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School: HKU
Course: Engg1010
A jet of water injected into stationary water: Upon emerging from the slit at the left, the jet of fluid loses some of its momentum to the surrounding fluid. This causes the jet to slow down and its width to increase (air bubbles in water). (Photograph co
School: HKU
Course: Engg1010
THE UNTVERSITYOF HONG KONG Facultyof Engireerirg ( m o Foundations f engineering echanics ENGGI 010) Tutorial 4: Torsion 1. The solid cylindricalshaftof variablesize,as shownin mm in Figure f is actedupon t by the torquesindicated.What is the maximum orsi
School: HKU
Course: Engg1010
E ."b N .g 00 g. 00 ( n =. 0 ? . :T ( n . =. 0 :3 'a 3 c: 3 3 3 9 0 o n 00 2. :3 'g ( < ( 0  ( 00 . . O '"1 ( :T ( '"1 ( . :T (  > 00 ( n =. O :3 N cr" (D 8 < (D =r '< 3 0 = C/) C/) (D (') [ C/) (D < (D = (') (D C/) 3 c: 3 C/) 3 g. (D C/) (D C/) ('
School: HKU
Course: Engg1010
An image of hurricane Allen viewed via satellite: Although there is considerable motion and structure to a hurricane, the pressure variation in the vertical direction is approximated by the pressuredepth relationship for a static fluid. 1Visible and infr
School: HKU
Course: Engg1010
Fluid Mechanics Problem 6 The diameter of a pipe bend is 300 mm at inlet and 150 mm at outlet and the flow is turned through 60o in a vertical plane. The axis at inlet is horizontal and the centre of the outlet section is 1.5 m above the centre of the inl
School: HKU
ENGG1010 Foundations of Engineering Mechanics Examination: Dec 14, 2010, 9:30am12:30pm, Flora Ho Sports Centre Exam paper is divided into Sections A and B, each containing 3 questions. You are required to attempt 5 out of the 6 questions. Section B two q
School: HKU
School: HKU
School: HKU
School: HKU
University Number (please write very clearly) _ Surname_ Answers for ENGG1010 Quiz, Part B Dynamics, 20 October 2009 Answer both questions. Write your answers on this question sheet and submit this sheet only. QB1. Bar AB is hinged to the edge of a circu
School: HKU
School: HKU
Name: ENGG1010 (20102011) First Semester MidTerm Test Question on fluid mechanics U. No.: Instructions Write your answers in the space provided, and hand in this sheet upon completion of the test. As shown below, an underwater tunnel is closed by a circ
School: HKU
Midterm Quiz for ENGG1010 Foundations of Engineering Mechanics Faculty of Engineering, HKU Class A , 23 October 2008 Instructions: (1) The quiz has four questions. Attempt all questions. (2) Write down your solutions on the space below each question. (3)
School: HKU
Course: Engg1010
Fluid Mechanics Problem 6 The diameter of a pipe bend is 300 mm at inlet and 150 mm at outlet and the flow is turned through 60o in a vertical plane. The axis at inlet is horizontal and the centre of the outlet section is 1.5 m above the centre of the inl
School: HKU
Course: Engg1010
Fluid Mechanics Problems 5 Find the horizontal component of the force exerted by the flowing water on 1 m width of the (shaded) outflow structure, which discharges via a 0.2 m diameter circular duct to the atmosphere. Hint: consider the control volume as
School: HKU
Course: Engg1010
Fluid Mechanics Problems 4 Oil ( = 800 kg/m3) flows through a pipe line (see figure below), which contracts from 450 mm diameter at A to 300 mm diameter at B and then forks, one branch being 150 mm diameter discharging at C and the other branch 225 mm dia
School: HKU
Course: Engg1010
Fluid Mechanics Problems 3 1. For the tank shown below, find the pressure gauge reading if the mercury manometer reading H = 0.2 m. Relative density of mercury = 13.6. Pressure gauge Hg 2. For an Lshaped gate shown below, determine the height H (of the w
School: HKU
Course: Engg1010
Fluid Mechanics Problems 2 Consider a gate CD, which is shown below and can be (a) rectangular, (b) circular, and (c) triangular in shape. The gate is only hinged at the top, and has no support elsewhere. Find the moment to be applied at the hinge in each
School: HKU
Course: Engg1010
Fluid Mechanics Problems 1 1. Some people say that the seawater level will rise as a result of the melting of an iceberg. How true is this statement? 2. As is schematically shown below, a boat carrying a rock floats on water in a lake. What happens to the
School: HKU
Course: Engg1010
A jet of water injected into stationary water: Upon emerging from the slit at the left, the jet of fluid loses some of its momentum to the surrounding fluid. This causes the jet to slow down and its width to increase (air bubbles in water). (Photograph co
School: HKU
Course: Engg1010
THE UNTVERSITYOF HONG KONG Facultyof Engireerirg ( m o Foundations f engineering echanics ENGGI 010) Tutorial 4: Torsion 1. The solid cylindricalshaftof variablesize,as shownin mm in Figure f is actedupon t by the torquesindicated.What is the maximum orsi
School: HKU
Course: Engg1010
E ."b N .g 00 g. 00 ( n =. 0 ? . :T ( n . =. 0 :3 'a 3 c: 3 3 3 9 0 o n 00 2. :3 'g ( < ( 0  ( 00 . . O '"1 ( :T ( '"1 ( . :T (  > 00 ( n =. O :3 N cr" (D 8 < (D =r '< 3 0 = C/) C/) (D (') [ C/) (D < (D = (') (D C/) 3 c: 3 C/) 3 g. (D C/) (D C/) ('
School: HKU
Course: Engg1010
An image of hurricane Allen viewed via satellite: Although there is considerable motion and structure to a hurricane, the pressure variation in the vertical direction is approximated by the pressuredepth relationship for a static fluid. 1Visible and infr
School: HKU
Course: Engg1010
Fluid Mechanics Problem 6 The diameter of a pipe bend is 300 mm at inlet and 150 mm at outlet and the flow is turned through 60o in a vertical plane. The axis at inlet is horizontal and the centre of the outlet section is 1.5 m above the centre of the inl
School: HKU
Course: Engg1010
Fluid Mechanics Problems 5 Find the horizontal component of the force exerted by the flowing water on 1 m width of the (shaded) outflow structure, which discharges via a 0.2 m diameter circular duct to the atmosphere. Hint: consider the control volume as
School: HKU
Course: Engg1010
Fluid Mechanics Problems 4 Oil ( = 800 kg/m3) flows through a pipe line (see figure below), which contracts from 450 mm diameter at A to 300 mm diameter at B and then forks, one branch being 150 mm diameter discharging at C and the other branch 225 mm dia
School: HKU
Course: Engg1010
Fluid Mechanics Problems 3 1. For the tank shown below, find the pressure gauge reading if the mercury manometer reading H = 0.2 m. Relative density of mercury = 13.6. Pressure gauge Hg 2. For an Lshaped gate shown below, determine the height H (of the w
School: HKU
Course: Engg1010
Fluid Mechanics Problems 2 Consider a gate CD, which is shown below and can be (a) rectangular, (b) circular, and (c) triangular in shape. The gate is only hinged at the top, and has no support elsewhere. Find the moment to be applied at the hinge in each
School: HKU
Course: Engg1010
Fluid Mechanics Problems 1 1. Some people say that the seawater level will rise as a result of the melting of an iceberg. How true is this statement? 2. As is schematically shown below, a boat carrying a rock floats on water in a lake. What happens to the
School: HKU
Course: Engg1010
Time table for ENGG 1010: Foundations of Engineering Mechanics, Semester 1, 201112 Lecturers: C. O. Ng for fluid mechanics (cong@hku.hk, 2859 2622, office HW701) A. K. Soh for statics and mechanics of materials (aksoh@hku.hk, 2859 8061, office HW720) Uni
School: HKU
Course: Engg1010
What What is an Engineer? Engineering is the profession in which knowledge of the mathematical and natural sciences gained by study, experience, and practice is applied with judgement to develop ways to utilize, economically, the materials and forces of n
School: HKU
Course: Foundation Of Engineering
Vectors A real number is a point on the real line R To describe a point on a plane R2, we use two numbers, e.g., (3,1) In fact, this is just an expression of a point using rectangular coordinate system 3 If we put the coordinates as , we have a vecto
School: HKU
Course: Foundation Of Engineering
System of linear equations A simple example: x 2 x = 1 1 2 x1 + 3 x2 = 3 x1 = 3, x2 = 2 Physical meaning: Solution is the intersection point of two lines Other possibilities: No solution or infinite many solutions 1 Y.C. WU  HKUEEE Example of three
School: HKU
Course: Foundation Of Engineering
MATH 1851  Ordinary Differential Equations (ODEs) (A) First order equations A linear, first order ODE is always solvable by the method of integrating factor: dy p ( x) y q ( x) . dx Multiplying by (the integrating factor) exp p( x) dx will turn the le
School: HKU
Course: Foundation Of Engineering
Q1. Find the average rate of change of the function dened by P () = 3 42 + 5 on the interval [1, 2]. Ans. Average rate of change (check denition) is P/ which is P (2) P (1) =0 . 21 1 Q2. Find the tangent line to the curve y = 2 x3 at the point (1, 1). Ans
School: HKU
Course: Foundation Of Engineering
Math1851, Tutorial 2 (for 2/106/10) FLT Please go through all 10 problems before your tutorial session. 1. (Ex1.5: #60) If functions f (x) and g (x) are continuous for 0 x 1, could f (x)/g (x) possibly be discontinuous at a point of [0, 1]? Give reasons
School: HKU
Course: Foundation Of Engineering
Math1851, Tutorial 1 (for 24/928/9) FLT Please work on the following 10 problems before your tutorial session. 1. (Ex1.1: #6) Find the average rate of change of the function dened by P () = 3 42 + 5 on the interval [1, 2]. 2. (Ex1.1: #12) Find the tangen
School: HKU
Course: Foundation Of Engineering
Math1851, Assignment 1 (Due 8 Oct 5.00pm) FLT 1. Ex1.2: Q77 2. Ex1.4: Q3 3. Ex1.4: Q17 4. Ex1.4: Q25 5. Ex2.2: Q25 6. Ex2.5: Q31 7. Ex2.5: Q49 8. Ex2.7: Q9 9. Ex3.2: Q15b 10. Ex3.2: Q61 11. P.64: Q17 12. Evaluate 1 (1 + x) k 1 lim . x0 x 13. Evaluate cos
School: HKU
Course: Foundation Of Engineering
MATH 1851  Laplace Transforms (A) Basic concepts The Laplace transform of a function f (t) is defined by L ( f (t ) ) L( f ) 0 e st f (t ) dt F ( s) , f (t ) L1 ( F (s), where t is usually time. The convention here is that small letter is used for the f
School: HKU
Course: Foundation Of Engineering
Chapter 7 Transcendental Functions 7.5 Indeterminate Forms and L Hpitals Rule o Theorem 5  L Hpitals Rule [377] Suppose that f (a) = g (a) = 0, that f and g are o dierentiable on an open interval I containing a, and that g (x) = 0 on I if x = a. Then f (
School: HKU
Course: Foundation Of Engineering
Chapter 4 nates Parametric Equations and Polar Coordi Denition [200] If x and y are given as functions x = f (t), y = g (t) over an interval I of tvalues, (can think of t as the time), then the set of points (x, y ) = (f (t), g (t) dened by these equati
School: HKU
Course: Foundation Of Engineering
Chapter 3 Applications of Derivatives The number in [ ] refers to the page number of our textbook Theorem 1  The Extreme Value Theorem [139] If f is continuous on a closed interval [a, b], then f attains both an absolute maximum value M and an absolute m
School: HKU
Course: Foundation Of Engineering
Chapter 2 Dierentiation The number in [ ] refers to the page number of our textbook Denition [67] The derivative of a function f at a point x0 , denoted f (x0 ), is f (x0 + h) f (x0 ) h0 h f (x0 ) = lim provdied this limit exists. Read Eg.1 [67]. [68] If
School: HKU
Course: Foundation Of Engineering
Chapter 1 Limits and Continuity The number in [ ] refers to the page number of our textbook Eg. 1, 2 [4] An rock falls from a cli and the distance it falls through after t seconds is given by y = 4.9t2 . The average velocity of the rock for the rst 2 seco
School: HKU
Course: Foundation Of Engineering
Worked Examples July 31, 2012 1 Limits and continuity x2 + 3x 10 by cancelling a common factor . x 1 x2 + 5 x 1. Find the limit lim Ans : x2 + 3x 10 (x 2)(x + 5) (x 2) = lim = lim = 1 2 + 5x x1 x 1 x1 x x(x + 5) x lim x2 + 9 3 2. Find the limit lim by cre
School: HKU
Course: Foundation Of Engineering
MATH1851B: Calculus and Ordinary Differential Equations Tutorials Tue 15: 30 16: 20 17: 30 18: 20 207 103 Fri Week 4 18Sep 21Sep Week 5 25Sep 28Sep Week 6 X 05Oct Week 7 09Oct 12Oct Week 8 16Oct 19Oct Week 9 X Week 10 26Oct Reading Week Week 11
School: HKU
Course: Foundation Of Engineering
13 Bernoulli Experiment and Its Related Distributions An experiment is called a Bernoulli experiment if there are only two possible outcome: success with probability p and failure with probability (1 p) where 0 < p < 1. We say x is a Bernoulli random vari
School: HKU
Course: Foundation Of Engineering
Figure 1: St Augustine and Monica by Ary Scheer (1846). Taken from Wikipedia. 4 Some History of Probability* This section is optional and it aims to introduce some story of probability. With the advent of Christianity, the concept of random events develop
School: HKU
Course: Foundation Of Engineering
3 Complex Variables The imaginary number i is the solution of the equation x2 + 1 = 0. This idea of i was introduced to answer the above question. But then it results in many interesting results, beautiful theory and useful applications. In general a comp
School: HKU
Course: Foundation Of Engineering
MATH 1853 Probability Theory & Statistics (0) A Review on set theory and basic calculus. (1) Elementary Complex Variables: Arithmetic of complex numbers; Modulus argument and conjugate; Basic properties and operations of complex numbers; Argand diagram; A
School: HKU
Course: Foundation Of Engineering
MATH 1853 Homework (Fall 2012) 1. Determine the values of k so that the following system in unknowns x, y , z has: (i) a unique solution, (ii) no solution, (iii) an infinite number of solutions: x 2y =1 x y + kz = 2 ky + 4 z = 6 2. Let W be the solution s
School: HKU
Course: Foundation Of Engineering
Cross Product Recall we have seen the inner product (or called the dot product), with definition v1 v u v = u v = [u1 u2 . un ] 2 = u1v1 + u2 v2 + . + un vn vn T Notice that the input of this operation is two vectors (of same dimension), and the out
School: HKU
Course: Foundation Of Engineering
Inner Product For two vectors u, v in Rn, the inner product is defined as v1 v uT v = [u1 u2 . un ] 2 = u1v1 + u2 v2 + . + un vn vn It is obvious that uTv= vTu From inner product, we can define other attributes of a vector The length (or norm) of
School: HKU
Course: Foundation Of Engineering
Eigenvalues and Eigenvectors For an nn matirx A, if there is a nonzero vector x such that Ax=x for some scalar , then is an eigenvalue of A, and x is the eigenvector corresponding to If we view A as a mapping, Ax=x means that the mapping A acting on x
School: HKU
Course: Foundation Of Engineering
Determinants Recall the 22 matrix inverse equation A 1 = 1 d b ad bc c a Is it possible to extend the result to matrix of dimension nn ? We first look at the concept of determinant a11 a12 For 22 matrix A = , its determinant is det(A)=a11a22a12a21
School: HKU
Course: Foundation Of Engineering
Matrices As seen in the previous chapter, a matrix consists of vectors as columns: A=[a1 a2 an] For an mn matrix, its structure is Two matrices are equal iff they have the same size and their corresponding entries are equal Sum of two matrices is just
School: HKU
Course: FOUNDATION OF COMPUTER SCIENCE
ENGG1007 Foundations of Computer Science Introduction Professor Francis Chin and Dr SM Yiu , 1 Teaching Team ENGG1007 FCS Professor Francis Chin General Office CB301 Phone : 28592178, chin@cs.hku.hk Dr SM Yiu Office: CB402 Phone: 28578242, smyiu@cs.hku.h
School: HKU
Take This Blog and Shove It! When utopian ideals crash into human nature sloth triumphs. In the history of the web, last spring may figure as a tipping point. Thats when Wikipedia, the free encyclopedia that anyone can edita site that grew from 100,000 ar
School: HKU
University Number (please write very clearly) _ Surname_ Answers for ENGG1010 Quiz, Part B Dynamics, 20 October 2009 Answer both questions. Write your answers on this question sheet and submit this sheet only. QB1. Bar AB is hinged to the edge of a circu
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
School: HKU
(IV) FLUIDS IN MOTION Fluid motions manifest themselves in many different ways. Some can be described very easily, while others require a thorough understanding of physical laws. In engineering applications, it is important to describe the fluid motions a
School: HKU
Name: ENGG1010 (20102011) First Semester MidTerm Test Question on fluid mechanics U. No.: Instructions Write your answers in the space provided, and hand in this sheet upon completion of the test. As shown below, an underwater tunnel is closed by a circ
School: HKU
Midterm Quiz for ENGG1010 Foundations of Engineering Mechanics Faculty of Engineering, HKU Class A , 23 October 2008 Instructions: (1) The quiz has four questions. Attempt all questions. (2) Write down your solutions on the space below each question. (3)
School: HKU
ENGG1010 Foundations of Engineering Mechanics Examination: Dec 14, 2010, 9:30am12:30pm, Flora Ho Sports Centre Exam paper is divided into Sections A and B, each containing 3 questions. You are required to attempt 5 out of the 6 questions. Section B two q
School: HKU
School: HKU
School: HKU
School: HKU
University Number (please write very clearly) _ Surname_ Answers for ENGG1010 Quiz, Part B Dynamics, 20 October 2009 Answer both questions. Write your answers on this question sheet and submit this sheet only. QB1. Bar AB is hinged to the edge of a circu
School: HKU
School: HKU
Name: ENGG1010 (20102011) First Semester MidTerm Test Question on fluid mechanics U. No.: Instructions Write your answers in the space provided, and hand in this sheet upon completion of the test. As shown below, an underwater tunnel is closed by a circ
School: HKU
Midterm Quiz for ENGG1010 Foundations of Engineering Mechanics Faculty of Engineering, HKU Class A , 23 October 2008 Instructions: (1) The quiz has four questions. Attempt all questions. (2) Write down your solutions on the space below each question. (3)
School: HKU
Course: Engg1010
Fluid Mechanics Problem 6 The diameter of a pipe bend is 300 mm at inlet and 150 mm at outlet and the flow is turned through 60o in a vertical plane. The axis at inlet is horizontal and the centre of the outlet section is 1.5 m above the centre of the inl
School: HKU
Course: Engg1010
Fluid Mechanics Problems 5 Find the horizontal component of the force exerted by the flowing water on 1 m width of the (shaded) outflow structure, which discharges via a 0.2 m diameter circular duct to the atmosphere. Hint: consider the control volume as
School: HKU
Course: Engg1010
Fluid Mechanics Problems 4 Oil ( = 800 kg/m3) flows through a pipe line (see figure below), which contracts from 450 mm diameter at A to 300 mm diameter at B and then forks, one branch being 150 mm diameter discharging at C and the other branch 225 mm dia
School: HKU
Course: Engg1010
Fluid Mechanics Problems 3 1. For the tank shown below, find the pressure gauge reading if the mercury manometer reading H = 0.2 m. Relative density of mercury = 13.6. Pressure gauge Hg 2. For an Lshaped gate shown below, determine the height H (of the w
School: HKU
Course: Engg1010
Fluid Mechanics Problems 2 Consider a gate CD, which is shown below and can be (a) rectangular, (b) circular, and (c) triangular in shape. The gate is only hinged at the top, and has no support elsewhere. Find the moment to be applied at the hinge in each
School: HKU
Course: Engg1010
Fluid Mechanics Problems 1 1. Some people say that the seawater level will rise as a result of the melting of an iceberg. How true is this statement? 2. As is schematically shown below, a boat carrying a rock floats on water in a lake. What happens to the
School: HKU
Course: Foundation Of Engineering
MATH 1853 Homework (Fall 2012) 1. Determine the values of k so that the following system in unknowns x, y , z has: (i) a unique solution, (ii) no solution, (iii) an infinite number of solutions: x 2y =1 x y + kz = 2 ky + 4 z = 6 2. Let W be the solution s
School: HKU
Fluid Mechanics Problem 6 The diameter of a pipe bend is 300 mm at inlet and 150 mm at outlet and the flow is turned through 60o in a vertical plane. The axis at inlet is horizontal and the centre of the outlet section is 1.5 m above the centre of the inl
School: HKU
Fluid Mechanics Problem 5 Find the horizontal component of the force exerted by the flowing water on 1 m width of the (shaded) outflow structure, which discharges via a 0.2 m diameter circular duct to the atmosphere. Hint: consider the control volume as i
School: HKU
Fluid Mechanics Problem 4 Oil ( = 800 kg/m3) flows through a pipe line (see figure below), which contracts from 450 mm diameter at A to 300 mm diameter at B and then forks, one branch being 150 mm diameter discharging at C and the other branch 225 mm diam
School: HKU
Fluid Mechanics Problems 3 1. For the tank shown below, find the pressure gauge reading if the mercury manometer reading H = 0.2 m. Relative density of mercury = 13.6. Pressure gauge Hg 2. For an Lshaped gate shown below, determine the height H (of the w
School: HKU
Fluid Mechanics Problems 2 Consider a gate CD, which is shown below and can be (a) rectangular, (b) circular, and (c) triangular in shape. The gate is only hinged at the top, and has no support elsewhere. Find the moment to be applied at the hinge in each
School: HKU
Fluid Mechanics Problems 1 1. Some people say that the seawater level will rise as a result of the melting of an iceberg. How true is this statement? 2. As is schematically shown below, a boat carrying a rock floats on water in a lake. What happens to the