Civil Engineering Hydraulics: Essential Theory with Worked Examples

R. E. Featherstone, C. Nalluri

Chapter 2 Fluid Statics - all with Video Answers

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Chapter Questions

Problem 1

(a) A large storage tank contains a salt solution of variable density given by $\rho=1050+\mathrm{kh}$ in $\mathrm{kg} / \mathrm{m}^3$, where $\mathrm{k}=50 \mathrm{~kg} / \mathrm{m}^4$, at a depth h metres below the free surface. Calculate the pressure intensity at the bottom of the tank holding 5 m of the solution.
(b) A Bourdon type pressure gauge is connected to a hydraulic cylinder activated by a piston of 20 mm diameter. If the gauge balances a total mass of 10 kg placed on the piston, determine the gauge reading in metres of water.

Aadit Sharma

Numerade Educator

04:18

Problem 2

A closed cylindrical tank 4 m high is partly filled with oil of density $800 \mathrm{~kg} / \mathrm{m}^3$ to a depth of 3 m . The remaining space is filled with air under pressure. A U-tube containing mercury (relative density $\mathbf{1 3 . 6}$ ) is used to measure the air pressure, with one end open to atmosphere. Find the gauge pressure at the base of the tank when the mercury deflection in the open limb of the U-tube is (i) 100 mm above, and (ii) 100 mm below the level in the other limb.

Simran Hiranandani

Numerade Educator

03:24

Problem 3

A manometer consists of a glass tube, inclined at $30^{\circ}$ to horizontal, connected to a metal cylinder standing upright. The upper end of the cylinder is connected to a gas supply under pressure. Find the pressure in millimetres of water when the manometer fluid of relative density 0.8 reads a deflection of 80 mm along the tube. Take the ratio, $r$, of the diameters of the cylinder and the tube as 64 . What value of $r$ would you suggest so that the error due to disregarding the change in level in the cylinder will not exceed $0 \cdot 2 \%$ ?

Andreas Papavassiliou

Numerade Educator

01:36

Problem 4

In order to measure the pressure difference between two points in a pipeline carrying water, an inverted $U$-tube is connected to the points and air under atmospheric pressure is entrapped in the upper portion of the U-tube. If the manometer deflection is 0.8 m and the downstream tapping is 0.5 m below the upstream point, find the pressure difference between the two points.

Kratika Bhadauria

Numerade Educator

04:12

Problem 5

A high pressure gas pipeline is connected to a macromanometer consisting of four U-tubes in series with one end open to atmosphere and a deflection of 500 mm of mercury (relative density 13-6) has been observed. If water is entrapped between the mercury columns of the manometer and the relative density of the gas is $1.2 \times 10^{-3}$, calculate the gas pressure in $\mathrm{N} / \mathrm{mm}^2$, the centre line of the pipeline being at a height of 0.50 m above the top mercury level.

Andreas Papavassiliou

Numerade Educator

03:21

Problem 6

A dock gate is to be reinforced with three identical horizontal beams. If the water stands to depth of 5 m and 3 m on either side, find the positions of the beams, measured above the floor level, so that each beam will carry an equal load, and calculate the load on each beam per unit length.

Kudakwashe Mapiki

Numerade Educator

02:59

Problem 7

A storage tank of a sewage treatment plant is to discharge excess sewage into the sea through a horizontal rectangular culvert 1 m deep and 1.3 m wide. The face of the discharge end of the culvert is inclined at $40^{\circ}$ to the vertical and the storage level is controlled by a flap-gate weighing 4.5 kN , hinged at the top edge and just covering the opening. When the sea water stands to the hinge level, to what height above the top of the culvert will the sewage be stored before a discharge occurs? Take the density of the sewage as $1000 \mathrm{~kg} / \mathrm{m}^3$ and of the sea water as $1025 \mathrm{~kg} / \mathrm{m}^3$.

Surendra Kumar

Numerade Educator

Problem 8

A radial gate, 2 m long, hinged about a horizontal axis, closes the rectangular sluice of a control dam by the application of a counter-weight W (see fig. 2.33).
Determine (i) the total hydrostatic thrust and its location on the gate when the storage depth is 4 m , and (ii) for the gate to be stable, the counterweight W. Explain what will happen if the storage increases beyond 4 m .
FIGURE CAN'T COPY.

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01:42

Problem 9

A sector gate of radius 3 m and length 4 m retains water as shown in fig. 2.34.
FIGURE CAN'T COPY.
Determine the magnitude, direction and location of the resultant hydrostatic thrust on the gate.

Kudakwashe Mapiki

Numerade Educator

04:30

Problem 10

The profile of the inner face of a dam is a parabola with equation $y$ $=0.30 \mathrm{x}^2$ (see fig. 2.35). The dam retains water to a depth of 30 m above the base. Determine the hydrostatic thrust on the dam per unit length, its inclination to the vertical and the point at which the line of action of this thrust intersects the horizontal base of the dam.

Andreas Papavassiliou

Numerade Educator

04:19

Problem 11

A homogeneous wooden cylinder of circular section, relative density 0.7 , is required to float in oil of density $900 \mathrm{~kg} / \mathrm{m}^3$. If d and h are the diameter and height of the cylinder respectively, establish the upper limiting value of the ratio $\mathrm{h} / \mathrm{d}$ for the cylinder to float with its axis vertical.

Adnan Gill

Numerade Educator

01:04

Problem 12

A conical buoy floating in water with its apex downwards has a diameter $d$ and a vertical height $h$. If the relative density of the material of the buoy is s , prove that for stable equilibrium,
FIGURE CAN'T COPY.

Narayan Hari

Numerade Educator

01:04

Problem 13

A cyclindrical buoy weighing 20 kN is to float in sea water whose density is $1020 \mathrm{~kg} / \mathrm{m}^3$. The buoy has a diameter of 2 m and is 2.5 m high. Prove that it is unstable.

If the buoy is anchored with a chain attached to the centre of its base, find the tension in the chain to keep the buoy in vertical position.

Narayan Hari

Numerade Educator

03:56

Problem 14

A floating platform for offshore drilling purposes is in the form of a square floor supported by 4 vertical cylinders at the corners. Determine the location of the centroid of the assembly in terms of the side $L$ of the floor and the depth of submergence $h$ of the cylinders, so as to float in neutral equilibrium under a uniformly distributed loading condition.

Linda Winkler

Numerade Educator

Problem 15

A platform constructed by joining two 10 m long wooden beams as shown in fig. 2.36 is to float in water. Examine the stability of a single beam and of the platform and determine their stability moments. Neglect the weight of the connecting pieces and take the density of wood as $600 \mathrm{~kg} / \mathrm{m}^3$.
FIGURE CAN'T COPY.

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01:39

Problem 16

A rectangular barge 10 m wide and 20 m long is 5 m deep and weighs 6 MN when loaded without any ballast. The barge has two compartments each 4 m wide and 20 m long, symmetrically placed about its central axis, and each containing 1 MN of water ballast. The water surface in each compartment is free to move. The centre of gravity without ballast is 3.0 m above the bottom and on the geometrical centre of the plan. (i) Calculate the metacentric height for rolling, and (ii) if 100 kN of the deck load is shifted 5 m laterally find the approximate heel angle of the barge.

Kratika Bhadauria

Numerade Educator

02:33

Problem 17

A U-tube acceleration meter consists of two vertical limbs connected by a horizontal tube of 400 mm long parallel to the direction of motion. Calculate the level difference of the liquid in the U-tube when it is subjected to a horizontal uniform acceleration of $6 \mathrm{~m} / \mathrm{s}^2$.

Keshav Singh

Numerade Educator

03:25

Problem 18

An open rectangular tank 4 m long and 3 m wide contains water up to a depth of 2 m . Calculate the slope of the free surface of water when the tank is accelerated at $2 \mathrm{~m} / \mathrm{s}^2$, (i) up a slope of $30^{\circ}$, and (ii) down a slope of $30^{\circ}$.

Chai Santi

Numerade Educator

01:05

Problem 19

Prove that, in the forced vortex motion (fluids subjected to rotation externally) of a liquid, the rate of increase of the pressure, $p$, with respect to the radius, r , at a point in liquid is given by $\mathrm{dp} / \mathrm{dr}=\rho \omega^2 \mathrm{r}$, in which $\omega$ is the angular velocity of the liquid and $\rho$ is its mass density. Hence calculate the thrust of the liquid on the top of a closed vertical cylinder of 450 mm diameter, completely filled with water under a pressure of $10 \mathrm{~N} / \mathrm{cm}^2$, when the cylinder rotates about its axis at 240 rpm .

Mayukh Banik

Numerade Educator