# Taylor Columns. Martha W. Buckley c May 24, 2004

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Taylor Columns Martha W. Buckley c May 24, 2004 Abstract. In this experiment we explore the effects of rotation on the dynamics of fluids. Using a rapidly rotating tank we demonstrate that an attempt to introduce gradients in velocity in the direction parallel to the axis of rotation in a geostrophically balanced, constant density fluid leads to the formation of Taylor columns. The columns are visualized by tracking flows by adding dye to the fluid. Additionally, the Rossby number of the flow was calculated to be Ro = 10 3 in order to verify that the fluid is in geostrophic balance. Next, we prove that geostrophic balance is a good approximation for most flows in the earth s atmosphere by demonstating that the Rossby number of typical flows in the atmosphere is very small and that the geostrophi wind closely matches the observed wind. Finally, we prove that Taylor columns are not observed in the atmosphere because the density is not constant. 1. Introduction. Imagine dragging a small object slowly through a stationary fluid. Our intuition tells us that the motion of the object should disturb the flow in all directions, creating a wake. The case of pulling an object through a rotating flow might appear to be equivalent, but as is shown in Figure 1-1, if certain conditions are met, the flow created in a rotating fluid is drastically different. The obstacle at the bottom moves slowly through a rotating fluid. Rather than disturbing the flow in all directions, as one would expect, the object pulls a column of fluid parallel to the axis of rotation with it as it moves. Thus, in a rotating environment, a slowly moving object (nearly) immobilizes an entire column of fluid parallel to the rotation axis, behaving nearly as a solid cylinder extended parallel to the rotation axis. This amazing and unintuitive phenomenon was first hypothesized by G.I. Taylor in 1923, and the columns formed are popularly known as Taylor columns. In this paper we will explore how geostrophic balance in an incompressible fluid leads to the condition that there can be no gradients in a direction parallel to the axis of rotation. We show how an attempt to create gradients in the vertical direction by disturbing the flow leads to the formation of Taylor columns. In our tank experiment, we visualize these Taylor columns by adding dye and tracking its movement. Next, we explore geostrophic balance in the atmosphere. We compare observed winds to the geostrophic wind and conclude that geostrophic balance is a very good approximation for the atmosphere. Finally, we explore the reasons why Taylor columns are not observed in the atmosphere although the atmosphere is in geostrophic balance. 2. The Taylor-Proudman Theorem. Taylor predicted the existence of columns of fluid when rotating fluids are disturbed based on a theorem proved by Joseph Proudman in The theorem, which is now known as the Taylor-Proudman Theorem, states that in a geostrophically balanced flow in an incompressible fluid, there can be no gradients in velocity perpendicular to the rotation axis. We will now prove the Taylor- 1

2 Figure 1-1: The obstacle at the bottom moves slowly through a rotating fluid. Rather than disturbing the flow in all directions, as one would expect, the object pulls a column of fluid parallel to the axis of rotation with it as it moves. Proudman Theorem by considering the dominant forces which must balance in the Navier-Stoke s Equations. The Navier-Stoke s Equations for a fluid of density ρ and kinematic viscosity ν rotating with angular velocity Ω = (0, 0, ω) in an external gravity field g = (0, 0, g) are The Continuity Equation: ρ + (ρu) = 0 (2-1) t The Momentum Equation: u + (u )u } t {{} interia + Ω (Ω r) = }{{} centrifugal 1 ρ P }{{} pressure gradients + ν }{{ 2 u } 2Ω u }{{} viscous coriolis g }{{} gravity (2-2) where u = (u, v, w) is the velocity vector and r is the position vector. The centrifugal and coriolis forces are fictitious forces which arise because we are attempting to apply Newton s Law (F = ma) in a non-inertial (rotating) reference frame. The momentum equation can be simplified by noting that since gravity is a potential field, it can be written as g = φ. The centrifugal force term can also be rewritten as a gradient of a potential via the vector identity Ω (Ω r) = 1/2 (Ω r) 2. Therefore, we can combine gravity and the centrifugal force into a single term Φ (φ + 1/2(Ω r) 2 ), which we consider as a modified gravitational force. The second step in simplifying the equation is to consider the relative importance of the different forces in the types of flows that interest us. We define the characteristic 2

3 velocity of the system U, the characteristic length scale L, and the characteristic timescale τ = L/U. Since the Reynolds number of the flow, which is defined as the ratio of inertial forces to viscous forces, is large for all flows on the length scales that we are considering, viscosity can be neglected. Both inertial terms scale as U 2 L, and the coriolis term scales as ωu. If the flows are much weaker than the rotation of the system, then the Rossby number Ro intertia coriolis = U (2-3) ωl is much less than 1. Therefore, the inertial terms can be neglected, and the equations of motion reduce to 1 P = 2Ω u Φ. (2-4) ρ The vertical component of this equation is termed hydrostatic balance and the vertical components are termed geostrophic balance. Taking the curl of both sides yields 1 P = (2Ω u Φ). ρ If ρ is constant the left-hand side vanishes because the curl of a gradient is always zero. For constant Ω, the right-hand side reduces to 2(Ω u (Ω )u). For an incompressible fluid (ρ = constant) the continuity equation reduces to u = 0, and therefore the first term on the right hand side vanishes. Therefore, Since Ω = ωẑ, Ω u = 0. Ω u = u z = 0, which implies that the fluid velocity is independent of z. Hence, we have proved that in an incompressible fluid in geostrophic balance, the velocity in the direction parallel to the axis of rotation must be constant. 3. Our Experiment. In our experiment we form Taylor columns by creating a flow over a small object at the bottom of a rotating tank. The experimental setup is shown in Figure 3-1. A 10 o arc is drawn on the bottom of the tank in order to aid us in measuring velocities. A small disk is placed at the bottom of a large cylindrical tank approximately 2/3 of the way from the center of the tank, and the tank is then filled with water. The tank is placed on a rotating table which is equipped with a camera that co-rotates with the table. The table is set to rotate at a relatively high rotation rate (10.52 rpm), and the fluid is allowed to spin up until it is undergoing solid body rotation. The mechanism for the spin-up of the fluid is the diffusion of viscosity from the bottom and walls of the tank. In order to ensure that the fluid is undergoing solid body rotation, a small paper dot is placed on the surface and observed in the rotating frame of reference with the camera. If the dot is stationary in the rotating frame, the fluid is undergoing solid body rotation. Next, the rotation rate of the table is slowed down very slightly. Since the water will continue to move at its initialrotation rate for a finite amount of time, there will be a small relative velocity between the disk sitting on the bottom of the tank and the water in the tank. If we move into the frame of reference of the rotating tank (as observed 3

4 Figure 3-1: This figure shows the setup of the experiment. The tank has a radius R=29.6 cm, and the disk was placed 16.9 cm from the center of the tank. The tank is set to rotate about a vertical axis at a speed of rpm. The black lines on the bottom of the tank subtends an angle of 10 o and are used in order to estimate the Rossby number for the flow. 4

5 Figure 3-2: This figure illustrates the velocity of the flow relative to the obstacle after the rotation rate of the tank has been slowed slightly. If the relative flow is sufficiently small, the Taylor-Prundam theorem can be applied and Taylor columns will be observed in the flow, as illustrated. Figure taken from Notes on the Taylor Column Experiment (2004). by the camera) the water is seen to slowly flow past the disk, as is illustrated in Figure 3-2. Before describing the experiment further, I will outline why we expect to observe Taylor columns in our experiment. Since the water is incompressible and the velocity of the relative flow is much smaller than the rotation rate, we can apply the Taylor- Proudman theorem. Therefore, there can be no gradients in velocity in the vertical direction. At the bottom of the tank the flow must go around the obstacle. However, since there can be no velocity gradients in the vertical direction, the flow must also go around the obstacle even above the top of the obstacle, as if the obstacle is extended to the top of the tank. This column of stationary fluid above the obstacle is the Taylor column which we hope to observe. In order to observe the Taylor columns we inject dye into the fluid at the leading edge of the obstacle, see Figure 3-3. If no Taylor columns are present, we expect the dyed fluid above the top of the obstacle to flow over the obstacle. However, as is observed in Figure 3-3, the dye (blue) goes around the obstacle, as if the obstacle was extended to the top of the tank. We also have placed red dye above the obstacle, and as predicted by the Taylor-Proudman theorem, the dye remains stationary over the top of the obstacle. 4. Measurement of Rossby Number. In order to obtain some quantitative information about our flow, we attempt to measure the Rossby number of the flow. We determine the velocity of the fluid relative to the obstacle by measuring how long it takes for a paper dot on the surface to traverse the arc marked on the bottom of the tank. The angle of the arc is θ = π/18, and it takes 3/4 minutes for the dot to travel this angular distance. Therefore, the relative angular velocity ω rel = 2π/27. Since the puck was located at a distance R = 16.5 cm from the center of the tank, the relative velocity of the fluid past the puck U is U = Rω rel = 1.2π. The rotation rate of the tank is Ω = 10.52rpm = 21.04π radians/min. Taking the characteristic length scale L to be the radius of the tank, which is 29.6 cm, the Rossby number is Ro = U 2ΩL =

6 Figure 3-3: This figure shows the Taylor columns created in our flow. Blue dye was placed on the leading edge of the obstacle ino order to show that the flow goes around the obstacle, as if the obstacle was extended to the top of the tank. Red dye was placed on the top of the obstacle, and as predicted by the Taylor-Proudman theorem, the dye remains stationary over the top of the obstacle. The red dye at the far side of the tank is leftover from a previous attempt to use dye to visualize the Taylor columns. 6

7 Since Taylor columns are only expected for Rossby numbers much less than 1, our Rossby number is consistent with the fact that we observed Taylor columns in our flow. 5. The Real World: Atmospheric Data. Just as the water in our tank, the atmosphere is in geostrophic balance. This can be seen by proving that the typical Rossby number for flows in the atmosphere is small. Since the earth rotates once in 24 hours, the rotation rate is Ω = 2π/( ) = sec 1. If we take the typical length-scale of the flow L to be on the order of the radius of the earth L = m/s. Typical wind speeds U are on the order of tens of meters per second. Therefore, the Rossby number Ro = U ΩL 0.1. Since the Rossby number is significantly less than 1, inertial terms can be ignored and geostrophic balance is a good approximation for the flow. In this section we will calculate the geostrphic wind in the atmosphere at the 500 mbar pressure surface, and compare them to the observed winds to demostrate that the atmosphere is in geostrophic balance. Then, we will explain why Taylor columns are not observed in the atmosphere although it is in geostrophic balance. 6. The Geostrophic Wind. As discussed in Section 2, the equation for geostrophic motion is fẑ u + 1 P. (6-1) ρ where f = 2 Ω sin φ is the Corliolis parameter, which quantifies the vertical component of the earth s rotation rate at latitude φ, u is the velocity, P is the pressure, and ẑ is the unit vector in the vertical direction. The velocity defined by the geostrophic flow equation is u g = 1 P (6-2) fρẑ In component form the equation is u g = (u g, v g, w g ) = 1 ( P fρ y, P ) x, 0. (6-3) The geostrophic flow is perpendicular to the pressure gradient, and therefore it is along the isobars of the flow. The speed of the geostrophic flow is proportional to the pressure gradient. However, it is convenient to express the geostrphic wind in terms of pressure coordinates because atmospheric data is generally measured on a constant pressure surface rather than on a constant height surface. The geostrophic wind equation expressed in pressure coordinates is (u g, v g ) = ( g z f y, g f z ) x (6-4) Figure 5 is a plot of the observed winds at the 500 mbar pressure surface over Japan. The contours show the height of the 500 mbar pressure surface in meters and the vectors show the wind velocity. Each quiver represents a wind speed of 10 m/s and the direction of the wind blows from the side of the vector with the quiver. Figure 6 is a plot of the same area, but now the geostrophic wind (as calulated by Equation (6-4) rather than the observed wind is plotted. As can be seen by comparing Figures 5 and 6, the geostrophic wind is very close to the observed wind, which indicates that geostrophic balance is a good approximation for the atmosphere. The similarity of the geostrophic wind to the 7

8 observed wind can be made even more explicit by calculating the ageostrophic wind, which is the difference of the observed wind and the geostrophic wind, as is done in Figure 7. The deviations between the observed wind and the geostrophic wind are less than 5 m/s. Therefore, geostrophic balance is a good approximation for the earth s atmosphere. 7. The Atmosphere: A Compressible Fluid. Since the atmosphere is in geostrophic balance, it begs the question of whether Taylor columns are seen in the atmosphere. It might be quite wild to imagine that a light wind blowing over an obstacle might cause a stationary column of air above the obstacle. However, Taylor columns are not seen in the atmosphere. This is due to the fact that the density is not constant in the earth s atmosphere. In the atmosphere density is a function of pressure and temperature, ρ = ρ(p, T ). Due to the density variations with temperature, an expression comparable to the Taylor- Proudman Theorem is untidy when height is used as the vertical coordinate. The equations are much simpler if we use pressure as our vertical coordinate. Pressure is also a convenient coordinate for atmospheric observations because atmospheric data is generally measured on a constant pressure surface rather than on a constant height surface. We saw that geostrophic balance holds in the atmosphere both by calculating the Rossby number of typical flows in the atmosphere We saw that geostrophic balance holds in the atmosphere both by calculating the Rossby number of typical flows in the atmosphere In pressure coordinates, the hydrostatic relation can be expressed as z P = 1 gρ (7-1) As seen in section (6), the geostrophic wind can also be expressed in pressure coordinates. Taking the P-derivative of the x-component of Equation (6-4) and using the hydrostatic relation, yields u P = 1 f ( 1 ) y ρ P Approximating the atmosphere as an ideal gas, we use the equation of state 1/ρ = RT/P where T is the temperature and R = 287Jkg 1 K 1 is the is the gas constant for dry air. Substituting the equation of state into the above equation yields u P = R ( T ) fp y P (7-2) Similarly, for v we find v P = R ( T ) fp x P (7-3) Equations (6-3) and (6-4) are the thermal wind equations in pressure coordinates. They the analogue of the Taylor-Proudman Theorefor a fluid where density is not constant. We can see that horizontal variations in temperature cause vertical gradients in velocity. 8

9 8. Conclusions. We proved the Taylor-Proundman Theorem: for a geostrophically balanced flow in a rotating fluid with a constant density, there can be no gradients in the direction parallel to the axis of rotation. We demonstrated in the laboratory that an attempt to introduce gradients in velocity in the vertical direction leads to the formation of Taylor columns. Finally, we proved that geostrophic balance is a good approximation for most flows in the earth s atmosphere. However, Taylor columns are not observed in the atmosphere because the density is not constant. 9. Further Work. One aspect of the experiment that could be improved significantly is the method of visualization of the Taylor columns. Using a high molecular weight dye, such as phenol blue, would significantly reduce diffusive flows and make visualization easier. An interesting extension to the experiment might be to try the experiment with a fluid in which the density is not constant, such as in a fluid with a gradient in salt concentration. References Peacock, Thomas Notes of Rotating Flows. (2003) Notes on the Taylor Column Experiment (2004) Project 1 Notes (2004) 9

### ATM 316: Dynamic Meteorology I Final Review, December 2014

ATM 316: Dynamic Meteorology I Final Review, December 2014 Scalars and Vectors Scalar: magnitude, without reference to coordinate system Vector: magnitude + direction, with reference to coordinate system

### Chapter 4 Atmospheric Pressure and Wind

Chapter 4 Atmospheric Pressure and Wind Understanding Weather and Climate Aguado and Burt Pressure Pressure amount of force exerted per unit of surface area. Pressure always decreases vertically with height

### 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

### Physics of the Atmosphere I

Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de heidelberg.de Last week The conservation of mass implies the continuity equation:

### Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

### Chapter 24 Physical Pendulum

Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

### 5.2 Rotational Kinematics, Moment of Inertia

5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position

### CH-205: Fluid Dynamics

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

### 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

### Dynamics IV: Geostrophy SIO 210 Fall, 2014

Dynamics IV: Geostrophy SIO 210 Fall, 2014 Geostrophic balance Thermal wind Dynamic height READING: DPO: Chapter (S)7.6.1 to (S)7.6.3 Stewart chapter 10.3, 10.5, 10.6 (other sections are useful for those

### 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.

### Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture No. # 36 Pipe Flow Systems

Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture No. # 36 Pipe Flow Systems Welcome back to the video course on Fluid Mechanics. In today

### Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

### Let s first see how precession works in quantitative detail. The system is illustrated below: ...

lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

Week 8 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

### Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular

### NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES

Vol. XX 2012 No. 4 28 34 J. ŠIMIČEK O. HUBOVÁ NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES Jozef ŠIMIČEK email: jozef.simicek@stuba.sk Research field: Statics and Dynamics Fluids mechanics

### Swissmetro travels at high speeds through a tunnel at low pressure. It will therefore undergo friction that can be due to:

I. OBJECTIVE OF THE EXPERIMENT. Swissmetro travels at high speeds through a tunnel at low pressure. It will therefore undergo friction that can be due to: 1) Viscosity of gas (cf. "Viscosity of gas" experiment)

### 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

### Lecture 24 - Surface tension, viscous flow, thermodynamics

Lecture 24 - Surface tension, viscous flow, thermodynamics Surface tension, surface energy The atoms at the surface of a solid or liquid are not happy. Their bonding is less ideal than the bonding of atoms

### Torque Analyses of a Sliding Ladder

Torque Analyses of a Sliding Ladder 1 Problem Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 6, 2007) The problem of a ladder that slides without friction while

### Practice Problems on Boundary Layers. Answer(s): D = 107 N D = 152 N. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Nov 22

BL_01 A thin flat plate 55 by 110 cm is immersed in a 6 m/s stream of SAE 10 oil at 20 C. Compute the total skin friction drag if the stream is parallel to (a) the long side and (b) the short side. D =

### NONINERTIAL FRAMES. path of the pendulum across the. bearing. table

LAB 9b NONINERTIAL FRAMES EXPERIMENTAL QUESTION In this lab, you will explore the distinction between inertial and non-inertial reference frames. According to chapter N9, Newton s second law fails to hold

### State Newton's second law of motion for a particle, defining carefully each term used.

5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

### 15. The Kinetic Theory of Gases

5. The Kinetic Theory of Gases Introduction and Summary Previously the ideal gas law was discussed from an experimental point of view. The relationship between pressure, density, and temperature was later

### 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

### Contents. Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 1

Contents 1. Introduction 2. Fluids 3. Physics of Microfluidic Systems 4. Microfabrication Technologies 5. Flow Control 6. Micropumps 7. Sensors 8. Ink-Jet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors

### Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is

Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of

### CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies

CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at

### Basic Fluid Dynamics

Basic Fluid Dynamics Yue-Kin Tsang February 9, 2011 1 ontinuum hypothesis In the continuum model of fluids, physical quantities are considered to be varying continuously in space, for example, we may speak

### CHARGE TO MASS RATIO OF THE ELECTRON

CHARGE TO MASS RATIO OF THE ELECTRON In solving many physics problems, it is necessary to use the value of one or more physical constants. Examples are the velocity of light, c, and mass of the electron,

### 11 Navier-Stokes equations and turbulence

11 Navier-Stokes equations and turbulence So far, we have considered ideal gas dynamics governed by the Euler equations, where internal friction in the gas is assumed to be absent. Real fluids have internal

### Chapter 4 Dynamics: Newton s Laws of Motion. Copyright 2009 Pearson Education, Inc.

Chapter 4 Dynamics: Newton s Laws of Motion Force Units of Chapter 4 Newton s First Law of Motion Mass Newton s Second Law of Motion Newton s Third Law of Motion Weight the Force of Gravity; and the Normal

### Thought Questions on the Geostrophic Wind and Real Winds Aloft at Midlatitudes

Thought Questions on the Geostrophic Wind and Real Winds Aloft at Midlatitudes (1) The geostrophic wind is an idealized, imaginary wind that we define at each point in the atmosphere as the wind that blows

### Lecture 11: Introduction to Constitutive Equations and Elasticity

Lecture : Introduction to Constitutive quations and lasticity Previous lectures developed the concepts and mathematics of stress and strain. The equations developed in these lectures such as Cauchy s formula

### Physics 200A FINALS Shankar 180mins December 13, 2005 Formulas and figures at the end. Do problems in 4 books as indicated

1 Physics 200A FINALS Shankar 180mins December 13, 2005 Formulas and figures at the end. Do problems in 4 books as indicated I. Book I A camper is trying to boil water. The 55 g aluminum pan has specific

### Mechanics 1: Conservation of Energy and Momentum

Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

### Bead moving along a thin, rigid, wire.

Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing

### Introduction to basic principles of fluid mechanics

2.016 Hydrodynamics Prof. A.H. Techet Introduction to basic principles of fluid mechanics I. Flow Descriptions 1. Lagrangian (following the particle): In rigid body mechanics the motion of a body is described

### EXAMPLE: Water Flow in a Pipe

EXAMPLE: Water Flow in a Pipe P 1 > P 2 Velocity profile is parabolic (we will learn why it is parabolic later, but since friction comes from walls the shape is intuitive) The pressure drops linearly along

### Determination of Acceleration due to Gravity

Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two

### Chapter 13, example problems: x (cm) 10.0

Chapter 13, example problems: (13.04) Reading Fig. 13-30 (reproduced on the right): (a) Frequency f = 1/ T = 1/ (16s) = 0.0625 Hz. (since the figure shows that T/2 is 8 s.) (b) The amplitude is 10 cm.

### Chapter 13 Newton s Theory of Gravity

Chapter 13 Newton s Theory of Gravity The textbook gives a good brief account of the period leading up to Newton s Theory of Gravity. I am not going to spend much time reviewing the history but will show

### Acceleration due to Gravity

Acceleration due to Gravity 1 Object To determine the acceleration due to gravity by different methods. 2 Apparatus Balance, ball bearing, clamps, electric timers, meter stick, paper strips, precision

### Chapter 3.8 & 6 Solutions

Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled

### 3 Vorticity, Circulation and Potential Vorticity.

3 Vorticity, Circulation and Potential Vorticity. 3.1 Definitions Vorticity is a measure of the local spin of a fluid element given by ω = v (1) So, if the flow is two dimensional the vorticity will be

### Lecture 4: Pressure and Wind

Lecture 4: Pressure and Wind Pressure, Measurement, Distribution Forces Affect Wind Geostrophic Balance Winds in Upper Atmosphere Near-Surface Winds Hydrostatic Balance (why the sky isn t falling!) Thermal

### The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.

260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this

### Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.

Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.

### How do we predict Weather and Climate?

Explanation and follow-on work Introduction and aim of activity In this activity, students were asked to imagine themselves as human computers in a vast forecast factory. This idea was originally conceived

### Measuring the Earth s Diameter from a Sunset Photo

Measuring the Earth s Diameter from a Sunset Photo Robert J. Vanderbei Operations Research and Financial Engineering, Princeton University rvdb@princeton.edu ABSTRACT The Earth is not flat. We all know

### Civil Engineering Hydraulics Mechanics of Fluids. Flow in Pipes

Civil Engineering Hydraulics Mechanics of Fluids Flow in Pipes 2 Now we will move from the purely theoretical discussion of nondimensional parameters to a topic with a bit more that you can see and feel

### Angular acceleration α

Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-0 Linear and Circular Motion Compared Slide 7- Linear and Circular Kinematics Compared Slide 7-

### Orbital Mechanics. Angular Momentum

Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely

### Rotational inertia (moment of inertia)

Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment

### Physics 1A Lecture 10C

Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

### u = φ where 2 φ = 0. (1.1) (y εh) = Dt y ε h t ε φ h h x and

Kelvin-Helmholtz and Rayleigh-Taylor Instabilities Consider two fluid regions, y > 0 and y < 0. In y > 0 let the fluid have uniform velocity u = (U, 0, 0 and constant density ρ =, whereas in y < 0 let

### * Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying

### State Newton's second law of motion for a particle, defining carefully each term used.

5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

### No Brain Too Small PHYSICS. 2 kg

MECHANICS: ANGULAR MECHANICS QUESTIONS ROTATIONAL MOTION (2014;1) Universal gravitational constant = 6.67 10 11 N m 2 kg 2 (a) The radius of the Sun is 6.96 10 8 m. The equator of the Sun rotates at a

### Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows

Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical

### Response to Harmonic Excitation Part 2: Damped Systems

Response to Harmonic Excitation Part 2: Damped Systems Part 1 covered the response of a single degree of freedom system to harmonic excitation without considering the effects of damping. However, almost

### Newton s Second Law. ΣF = m a. (1) In this equation, ΣF is the sum of the forces acting on an object, m is the mass of

Newton s Second Law Objective The Newton s Second Law experiment provides the student a hands on demonstration of forces in motion. A formulated analysis of forces acting on a dynamics cart will be developed

### 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

### Chapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations

Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.

### Uniformly Accelerated Motion

Uniformly Accelerated Motion Under special circumstances, we can use a series of three equations to describe or predict movement V f = V i + at d = V i t + 1/2at 2 V f2 = V i2 + 2ad Most often, these equations

### 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:

### Chapter 1. Governing Equations of Fluid Flow and Heat Transfer

Chapter 1 Governing Equations of Fluid Flow and Heat Transfer Following fundamental laws can be used to derive governing differential equations that are solved in a Computational Fluid Dynamics (CFD) study

### Experiment 7: Forces and Torques on Magnetic Dipoles

MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics 8. Spring 5 OBJECTIVES Experiment 7: Forces and Torques on Magnetic Dipoles 1. To measure the magnetic fields due to a pair of current-carrying

### Physics in the Laundromat

Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized

### Lab 7: Rotational Motion

Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### skiing and external forces 3.1 force internal forces external forces 3.1.1 managing the forces

skiing and external forces 3 Ski instructors should not underestimate the importance of understanding the principles that make a ski speed up, slow down, or change direction. This chapter defines some

### Typhoon Haiyan 1. Force of wind blowing against vertical structure. 2. Destructive Pressure exerted on Buildings

Typhoon Haiyan 1. Force of wind blowing against vertical structure 2. Destructive Pressure exerted on Buildings 3. Atmospheric Pressure variation driving the Typhoon s winds 4. Energy of Typhoon 5. Height

### Exemplar Problems Physics

Chapter Eight GRAVITATION MCQ I 8.1 The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration

### 7. Kinetic Energy and Work

Kinetic Energy: 7. Kinetic Energy and Work The kinetic energy of a moving object: k = 1 2 mv 2 Kinetic energy is proportional to the square of the velocity. If the velocity of an object doubles, the kinetic

### HEAVY OIL FLOW MEASUREMENT CHALLENGES

HEAVY OIL FLOW MEASUREMENT CHALLENGES 1 INTRODUCTION The vast majority of the world s remaining oil reserves are categorised as heavy / unconventional oils (high viscosity). Due to diminishing conventional

### 1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids

1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases.

### Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur

Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 20 Conservation Equations in Fluid Flow Part VIII Good morning. I welcome you all

### Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

### 12.307. 1 Convection in water (an almost-incompressible fluid)

12.307 Convection in water (an almost-incompressible fluid) John Marshall, Lodovica Illari and Alan Plumb March, 2004 1 Convection in water (an almost-incompressible fluid) 1.1 Buoyancy Objects that are

### FREE FALL. Introduction. Reference Young and Freedman, University Physics, 12 th Edition: Chapter 2, section 2.5

Physics 161 FREE FALL Introduction This experiment is designed to study the motion of an object that is accelerated by the force of gravity. It also serves as an introduction to the data analysis capabilities

### G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M

G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD

### Fluids and Solids: Fundamentals

Fluids and Solids: Fundamentals We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation.

### FILTRATION. Introduction. Batch Filtration (Small Scale) Controlling parameters: Pressure drop, time for filtration, concentration of slurry

FILTRATION Introduction Separation of solids from liquids is a common operation that requires empirical data to make predictions of performance. These data are usually obtained from experiments performed

### Rotational Motion: Moment of Inertia

Experiment 8 Rotational Motion: Moment of Inertia 8.1 Objectives Familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body

### PSS 17.1: The Bermuda Triangle

Assignment 6 Consider 6.0 g of helium at 40_C in the form of a cube 40 cm. on each side. Suppose 2000 J of energy are transferred to this gas. (i) Determine the final pressure if the process is at constant

### PEDAGOGY: THE BUBBLE ANALOGY AND THE DIFFERENCE BETWEEN GRAVITATIONAL FORCES AND ROCKET THRUST IN SPATIAL FLOW THEORIES OF GRAVITY *

PEDAGOGY: THE BUBBLE ANALOGY AND THE DIFFERENCE BETWEEN GRAVITATIONAL FORCES AND ROCKET THRUST IN SPATIAL FLOW THEORIES OF GRAVITY * Tom Martin Gravity Research Institute Boulder, Colorado 80306-1258 martin@gravityresearch.org

### Viscous flow in pipe

Viscous flow in pipe Henryk Kudela Contents 1 Laminar or turbulent flow 1 2 Balance of Momentum - Navier-Stokes Equation 2 3 Laminar flow in pipe 2 3.1 Friction factor for laminar flow...........................

### Dimensional analysis is a method for reducing the number and complexity of experimental variables that affect a given physical phenomena.

Dimensional Analysis and Similarity Dimensional analysis is very useful for planning, presentation, and interpretation of experimental data. As discussed previously, most practical fluid mechanics problems

### Introduction to COMSOL. The Navier-Stokes Equations

Flow Between Parallel Plates Modified from the COMSOL ChE Library module rev 10/13/08 Modified by Robert P. Hesketh, Chemical Engineering, Rowan University Fall 2008 Introduction to COMSOL The following

### FILTRATION. Introduction

FILTRATION Introduction Separation of solids from liquids is a common operation that requires empirical data to make predictions of performance. These data are usually obtained from experiments performed

### Physics 113 Exam #4 Angular momentum, static equilibrium, universal gravitation, fluid mechanics, oscillatory motion (first part)

Physics 113 Exam #4 Angular momentum, static equilibrium, universal gravitation, fluid mechanics, oscillatory motion (first part) Answer all questions on this examination. You must show all equations,

### Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion. Physics is about forces and how the world around us reacts to these forces.

Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion Physics is about forces and how the world around us reacts to these forces. Whats a force? Contact and non-contact forces. Whats a

### AP1 Gravity. at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit?

1. A satellite of mass m S orbits a planet of mass m P at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit? (A) v = Gm P R (C)

### Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

### AS CHALLENGE PAPER 2014

AS CHALLENGE PAPER 2014 Name School Total Mark/50 Friday 14 th March Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Marks allocated for each question

### Simple Harmonic Motion

Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights