanda mixes with the river Bhagirathi and becomes river Ganga. Suppose Alaknanda has a width of 12 m, Bhagirathi has a width of 8 m and Ganga has a width of 16 m. Assume that the depth of water is same in the three rivers. Let the average speed of water in Alaknanda be 20 km/h and in Bhagirathi be 16 km/h. Find the average speed of water in the river Ganga.

27. 23 km/h

28. Water flows through a horizontal tube of variable cross-section (figure). The area of cross-section at A and B are 4 mm^{2} and 2 mm^{2} respectively. If 1 cc of water enters per second through A, find (a) the speed of water at A, (b) the speed of water at B and (c) the pressure difference P_{A} – P_{B}.

28. (a) 25 cm/s, (b) 50 cm/s (c) 94 N/m^{2}

29. Suppose the tube in the previous problem is kept vertical with A upward but the other conditions remain the same. The separation between the cross-section at A and B is 15/16 cm. Repeat parts (a), (b) and (c) of the previous problem. Take g = 10 m/s^{2}.

29. (a) 25 cm/s, (b) 50 cm/s (c) zero

30. Suppose the tube in the previous problem is kept vertical with B upward. Water enters through B at the rate of 1 cm^{3}/s. Repeat part (a), (b) and (c). Note that the speed decreases as the water falls down.

30. (a) 25 cm/s, (b) 50 cm/s (c) 188 N/m^{2}

31. Water flows through a tube shown in figure. The areas of cross-section at A and B are 1 cm^{2} and 0.5 cm^{2} respectively. The height difference between A and B is 5 cm. If the speed of water at A is 10 cm/s find (a) the speed at B and (b) the difference in pressures at A and B.

31. (a) 20 cm/s, (b) 485 N/m^{2}

32. Water flows through a horizontal tube as shown in figure. If the difference of heights of water column in the vertical tubes is 2 cm, and the areas of cross-section at A and B are 4 cm^{2} and 2 cm^{2} respectively, find the rate of flows of water across any section.

32. 146 cc/s,

33. Water flows through the tube shown in figure. The areas of cross-section of the wide and the narrow portion of the tube are 5 cm^{2} and 2 cm^{2} respectively. The rate of flow of water through the tube is 500 cm^{3}/s. Find the difference of mercury levels in the U-tube.

34. Water leaks out from an open tank through a hole of area 2 mm^{2} in the bottom. Suppose water is filled up to a height of 80 cm and the area of cross-section of the tank is 0.4 m^{2}.The pressure at the open surface and at the hole are equal to the atmospheric pressure. Neglect the small velocity of the water near the open surface in the tank. (a) Find the initial speed of water coming out of the hole. (b) Find the speed of water coming out when half of water has leaked out. (c) Find the volume of water leaked out during a time interval dt after the height remained is h. Thus find the decrease in height dh in terms of h and dt. (d) From the result of part (c) find the time required for half of the water to leak out.

35. H/2