# Solution of Chapter 13. Fluid Mechanics (HC Verma - Concepts of Physics Part 1 Book)

## Chapter Exercises

1

Is it 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.

3

In the derivation of P1 - P2 = ρgz, it was assumed that the liquid is incompressible. Why will this equation not be strictly valid for a compressible liquid?

4

Suppose the density of air at Madras is ρ0 and atmospheric pressure is P0. If we go up, the density and the pressure both decrease. Suppose we wish to calculate the pressure at a height 10 kin above Madras. If we use the equation P0 - P = ρ0gz, will we get a pressure more than the actual or less than the actual? Neglect the variation in g. Does your answer change if you also consider the variation in g?

5

The free surface of a liquid resting in an inertial frame is horizontal. Does the normal to the free surface pass through the center of the earth? Think separately if the liquid is (a) at the equator (b) at a pole (c) somewhere 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 (13-Q1). If a small hole is made at a point P in the barometer tube, will the mercury come out from this hole? 10

Is Archimedes’ principle valid in an elevator accelerating up? In a car accelerating on a level road?

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, its 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 cannot because

2

Consider the equations and .

In an elevator accelerating upward

3

The three vessels shown in figure (13-Q2) have same base area. Equal volumes of a liquid are poured in the three vessels. The force on the base will be 4

Equal mass of three liquids are kept in three identical cylindrical vessels A, B and C. The densities are ρA, ρB, ρC with ρA < ρB < ρC. The force on the base will be

5

Figure (13-Q3) shows a siphon. The liquid shown is water. The pressure difference PB — PA between the points A and B is 16

A beaker 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

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

8

Suppose the pressure at the surface of mercury in a barometer tube is P1 and the pressure at the surface of mercury in the cup is P2.

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

10

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

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 and then closed by a cork. It is inverted in a mercury cup and the cork is removed. The height of mercury column in the tube over the surface in the cup will be

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

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

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

15

A wooden object floats in water kept in a beaker. The object is near a side of the beaker (figure 13-Q4). Let P1, P2, P3 be the pressures at the three points A, B and C of the bottom as shown in the figure. 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

17

Consider the situation of the previous problem. Let the water push the left wall by a force F1 and the right wall by a force F2.

18

Water enters through end A with a speed u1 and leaves through end B with a speed u2 of a cylindrical tube AB. The tube is always completely filled with water. In case I the tube is horizontal, in case II it is vertical with the end A upward and in case III it is vertical with the end B upward. We have v1 = v2 for

19

Bernoulli theorem is based on conservation of

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.

21

Water and mercury are filled in two cylindrical vessels up to same height. Both vessels have a hole in the wall near the bottom. The velocity of water and mercury coming out of the holes are u1 and u2 respectively.

22

A large 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 υ.

## Objective II

1

A solid float in a liquid in a partially dipped position.

2

The weight of an empty balloon on a spring balance is W1. The weight becomes W2 when the balloon is filled with air. Let the weight of the air itself be ω. 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.

3

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

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.

5

In a streamline flow,

6

Water flows through two identical tubes A and B. A volume V0 of water passes through the tube A and 2 V0 through B in a given time. Which of the following may be correct?

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.

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

## Exercises

1

The surface of water in a water tank on the top of a house is 4 m above the tap level. Find the pressure of water at the tap when the tap is closed, is it necessary to specify that the tap is closed? Take g = 10 m s-2.

2

The heights of mercury surfaces in the two arms of the manometer shown in figure (13-E1) are 2 cm and 8 cm. Atmospheric pressure = 101 × 10 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. 3

The area of cross section of the wider tube shown in figure (13-E2) is 900 cm2. If the boy standing on the piston weighs 45 kg, find the difference in the levels of water in the two tubes. 4

A glass full of water has a bottom of area 20 cm2, top of area 20 cm, height 20 cm and volume half a liter. (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 = 10 × 10 N m-2. Density of water = 1000 kg m4 and g = 10 m s-2. Take all numbers to be exact. 5

Suppose the glass of the previous problem is covered 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.

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)?

7

Find the force exerted by the water on a 2 m2 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?

8

Water is filled in a rectangular tank of size 3 m × 2 m × 1 m. (a) Find the total force exerted by the water on the bottom surface of the tank. (b) Consider a vertical side of area 2 m × 1 m. Take a horizontal strip of width δx meter in this side, situated at a depth of x meter from the surface of water. Find the force by the water on this strip (c) Find the torque of the force calculated in part (b) about the bottom edge of this side. (d) Find the total force by the water on this side. (e) Find the total torque by the water on the side about the bottom edge. Neglect the atmospheric pressure and take g = 10 m s-2.

9

An ornament 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 of copper is 8.9.

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.

11

A metal piece of mass 160 g lies in equilibrium inside a glass of water (figure 13-E4). The piece touches the bottom of the glass at a small number of points. If the density of the metal is 8000 kg m4, find the normal force exerted by the bottom of the glass on the metal piece. 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?

13

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

14

A cube of ice floats partly in water and partly in K.oil (figure 13-E5). Find the ratio of the volume of ice immersed in water to that in K.oil. Specific gravity of K.oil is 0.8 and that of ice is 0.9.

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.

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’S and that of lead is 11.3.

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.

18

A cubical metal block of edge 12 cm floats in mercury with one fifth of the height inside 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.

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.

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.

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 = 1293 kg m-3.

22

A cylindrical object of outer diameter 20 cm and mass 2 kg floats in water with its axis vertical. 1f it is slightly depressed and then released, find the time period of the resulting simple harmonic motion of the object.

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(13-E6). (a) Find the elongation of the spring in equilibrium condition. (b) 1f the object is slightly depressed and released, find the time period of resulting oscillations of the object. The spring constant =500 N m-1. 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-1 fixed at the bottom. If the entire system is completely immersed in water find (a) the elongation (for compression) of the spring in equilibrium and (b) the time-period of vertical oscillations of the block when it is slightly depressed and released.

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.

26

A U-tube containing a liquid is accelerated horizontally with a constant acceleration α0. If the separation between the vertical limbs is l, find the difference in the heights of the liquid in the two arms.

27

At Deoprayag (Garhwal, UP) river Alaknanda mixes with the river Bhagjrathi 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-1 and in Bhagirathi be 16 km h-1. Find
the average speed of water in the river Ganga.

28

Water flows through a horizontal tube of variable cross section (figure 13-E7). The area of cross section at A and B are 4 mm2 and 2 mm2 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 PA – PB. 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 sections at A and B is 15/16 cm. Repeat parts (a), (b) and (c) of the previous problem. Take g = 10 m s-2.

30

Suppose the tube in the previous problem is kept vertical with B upward. Water enters through B at the rate of 1 cm3 s-1. Repeat parts (a), (b) and (c). Note that the Speed decreases as the water falls down.

31

Water flows through a tube shown in figure (13-E8). The areas of cross section at A and B are 1 cm2 and 0.5 cm2 respectively. The height difference between A and B is 5 Cm. If the speed of water at A is 10 cm s-1, find (a) the speed at B and (b) the difference in pressures at A and B. 32

Water flows through a horizontal tube as shown in figure (13-E9). 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 cm2 and 2 cm2 respectively, find the rate of flow of water across any section. 33

Water flows through the tube shown in figure (13-E10). The areas of cross section of the wide and the narrow portions of the tube are 5 cm2 and 2 cm2 respectively. The rate of flow of water through the tube is 500 cm3 s-1. Find the difference of mercury levels in the U-tube.  