CH-205: Fluid Dynamics

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "CH-205: Fluid Dynamics"

Transcription

1 CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) Solutions of Mid Semester Examination Data Given: Density of water, ρ = 1000 kg/m 3, gravitational acceleration, g = 9.81 m/s Question # 1: [4+4+] Consider the flow of water through a clear tube (cross-section area A 1 ). It is sometimes possible to observe cavitation in the throat created by pinching off the tube to a very small diameter of cross-section area A. Let the average velocities and pressures at points 1 (inlet) and (throat) are (V 1, P 1 ) and (V, P ), respectively. Both the maximum velocity and minimum pressure occur at throat. Consider that the throat diameter is 1/0th of the inlet diameter. (a) estimate the minimum average inlet velocity at which cavitation is likely to occur for water entering at pressure of kpa and temperature of (a) 0 o C and (b) 50 o C, respectively. Explain why the required velocity in part (b) is higher or lower than that part (a). Under the incompressible flow with negligible gravitational effects and negligible irreversibilities conditions, both (V A) and [P +ρ(v /)] may be taken as constant along the flow. The saturation pressure of water may be taken as.34 and 1.35 kpa at 0 o C and 50 o C, respectively. Solution # 1 Given that the flow is incompressible with negligible gravitational effects and negligible irreversibilities, V A and [P + ρ(v /)] may be taken as constant along the flow. Therefore, applying the prescribed balances in between inlet (point 1) and throat (point ), we get and V 1 A 1 = V A or V = V 1 (A 1 /A ) or V = V 1 (D 1 /D ) (1) On substituting Eq. (1) in Eq. (), we get (P 1 P ) = ρ V 1 ( D 4 1 D 4 P 1 + ρ V 1 = P + ρ V ) 1 or V 1 = () ( (P 1 P ) D ρ D 4 1) (3) Given P 1 = kpa, D 1 /D = 0, ρ = 1000 kg/m 3 We understand that the the pressure (P ) anywhere in flow should not be allowed to drop below the vapor pressure (P v ) at the given temperature to avoid the cavitation. For a pure substance, the vapor pressure (P v ) equals to the saturation pressure (P sat ). Therefore, the cavitation is likely to occur at the throat when P P sat. (a) At 0 o C, P = P sat =.34 kpa After substituting the numerical values in Eq. (3), we get ( ) 10 3 V 1 = ( ) 1 = m/s (4) (b) At 50 o C, P = P sat = 1.35 kpa After substituting the numerical values in Eq. (3), we get ( ) 10 3 V 1 = ( ) 1 = m/s (5) Instructors: RPB & SC Page 1 of 6 Sept 11, 013 (10 a.m. - 11:30 a.m.)

2 CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) The results show that at 50 o C, cavitation may occur at lower value of the inlet average velocity compared to that at 0 o C. It is simply because for the given decrease in the duct cross-sectional area, the velocity increases and the pressure decreases according to given flow continuity and mechanical energy balances. We also know that the vapor pressure increases with increasing value of temperature. Therefore, the pressure difference (P 1 P ) to be maintained to avoid the cavitation decreases with increasing temperature, In first case, the pressure (P 1 P ) is very large and, thus, the corresponding inflow velocity is larger compared to the second case. Question # : [ ] A metal cylinder (diameter D = 3 m and height L = 3 m) of specific weight of 5886 N/m 3 is required to float in water with its axis vertical. Determine the (i) depth of immersion h of cylinder in water, (ii) distance of center of buoyancy B and center of gravity G from the bottom point of cylinder O (iii) distance of center of gravity G from the center of buoyancy B (iv) volume of the cylinder submerged in the water V sub, (v) metacentric height (GM). Based on the above results, state whether the cylinder is in stable equilibrium. Solution # Given, the length of cylinder, L = 3 m, diameter of the cylinder D = 3m, specific weight of cylinder w c = 5886 N/m 3 Consider that h height of the cylinder is immersed in the liquid. Taking the force balance, Weight of cylinder = weight of water displaced where w is the specific weight of water. (i) the depth of immersion (h) of cylinder in water π 4 D L w c = π 4 D h w h = L w c w = = 1.8 m 9810 (ii) Distance of center of buoyancy (B) from the bottom center (O) OB = h = 0.9 m Distance of center of gravity (G) from the bottom center (O) OG = L = 1.5 m (iii) Distance of center of gravity (G) from center of buoyancy (B) (iv) volume of cylinder submerged in water BG = OG OB = 0.6 m V sub = π 4 D h = π = 1.73 m Instructors: RPB & SC Page of 6 Sept 11, 013 (10 a.m. - 11:30 a.m.)

3 CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) (v) metacentric height, GM is a measure of stability for the floating bodies. It is the distance between the center of gravity (G) and the metacenter (M, the intersection point of the lines of action of buoyancy force through the body before and after rotations). where BM is metacentric radius. GM = BM BG The second moment of inertia (I xx,c ) would be related to the buoyant force for the the small angle of rotations due to displacement as follow w I xx,c = BM F B w I xx,c = BM w V sub BM = I xx,c V sub For the circular cross-sections, the second moment of inertia, I xx,c = πd4 64 BM = I xx,c V sub = D 16h = m GM = = m The negative value of the GM indicates that the metacentre is below the center of gravity. Thus, the cylinder is in unstable equilibrium. Question # 3(A): [3] Consider two identical glasses of water, one stationary and other moving on a horizontal plane with constant acceleration. Assuming no splashing or spilling occurs, give your to-the-point explanation that which glass will have a higher pressure at the (i) front (ii) midpoint and (iii) back of the bottom surface? Solution # 3(A) We know that the pressure in all cases is the hydrostatic pressure, which is directly proportional to the fluid height. The pressure at the bottom surface is constant when the glass is stationary. For a glass moving on a horizontal plane with constant acceleration, water will collect at the back but the water depth will remain constant at the center. Therefore, the pressure at the midpoint will be the same for both glasses. But the bottom pressure will be low at the front relative to the stationary glass, and high at the back (again relative to the stationary glass). Question # 3(B): [3++] A water tank is being towed on an uphill road (inclined by 0 o with the horizontal) with constant acceleration of 5 m/s in the direction of motion. Determine the angle the free surface of water makes with horizontal. What would your answer be if the direction of motion were downward on the same road with the same acceleration? If the water is replaced by oil (specific gravity 0.8) then what would be the new values of angle in both the cases? Solution # 3(B) The effects of splashing, breaking, driving over bumps, and climbing hills are assumed to be negligible and the acceleration remains constant. Instructors: RPB & SC Page 3 of 6 Sept 11, 013 (10 a.m. - 11:30 a.m.)

4 CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) From geometrical considerations, the horizontal and vertical components of acceleration are a x = a cos α a z = a sin α The tangent of the angle the free surface makes with the horizontal is tan θ = a x g + a z = a cos α g + a sin α = 5 cos 0 o = θ =.0o sin 0o When the direction of motion is reversed, both a x and a z are in negative x- and z-directions, respectively, and thus become negative quantities, a x = a cos α a z = a sin α Then the tangent of the angle the free surface makes with the horizontal becomes tan θ = a x g + a z = a cos α g a sin α = 5 cos 0o = θ = 30.1o sin 0o The analysis is valid for any fluid with constant density, not just water. Since we used no information that pertains to water in the solution and so there will not be any change in the results after changing the fluid from water to oil. Question # 4(A): [++] Consider steady, incompressible, two-dimensional flow through a converging duct. A simple approximate velocity and pressure field for this flow are given as V = (u, v) = (U 0 + bx) i by j (6) P = P 0 ρ [ U0 bx + b (x + y ) ] (7) where U 0 and P 0 are the horizontal velocity and pressure, respectively, at x = 0. Note that this equation ignores viscous effects along the walls but is a reasonable approximation throughout the majority of the flow field. Calculate the material acceleration for fluid particles passing through this duct. Further, give your answer in two ways: (i) as acceleration components a x and a y, and (ii) as acceleration vector a. Also, obtain an expression for the rate of change of pressure following a fluid particle. Instructors: RPB & SC Page 4 of 6 Sept 11, 013 (10 a.m. - 11:30 a.m.)

5 CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) Solution # 4(A) The velocity and pressure fields for steady, incompressible, two-dimensional flow through a converging duct is given as, V = (u, v) = (U 0 + bx) i by j P = P 0 ρ [ U0 bx + b (x + y ) ] The acceleration field components are obtained from its definition (the material acceleration) in Cartesian coordinates, The material acceleration components D V Dt = V t + V. V a x = Du Dt = u t + u u x + v u y + w u z = (U 0 + bx)b + ( by)0 = (U 0 + bx)b a y = Dv Dt = v t + u v x + v v y + w v z = (U 0 + bx)0 + ( by)( b) = b y In terms of the vector notations, material acceleration vector a = ax i + ay j = b(u0 + bx) i + b y j By definition, the material derivative, when applied to pressure, produces the rate of change of pressure following a fluid particle. Therefore, the rate of change of pressure following a fluid particle DP Dt = P t +u P P +v x y + w P z = (U 0+bx)( ρu 0 b ρb x)+( by)( ρb y) = ρ [ U0 b U 0 b x + b 3 (y x ) ] Question # 4(B): [+] Consider the following steady, two-dimensional velocity field V = (u, v) = [a (b cx) ] i + (c xy cby) j (8) Is there a stagnation point in this flow field? If so, calculate the location of stagnation point. The given velocity field, Solution # 4(B) V = (u, v) = [a (b cx) ] i + (c xy cby) j At a stagnation point, both u and v must equal zero. At any point (x,y) in the flow field, the velocity components u and v are given as u = a (b cx) v = c xy cby Instructors: RPB & SC Page 5 of 6 Sept 11, 013 (10 a.m. - 11:30 a.m.)

6 CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) Setting these to zero and solving simultaneously for x and y yields the stagnation point u = 0 = a (b cx) x = b a c v = 0 = c xy cby y = 0 So, yes there is a stagnation point; its location is x = (b - a)/c, y = 0. If the flow were three-dimensional, then set w = 0 as well to determine the location of the stagnation point. Some flow fields may have more than one stagnation point. Instructors: RPB & SC Page 6 of 6 Sept 11, 013 (10 a.m. - 11:30 a.m.)

8 Buoyancy and Stability

8 Buoyancy and Stability Jianming Yang Fall 2012 16 8 Buoyancy and Stability 8.1 Archimedes Principle = fluid weight above 2 ABC fluid weight above 1 ADC = weight of fluid equivalent to body volume In general, ( = displaced fluid

More information

Figure 1- Different parts of experimental apparatus.

Figure 1- Different parts of experimental apparatus. Objectives Determination of center of buoyancy Determination of metacentric height Investigation of stability of floating objects Apparatus The unit shown in Fig. 1 consists of a pontoon (1) and a water

More information

When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

More information

F mg (10.1 kg)(9.80 m/s ) m

F mg (10.1 kg)(9.80 m/s ) m Week 9 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

More information

Fluid Mechanics. Fluid Statics [3-1] Dr. Mohammad N. Almasri. [3] Fall 2010 Fluid Mechanics Dr. Mohammad N. Almasri [3-1] Fluid Statics

Fluid Mechanics. Fluid Statics [3-1] Dr. Mohammad N. Almasri. [3] Fall 2010 Fluid Mechanics Dr. Mohammad N. Almasri [3-1] Fluid Statics 1 Fluid Mechanics Fluid Statics [3-1] Dr. Mohammad N. Almasri Fluid Pressure Fluid pressure is the normal force exerted by the fluid per unit area at some location within the fluid Fluid pressure has the

More information

Chapter (1) Fluids and their Properties

Chapter (1) Fluids and their Properties Chapter (1) Fluids and their Properties Fluids (Liquids or gases) which a substance deforms continuously, or flows, when subjected to shearing forces. If a fluid is at rest, there are no shearing forces

More information

EXPERIMENT (2) BUOYANCY & FLOTATION (METACENTRIC HEIGHT)

EXPERIMENT (2) BUOYANCY & FLOTATION (METACENTRIC HEIGHT) EXPERIMENT (2) BUOYANCY & FLOTATION (METACENTRIC HEIGHT) 1 By: Eng. Motasem M. Abushaban. Eng. Fedaa M. Fayyad. ARCHIMEDES PRINCIPLE Archimedes Principle states that the buoyant force has a magnitude equal

More information

Tutorial 4. Buoyancy and floatation

Tutorial 4. Buoyancy and floatation Tutorial 4 uoyancy and floatation 1. A rectangular pontoon has a width of 6m, length of 10m and a draught of 2m in fresh water. Calculate (a) weight of pontoon, (b) its draught in seawater of density 1025

More information

CHAPTER 2.0 ANSWER B.20.2

CHAPTER 2.0 ANSWER B.20.2 CHAPTER 2.0 ANSWER 1. A tank is filled with seawater to a depth of 12 ft. If the specific gravity of seawater is 1.03 and the atmospheric pressure at this location is 14.8 psi, the absolute pressure (psi)

More information

These slides contain some notes, thoughts about what to study, and some practice problems. The answers to the problems are given in the last slide.

These slides contain some notes, thoughts about what to study, and some practice problems. The answers to the problems are given in the last slide. Fluid Mechanics FE Review Carrie (CJ) McClelland, P.E. cmcclell@mines.edu Fluid Mechanics FE Review These slides contain some notes, thoughts about what to study, and some practice problems. The answers

More information

FLUID MECHANICS. Problem 2: Consider a water at 20 0 C flows between two parallel fixed plates.

FLUID MECHANICS. Problem 2: Consider a water at 20 0 C flows between two parallel fixed plates. FLUID MECHANICS Problem 1: Pressures are sometimes determined by measuring the height of a column of liquid in a vertical tube. What diameter of clean glass tubing is required so that the rise of water

More information

Chapter 4. Motion in two & three dimensions

Chapter 4. Motion in two & three dimensions Chapter 4 Motion in two & three dimensions 4.2 Position and Displacement Position The position of a particle can be described by a position vector, with respect to a reference origin. Displacement The

More information

ME 305 Fluid Mechanics I. Part 4 Integral Formulation of Fluid Flow

ME 305 Fluid Mechanics I. Part 4 Integral Formulation of Fluid Flow ME 305 Fluid Mechanics I Part 4 Integral Formulation of Fluid Flow These presentations are prepared by Dr. Cüneyt Sert Mechanical Engineering Department Middle East Technical University Ankara, Turkey

More information

Higher Technological Institute Civil Engineering Department. Lectures of. Fluid Mechanics. Dr. Amir M. Mobasher

Higher Technological Institute Civil Engineering Department. Lectures of. Fluid Mechanics. Dr. Amir M. Mobasher Higher Technological Institute Civil Engineering Department Lectures of Fluid Mechanics Dr. Amir M. Mobasher 1/14/2013 Fluid Mechanics Dr. Amir Mobasher Department of Civil Engineering Faculty of Engineering

More information

Experiment (2): Metacentric height of floating bodies

Experiment (2): Metacentric height of floating bodies Experiment (2): Metacentric height of floating bodies Introduction: The Stability of any vessel which is to float on water, such as a pontoon or ship, is of paramount importance. The theory behind the

More information

1.4 Review. 1.5 Thermodynamic Properties. CEE 3310 Thermodynamic Properties, Aug. 26,

1.4 Review. 1.5 Thermodynamic Properties. CEE 3310 Thermodynamic Properties, Aug. 26, CEE 3310 Thermodynamic Properties, Aug. 26, 2011 11 1.4 Review A fluid is a substance that can not support a shear stress. Liquids differ from gasses in that liquids that do not completely fill a container

More information

Chapter 4: Buoyancy & Stability

Chapter 4: Buoyancy & Stability Chapter 4: Buoyancy & Stability Learning outcomes By the end of this lesson students should be able to: Understand the concept of buoyancy hence determine the buoyant force exerted by a fluid to a body

More information

Applied Fluid Mechanics

Applied Fluid Mechanics Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and

More information

Chapter 14 - Fluids. -Archimedes, On Floating Bodies. David J. Starling Penn State Hazleton PHYS 213. Chapter 14 - Fluids. Objectives (Ch 14)

Chapter 14 - Fluids. -Archimedes, On Floating Bodies. David J. Starling Penn State Hazleton PHYS 213. Chapter 14 - Fluids. Objectives (Ch 14) Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced. -Archimedes, On Floating Bodies David J.

More information

Homework 4. problems: 5.61, 5.67, 6.63, 13.21

Homework 4. problems: 5.61, 5.67, 6.63, 13.21 Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find

More information

AP2 Fluids. Kinetic Energy (A) stays the same stays the same (B) increases increases (C) stays the same increases (D) increases stays the same

AP2 Fluids. Kinetic Energy (A) stays the same stays the same (B) increases increases (C) stays the same increases (D) increases stays the same A cart full of water travels horizontally on a frictionless track with initial velocity v. As shown in the diagram, in the back wall of the cart there is a small opening near the bottom of the wall that

More information

People s Physics book 3e Ch 25-1

People s Physics book 3e Ch 25-1 The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate

More information

RELATIVE MOTION ANALYSIS: VELOCITY

RELATIVE MOTION ANALYSIS: VELOCITY RELATIVE MOTION ANALYSIS: VELOCITY Today s Objectives: Students will be able to: 1. Describe the velocity of a rigid body in terms of translation and rotation components. 2. Perform a relative-motion velocity

More information

CHAPTER 4 Motion in 2D and 3D

CHAPTER 4 Motion in 2D and 3D General Physics 1 (Phys : Mechanics) CHAPTER 4 Motion in 2D and 3D Slide 1 Revision : 2. Displacement vector ( r): 1. Position vector (r): r t = x t i + y t j + z(t)k Particle s motion in 2D Position vector

More information

Chapter 28 Fluid Dynamics

Chapter 28 Fluid Dynamics Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example

More information

Fluids flow conform to shape of container. Mass: mass density, Forces: Pressure Statics: Human body 50-75% water, live in a fluid (air)

Fluids flow conform to shape of container. Mass: mass density, Forces: Pressure Statics: Human body 50-75% water, live in a fluid (air) Chapter 11 - Fluids Fluids flow conform to shape of container liquids OR gas Mass: mass density, Forces: Pressure Statics: pressure, buoyant force Dynamics: motion speed, energy friction: viscosity Human

More information

Problem Set 6: Solutions

Problem Set 6: Solutions UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 15 / LeClair Spring 009 Problem Set 6: Solutions Solutions not yet completed: Halliday, Resnick, & Walker, problems 9.48, 9.53, 9.59, and 9.68

More information

Fluid Mechanics Definitions

Fluid Mechanics Definitions Definitions 9-1a1 Fluids Substances in either the liquid or gas phase Cannot support shear Density Mass per unit volume Specific Volume Specific Weight % " = lim g#m ( ' * = +g #V $0& #V ) Specific Gravity

More information

Chapter 8 Steady Incompressible Flow in Pressure Conduits

Chapter 8 Steady Incompressible Flow in Pressure Conduits Chapter 8 Steady Incompressible Flow in Pressure Conduits Outline 8.1 Laminar Flow and turbulent flow Reynolds Experiment 8.2 Reynolds number 8.3 Hydraulic Radius 8.4 Friction Head Loss in Conduits of

More information

Forces. -using a consistent system of units, such as the metric system, we can define force as:

Forces. -using a consistent system of units, such as the metric system, we can define force as: Forces Force: -physical property which causes masses to accelerate (change of speed or direction) -a push or pull -vector possessing both a magnitude and a direction and adds according to the Parallelogram

More information

Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

More information

Unit 4: Science and Materials in Construction and the Built Environment. Chapter 14. Understand how Forces act on Structures

Unit 4: Science and Materials in Construction and the Built Environment. Chapter 14. Understand how Forces act on Structures Chapter 14 Understand how Forces act on Structures 14.1 Introduction The analysis of structures considered here will be based on a number of fundamental concepts which follow from simple Newtonian mechanics;

More information

Hydrostatic Force on a Curved Surfaces

Hydrostatic Force on a Curved Surfaces Hydrostatic Force on a Curved Surfaces Henryk Kudela 1 Hydrostatic Force on a Curved Surface On a curved surface the forces pδa on individual elements differ in direction, so a simple summation of them

More information

Glossary of Physics Formulas

Glossary of Physics Formulas Glossary of Physics Formulas 1. Kinematic relations in 1-D at constant velocity Mechanics, velocity, position x - x o = v (t -t o ) or x - x o = v t x o is the position at time = t o (this is the beginning

More information

Lecture L22-2D Rigid Body Dynamics: Work and Energy

Lecture L22-2D Rigid Body Dynamics: Work and Energy J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

More information

p atmospheric Statics : Pressure Hydrostatic Pressure: linear change in pressure with depth Measure depth, h, from free surface Pressure Head p gh

p atmospheric Statics : Pressure Hydrostatic Pressure: linear change in pressure with depth Measure depth, h, from free surface Pressure Head p gh IVE1400: n Introduction to Fluid Mechanics Statics : Pressure : Statics r P Sleigh: P..Sleigh@leeds.ac.uk r J Noakes:.J.Noakes@leeds.ac.uk January 008 Module web site: www.efm.leeds.ac.uk/ive/fluidslevel1

More information

Physics 103 CQZ1 Solutions and Explanations. 1. All fluids are: A. gases. B. liquids. C. gases or liquids. D. non-metallic. E.

Physics 103 CQZ1 Solutions and Explanations. 1. All fluids are: A. gases. B. liquids. C. gases or liquids. D. non-metallic. E. Physics 03 CQZ Solutions and Explanations. All fluids are: A. gases B. liquids C. gases or liquids D. non-metallic E. transparent Matter is classified as solid, liquid, gas, and plasma. Gases adjust volume

More information

1.7 Cylindrical and Spherical Coordinates

1.7 Cylindrical and Spherical Coordinates 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

More information

du u U 0 U dy y b 0 b

du u U 0 U dy y b 0 b BASIC CONCEPTS/DEFINITIONS OF FLUID MECHANICS (by Marios M. Fyrillas) 1. Density (πυκνότητα) Symbol: 3 Units of measure: kg / m Equation: m ( m mass, V volume) V. Pressure (πίεση) Alternative definition:

More information

Chapter 3: Pressure and Fluid Statics

Chapter 3: Pressure and Fluid Statics Pressure Pressure is defined as a normal force exerted by a fluid per unit area. Units of pressure are N/m 2, which is called a pascal (Pa). Since the unit Pa is too small for pressures encountered in

More information

Mercury is poured into a U-tube as in Figure (14.18a). The left arm of the tube has crosssectional

Mercury is poured into a U-tube as in Figure (14.18a). The left arm of the tube has crosssectional Chapter 14 Fluid Mechanics. Solutions of Selected Problems 14.1 Problem 14.18 (In the text book) Mercury is poured into a U-tube as in Figure (14.18a). The left arm of the tube has crosssectional area

More information

General Physics (PHY 2130)

General Physics (PHY 2130) General Physics (PHY 30) Lecture 3 Solids and fluids buoyant force Archimedes principle Fluids in motion http://www.physics.wayne.edu/~apetrov/phy30/ Lightning Review Last lecture:. Solids and fluids different

More information

Chapter 9 Rotation of Rigid Bodies

Chapter 9 Rotation of Rigid Bodies Chapter 9 Rotation of Rigid Bodies 1 Angular Velocity and Acceleration θ = s r (angular displacement) The natural units of θ is radians. Angular Velocity 1 rad = 360o 2π = 57.3o Usually we pick the z-axis

More information

physics 111N rotational motion

physics 111N rotational motion physics 111N rotational motion rotations of a rigid body! suppose we have a body which rotates about some axis! we can define its orientation at any moment by an angle, θ (any point P will do) θ P physics

More information

Lecture 4. Vectors. Motion and acceleration in two dimensions. Cutnell+Johnson: chapter ,

Lecture 4. Vectors. Motion and acceleration in two dimensions. Cutnell+Johnson: chapter , Lecture 4 Vectors Motion and acceleration in two dimensions Cutnell+Johnson: chapter 1.5-1.8, 3.1-3.3 We ve done motion in one dimension. Since the world usually has three dimensions, we re going to do

More information

Vectors; 2-D Motion. Part I. Multiple Choice. 1. v

Vectors; 2-D Motion. Part I. Multiple Choice. 1. v This test covers vectors using both polar coordinates and i-j notation, radial and tangential acceleration, and two-dimensional motion including projectiles. Part I. Multiple Choice 1. v h x In a lab experiment,

More information

Projectile Motion. directions simultaneously. deal with is called projectile motion. ! An object may move in both the x and y

Projectile Motion. directions simultaneously. deal with is called projectile motion. ! An object may move in both the x and y Projectile Motion! An object may move in both the x and y directions simultaneously! The form of two-dimensional motion we will deal with is called projectile motion Assumptions of Projectile Motion! The

More information

Practice Problems on Bernoulli s Equation. V car. Answer(s): p p p V. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Sep 15

Practice Problems on Bernoulli s Equation. V car. Answer(s): p p p V. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Sep 15 bernoulli_0 A person holds their hand out of a car window while driving through still air at a speed of V car. What is the maximum pressure on the person s hand? V car 0 max car p p p V C. Wassgren, Purdue

More information

Min-218 Fundamentals of Fluid Flow

Min-218 Fundamentals of Fluid Flow Excerpt from "Chap 3: Principles of Airflow," Practical Mine Ventilation Engineerg to be Pubished by Intertec Micromedia Publishing Company, Chicago, IL in March 1999. 1. Definition of A Fluid A fluid

More information

2 rad c. π rad d. 1 rad e. 2π rad

2 rad c. π rad d. 1 rad e. 2π rad Name: Class: Date: Exam 4--PHYS 101--F14 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel, initially at rest, rotates with a constant acceleration

More information

Chapter 8 Fluid Flow

Chapter 8 Fluid Flow Chapter 8 Fluid Flow GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms, and use it in an operational

More information

CHAPTER 1 PROPERTIES OF FLUID

CHAPTER 1 PROPERTIES OF FLUID FLUID MECHANICS (2141906) CHAPTER 1 PROPERTIES OF FLUID Theory 1. Explain following terms in brief: a) Density or Mass density f) Kinematic viscosity b) Specific weight or Weight density g) Viscosity or

More information

9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration

9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Ch 9 Rotation 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Q: What is angular velocity? Angular speed? What symbols are used to denote each? What units are used? Q: What is linear

More information

Chapter 8- Rotational Motion

Chapter 8- Rotational Motion Chapter 8- Rotational Motion Textbook (Giancoli, 6 th edition): Assignment 9 Due on Thursday, November 26. 1. On page 131 of Giancoli, problem 18. 2. On page 220 of Giancoli, problem 24. 3. On page 221

More information

v v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )

v v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( ) Week 3 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

More information

Fluid Statics. [Ans.(c)] (iv) How is the metacentric height, MG expressed? I I

Fluid Statics. [Ans.(c)] (iv) How is the metacentric height, MG expressed? I I Fluid Statics Q1. Coose te crect answer (i) Te nmal stress is te same in all directions at a point in a fluid (a) only wen te fluid is frictionless (b) only wen te fluid is frictionless and incompressible

More information

ERBIL PLOYTECHNIC UNIVERSITY ERBIL TECHNICAL ENGINEERING COLLEGE. Fluid Mechanics. Lecture 3 - Solved Examples (7 examples) - Home works

ERBIL PLOYTECHNIC UNIVERSITY ERBIL TECHNICAL ENGINEERING COLLEGE. Fluid Mechanics. Lecture 3 - Solved Examples (7 examples) - Home works ERBIL PLOYTECHNIC UNIVERSITY ERBIL TECHNICAL ENGINEERING COLLEGE Fluid Mechanics Lecture 3 - Solved Examples (7 examples) - Home works By Dr. Fahid Abbas Tofiq 1 Example 1: A plate 0.025 mm distant from

More information

A Guide to Calculate Convection Coefficients for Thermal Problems Application Note

A Guide to Calculate Convection Coefficients for Thermal Problems Application Note A Guide to Calculate Convection Coefficients for Thermal Problems Application Note Keywords: Thermal analysis, convection coefficients, computational fluid dynamics, free convection, forced convection.

More information

Solutions 2.4-Page 140

Solutions 2.4-Page 140 Solutions.4-Page 4 Problem 3 A mass of 3 kg is attached to the end of a spring that is stretched cm by a force of 5N. It is set in motion with initial position = and initial velocity v = m/s. Find the

More information

Yimin Math Centre. 2/3 Unit Math Homework for Year Motion Part Simple Harmonic Motion The Differential Equation...

Yimin Math Centre. 2/3 Unit Math Homework for Year Motion Part Simple Harmonic Motion The Differential Equation... 2/3 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 9 Motion Part 3 1 9.1 Simple Harmonic Motion The Differential Equation................... 1 9.2 Projectile Motion

More information

WebAssign Lesson 3-3 Applications (Homework)

WebAssign Lesson 3-3 Applications (Homework) WebAssign Lesson 3-3 Applications (Homework) Current Score : / 27 Due : Tuesday, July 15 2014 11:00 AM MDT Jaimos Skriletz Math 175, section 31, Summer 2 2014 Instructor: Jaimos Skriletz 1. /3 points A

More information

Solution Derivations for Capa #10

Solution Derivations for Capa #10 Solution Derivations for Capa #10 1) The flywheel of a steam engine runs with a constant angular speed of 172 rev/min. When steam is shut off, the friction of the bearings and the air brings the wheel

More information

RELATIVE MOTION ANALYSIS: VELOCITY

RELATIVE MOTION ANALYSIS: VELOCITY RELATIVE MOTION ANALYSIS: VELOCITY Today s Objectives: Students will be able to: 1. Describe the velocity of a rigid body in terms of translation and rotation components. 2. Perform a relative-motion velocity

More information

Physics 271 FINAL EXAM-SOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath

Physics 271 FINAL EXAM-SOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath Physics 271 FINAL EXAM-SOLUTIONS Friday Dec 23, 2005 Prof. Amitabh Lath 1. The exam will last from 8:00 am to 11:00 am. Use a # 2 pencil to make entries on the answer sheet. Enter the following id information

More information

Lesson 5 Rotational and Projectile Motion

Lesson 5 Rotational and Projectile Motion Lesson 5 Rotational and Projectile Motion Introduction: Connecting Your Learning The previous lesson discussed momentum and energy. This lesson explores rotational and circular motion as well as the particular

More information

Fluids in Motion Supplement I

Fluids in Motion Supplement I Fluids in Motion Supplement I Cutnell & Johnson describe a number of different types of flow: Compressible vs incompressible (most liquids are very close to incompressible) Steady vs Unsteady Viscous or

More information

Physics-1 Recitation-7

Physics-1 Recitation-7 Physics-1 Recitation-7 Rotation of a Rigid Object About a Fixed Axis 1. The angular position of a point on a wheel is described by. a) Determine angular position, angular speed, and angular acceleration

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

More information

γ [Increase from (1) to (2)] γ (1ft) [Decrease from (2) to (B)]

γ [Increase from (1) to (2)] γ (1ft) [Decrease from (2) to (B)] 1. The manometer fluid in the manometer of igure has a specific gravity of 3.46. Pipes A and B both contain water. If the pressure in pipe A is decreased by 1.3 psi and the pressure in pipe B increases

More information

25.2. Applications of PDEs. Introduction. Prerequisites. Learning Outcomes

25.2. Applications of PDEs. Introduction. Prerequisites. Learning Outcomes Applications of PDEs 25.2 Introduction In this Section we discuss briefly some of the most important PDEs that arise in various branches of science and engineering. We shall see that some equations can

More information

XI / PHYSICS FLUIDS IN MOTION 11/PA

XI / PHYSICS FLUIDS IN MOTION 11/PA Viscosity It is the property of a liquid due to which it flows in the form of layers and each layer opposes the motion of its adjacent layer. Cause of viscosity Consider two neighboring liquid layers A

More information

Chapter 13 - Solutions

Chapter 13 - Solutions = Chapter 13 - Solutions Description: Find the weight of a cylindrical iron rod given its area and length and the density of iron. Part A On a part-time job you are asked to bring a cylindrical iron rod

More information

www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

More information

Test - A2 Physics. Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements)

Test - A2 Physics. Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements) Test - A2 Physics Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements) Time allocation 40 minutes These questions were ALL taken from the June 2010

More information

Physics 1120: 2D Kinematics Solutions

Physics 1120: 2D Kinematics Solutions Questions: 1 2 3 4 5 6 7 8 9 10 11 Physics 1120: 2D Kinematics Solutions 1. In the diagrams below, a ball is on a flat horizontal surface. The inital velocity and the constant acceleration of the ball

More information

Lecture 15. Torque. Center of Gravity. Rotational Equilibrium. Cutnell+Johnson:

Lecture 15. Torque. Center of Gravity. Rotational Equilibrium. Cutnell+Johnson: Lecture 15 Torque Center of Gravity Rotational Equilibrium Cutnell+Johnson: 9.1-9.3 Last time we saw that describing circular motion and linear motion is very similar. For linear motion, we have position

More information

PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. PHYS 101-4M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in

More information

Activity P13: Buoyant Force (Force Sensor)

Activity P13: Buoyant Force (Force Sensor) Activity P13: Buoyant Force (Force Sensor) Equipment Needed Qty Equipment Needed Qty Economy Force Sensor (CI-6746) 1 Mass and Hanger Set (ME-9348) 1 Base and Support Rod (ME-9355) 1 Ruler, metric 1 Beaker,

More information

UNIVERSITY OF ALABAMA Department of Physics and Astronomy. PH 125 / LeClair Spring Problem Set 2

UNIVERSITY OF ALABAMA Department of Physics and Astronomy. PH 125 / LeClair Spring Problem Set 2 UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 125 / LeClair Spring 2009 Instructions: Problem Set 2 1. Answer all questions below. Follow the problem-solving template provided. 2. Some problems

More information

Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle

More information

Rotational Dynamics. Luis Anchordoqui

Rotational Dynamics. Luis Anchordoqui Rotational Dynamics Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation ( O ). The radius of the circle is r. All points on a straight line

More information

Described by Isaac Newton

Described by Isaac Newton Described by Isaac Newton States observed relationships between motion and forces 3 statements cover aspects of motion for single objects and for objects interacting with another object An object at rest

More information

AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2000

AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2000 M31 AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2000 APPLIED MATHEMATICS - ORDINARY LEVEL FRIDAY, 23 JUNE - AFTERNOON, 2.00 to 4.30 Six questions to be answered. All questions

More information

Ch 6 Forces. Question: 9 Problems: 3, 5, 13, 23, 29, 31, 37, 41, 45, 47, 55, 79

Ch 6 Forces. Question: 9 Problems: 3, 5, 13, 23, 29, 31, 37, 41, 45, 47, 55, 79 Ch 6 Forces Question: 9 Problems: 3, 5, 13, 23, 29, 31, 37, 41, 45, 47, 55, 79 Friction When is friction present in ordinary life? - car brakes - driving around a turn - walking - rubbing your hands together

More information

Chapter 5: Distributed Forces; Centroids and Centers of Gravity

Chapter 5: Distributed Forces; Centroids and Centers of Gravity CE297-FA09-Ch5 Page 1 Wednesday, October 07, 2009 12:39 PM Chapter 5: Distributed Forces; Centroids and Centers of Gravity What are distributed forces? Forces that act on a body per unit length, area or

More information

Ground Rules. PC1221 Fundamentals of Physics I. Force. Zero Net Force. Lectures 9 and 10 The Laws of Motion. Dr Tay Seng Chuan

Ground Rules. PC1221 Fundamentals of Physics I. Force. Zero Net Force. Lectures 9 and 10 The Laws of Motion. Dr Tay Seng Chuan PC1221 Fundamentals of Physics I Lectures 9 and 10 he Laws of Motion Dr ay Seng Chuan 1 Ground Rules Switch off your handphone and pager Switch off your laptop computer and keep it No talking while lecture

More information

Experiment 8 ~ Rotational and Translational Energies

Experiment 8 ~ Rotational and Translational Energies Experiment 8 ~ Rotational and Translational Energies Purpose: The objective of this experiment is to examine the conversion of gravitational potential energy to different types of energy: translational,

More information

Assignment 3. Solutions. Problems. February 22.

Assignment 3. Solutions. Problems. February 22. Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.

More information

Figure 1.1 Vector A and Vector F

Figure 1.1 Vector A and Vector F CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

More information

MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION

MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 ASIC INTEGRATION This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning

More information

First Semester Learning Targets

First Semester Learning Targets First Semester Learning Targets 1.1.Can define major components of the scientific method 1.2.Can accurately carry out conversions using dimensional analysis 1.3.Can utilize and convert metric prefixes

More information

AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2001 APPLIED MATHEMATICS HIGHER LEVEL

AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2001 APPLIED MATHEMATICS HIGHER LEVEL M3 AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 00 APPLIED MATHEMATICS HIGHER LEVEL FRIDAY, JUNE AFTERNOON,.00 to 4.30 Six questions to be answered. All questions carry equal marks.

More information

Cartesian Coordinate System. Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y)

Cartesian Coordinate System. Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y) Physics 1 Vectors Cartesian Coordinate System Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y) Polar Coordinate System Origin and reference line

More information

Lesson 4 Rigid Body Statics. Taking into account finite size of rigid bodies

Lesson 4 Rigid Body Statics. Taking into account finite size of rigid bodies Lesson 4 Rigid Body Statics When performing static equilibrium calculations for objects, we always start by assuming the objects are rigid bodies. This assumption means that the object does not change

More information

Mechanics 1. Revision Notes

Mechanics 1. Revision Notes Mechanics 1 Revision Notes July 2012 MECHANICS 1... 2 1. Mathematical Models in Mechanics... 2 Assumptions and approximations often used to simplify the mathematics involved:... 2 2. Vectors in Mechanics....

More information

Davis Buenger Math Solutions September 8, 2015

Davis Buenger Math Solutions September 8, 2015 Davis Buenger Math 117 6.7 Solutions September 8, 15 1. Find the mass{ of the thin bar with density 1 x function ρ(x) 1 + x < x. Solution: As indicated by the box above, to find the mass of a linear object

More information

Lecture L13 - Conservative Internal Forces and Potential Energy

Lecture L13 - Conservative Internal Forces and Potential Energy S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L13 - Conservative Internal Forces and Potential Energy The forces internal to a system are of two types. Conservative forces, such as gravity;

More information

Sinking Bubble in Vibrating Tanks Christian Gentry, James Greenberg, Xi Ran Wang, Nick Kearns University of Arizona

Sinking Bubble in Vibrating Tanks Christian Gentry, James Greenberg, Xi Ran Wang, Nick Kearns University of Arizona Sinking Bubble in Vibrating Tanks Christian Gentry, James Greenberg, Xi Ran Wang, Nick Kearns University of Arizona It is experimentally observed that bubbles will sometimes sink to the bottom of their

More information

ENSC 283 Introduction and Properties of Fluids

ENSC 283 Introduction and Properties of Fluids ENSC 283 Introduction and Properties of Fluids Spring 2009 Prepared by: M. Bahrami Mechatronics System Engineering, School of Engineering and Sciences, SFU 1 Pressure Pressure is the (compression) force

More information

Physics 2101, First Exam, Fall 2007

Physics 2101, First Exam, Fall 2007 Physics 2101, First Exam, Fall 2007 September 4, 2007 Please turn OFF your cell phone and MP3 player! Write down your name and section number in the scantron form. Make sure to mark your answers in the

More information