DOCSLIB.ORG

Lesson 4 Equations for a Nonrotating Fluid Fundamental Forces Law of Motion (N2L)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lesson 4 Equations for a Nonrotating Fluid Fundamental Forces Law of Motion (N2L) Lesson 4 Equations for a Nonrotating Fluid Objectives The state of the atmosphere or ocean can be defined by 5 variables – Fundamental Forces – u, v, w, P and T • Inertial Reference Frame • Recall, using the EOS ρ can be found from P and T – Gravitational Force – Pressure Gradient Force (PGF) We therefore need 5 independent equations – Friction (Viscosity) Force (with boundary conditions) to determine the • Non-inertial Reference Frame evolution of a fluid – Centrifugal Force – The Laws of motion (3 directions) – Coriolis Force – Conservation of Mass – Conservation of Mass – The 1st Law of Thermodynamics • Continuity Equation Fundamental Forces Law of Motion (N2L) In lesson 1 we made an assumption - that the Inertial frame of reference atmosphere and ocean were continuous in – A non-accelerating coordinate system structure and properties at our scale of – Follow the same parcel around - therefore interest the time derivative is a total derivative – This means they are Newtonian fluids and therefore = can use N2L, F m a , to describe acceleration N2L becomes: • The approach is slightly different than we saw in Physics DU – In Physics we dealt with discrete objects of constant mass F = ρδxδyδz – In Fluids we deal with a continuous field of matter with varying Dt density – where: DU ∂U = + (U •∇)U Dt ∂t Gravitational Force (g*) Pressure Gradient Force (PGF) Newton’s Law of Universal Gravitation P = F/A Every particle in the universe attracts every other particle wit h a force that is proportional to the product of their masses and inversely propor tional to the Force due to variation of pressure within square distance between them. The force acts along the line joi ning the two particles. a fluid P (x ,y ,z ) Mm 0 0 0 0 F ≡ g* = −G r g r 2 P P+ δP z δ z δy y m F M δ r g ≡ = − x g* G 2 r m a x F 1 ∂P ∂P ∂P 1 P = − ( iˆ + ˆj + kˆ ) = − ∇P ρ ∂ ∂ ∂ ρ M m x y z Pressure Gradient Force (PGF) Frictional (Viscous) Force Molecular friction Example – Accelerates (±) a body that is in contact with another – Consider a SLP of 1000.3 hPa from a body moving at a different speed station near the center of a low, with a – Molecules collide with those of another • If one object moves (has momentum: p = mv), the other temperature of 9 C. 500 km away a object will receive some of it and move second station reports a SLP of 1000.9 Eddies can exchange momentum and therefore hPa. What is the pressure gradient force? cause frictional forces • Hint: IGL is needed – Wind experiences a retarding force: F fr (ground) – The ground experiences an accelerating force • Free moving surface - waves, currents… Viscous Force Viscous Force Force on upper plate µAu ∂u A fluid is a state that continuously yields to F = 0 =µA any shear stress d ∂z µ Here a solid exerts a shear stress on a (“mu”): dynamic (or shear) viscosity fluid coefficient Property of the fluid This force must just equal the force z = d u0 exerted by the upper plate on the fluid u(l) = u 0 immediately below it (N3L) u(z) State of uniform motion u(0) = 0 Every horizontal layer of fluid must exert the z = 0 same force on the fluid below it Viscous Force Shear Stress The shear stress - τ (“tau”) - exerted on the Defined steady state flow - no fluid by the upper plate is proportional to the acceleration and therefore no net velocity of the plate and inversely proportional viscous force to the distance between the plates (d) – The shear stress (internal friction) does not The x-component of the shear stress can be cause a net force written: ∂u τ iˆ =µ ∂z So, there must be a gradient in the – It is a flux of momentum shear stress ∂τ 2 • Momentum per unit time per unit area 1 µ ∂ ∂µ ∂ u = ( ) = υ υ = µ / ρ ρ ∂z ρ ∂z ∂z ∂z2 Shear Stress Apparent Forces υ (“nu”) is the kinematic viscosity coefficient – Atmosphere: 1.46 x 10 -5 m2/s Non-inertial reference frame – Ocean: 1.0 x 10 -6 m2/s – An accelerating reference frame Shear stress is a vector – Uniform circular motion – Shear stress has nine components (3D) • Centrifugal force • Coriolis force ∂ 2 ∂ 2 ∂ 2 = υ∇ 2 ˆ = υ u + u + u F u Fshear i ( ) x ∂x 2 ∂y 2 ∂z2 = υ∇2 Fy v = υ∇ 2 Fz w Centrifugal Acceleration Effective Gravity Appears to deflect the moving body outward from center of curvature Resultant of gravitational and centrifugal – Equal and opposite to centripetal acceleration forces – Always perpendicular to Earth’s axis; directed Directed perpendicular to local tangent outward of Earth, not toward the center F cf = Ω2R m Coriolis Force Coriolis Acceleration The force that appears to act on a body moving relative to a rotating system, when The coriolis acceleration describes the viewed from that system tendency for fluid parcels to turn In the absence of other forces, a parcel Merry-Go-Round Example would accelerate as: DU = − Ω× 2 2 U 1 Ω Dt Coriolis Force By definition: CF does no work, it only 3 acts to change the direction of flow Coriolis Acceleration: Derivation Coriolis Acceleration: Derivation Recall the coriolis parameter: f = 2Ωsin φ dw = Ω φ 2 ucos φ dt du = Ω φ d = dv Horizontal 2 sin (a ) fv = −2Ωusin φ = − fu dt dt dt Vertical du dz = −2Ωcos φ( ) = −2Ωcos φw dt dt Coriolis Acceleration: Full 3-D Coriolis Acceleration du = fv − 2Ωcos φw Example dt – The Coriolis acceleration can be calculated using observations. dv = − fu – City X, has a latitude of 40N and reports a dt wind of 5 kts from the North. – Calculate the Coriolis acceleration dw = 2Ωu cos φ dt Basic Laws of Physics Conservation of Mass Conservation of: Cannot create or destroy mass - 1 kg of a – Mass fluid will always be 1 kg of that fluid • Leads to Continuity Equation – If fluid is compressible then the shape will • SO335 and SO414 change – Momentum • Leads to EOM Continuity Equation • SO414 – Provides relationship between horizontal and – Energy vertical motions • Leads to 1st Law of Thermodynamics • SO345 Parcel A Parcel B Continuity Equation Continuity Equation Consider: ρ – A cube fixed in space thru which a fluid flows Mass Flux at the center of the cube is u – Mass is continually advected through the Using Taylor's expansion about the sides by the fluid motion center, we get in/out flow per unit area • Net inflow of mass thru sides = Rate of through sides A and B accumulation inside the volume ∂ ∂ ∂x A ρ B ∂ ∂x – Mass flux = M ρu− (ρu) u ρu+ (ρu) ∂x 2 δz ∂x 2 ∂t z δy y δx x Continuity Equation Continuity Equation Can get the net rate of flow into the volume due to the x component of velocity Mass Divergence form of mass continuity Then, the net rate of mass inflow per unit ∂ρ ∂ ∂ ∂ volume for all directions is: = −[ (ρu)+ (ρv)+ (ρw)] = −∇⋅(ρv ) ∂ t ∂x ∂y ∂z ∂M ∂ ∂ ∂ = −[ ( ρu ) + ( ρv) + ( ρw )] δxδyδz ∂t ∂x ∂y ∂z Velocity Divergence form of mass continuity 1 dρ + ∇⋅ v = 0 ρ dt The Boussinesq Approximation Incompressibility dρ If density is constant , = 0 then the fluid dt Joseph Boussinesq (1842-1929) noted is incompressible that we can safely assume density is 1 dρ constant except when it is multiplied by + ∇⋅ v = 0 and ρ becomes: gravity in calculations of pressure dt ∇ ⋅ v = 0 The assumption will greatly simplify the – Density MUST be constant (incompressible) equations of motion! – Don’t consider vertical length scales so large that hydrostatic pressure causes density variations Incompressibility Example - Ocean We can easily use ∇ ⋅ v = 0 to learn We have an offshore wind, and must about vertical motions solve for the resultant upwelling (i.e. vertical velocity) Expanded: ∂u ∂v ∂w Assume incompressible flow (why is this + + = 0 a good assumption?) ∂x ∂y ∂z N v=10 cm/s or 100 km ∂u ∂v ∂w + = − 100 m ∂x ∂y ∂z Example - Atmosphere Notes Low-level convergence (1m/s each), solve Conservation laws applied to flow in the for the vertical velocity 100 m from the atmosphere and ocean lead to equations of surface if v = 0 and dx = 10 m. motion Flow in the ocean can be assumed to be incompressible except when describing sound Density can be assumed to be constant except * when multiplied by gravity – Boussinesq approximation z Conservation of mass leads to the continuity 100 m y equation, which has an especially simple form u u 1 2 for an incompressible fluid (why?) x dx.
Load more

Recommended publications
  • PHYSICS of ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3
    Chapter 2 PHYSICS OF ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3 1 Ohio University, Athens, Ohio, USA 2 NASA Johnson Space Center, Houston, Texas, USA 3 Centre National de la Recherche Scientifique, Toulouse, France This chapter discusses potential technologies for achieving artificial gravity in a space vehicle. We begin with a series of definitions and a general description of the rotational dynamics behind the forces ultimately exerted on the human body during centrifugation, such as gravity level, gravity gradient, and Coriolis force. Human factors considerations and comfort limits associated with a rotating environment are then discussed. Finally, engineering options for designing space vehicles with artificial gravity are presented. Figure 2-01. Artist's concept of one of NASA early (1962) concepts for a manned space station with artificial gravity: a self- inflating 22-m-diameter rotating hexagon. Photo courtesy of NASA. 1 ARTIFICIAL GRAVITY: WHAT IS IT? 1.1 Definition Artificial gravity is defined in this book as the simulation of gravitational forces aboard a space vehicle in free fall (in orbit) or in transit to another planet. Throughout this book, the term artificial gravity is reserved for a spinning spacecraft or a centrifuge within the spacecraft such that a gravity-like force results. One should understand that artificial gravity is not gravity at all. Rather, it is an inertial force that is indistinguishable from normal gravity experience on Earth in terms of its action on any mass. A centrifugal force proportional to the mass that is being accelerated centripetally in a rotating device is experienced rather than a gravitational pull.
    [Show full text]
  • LECTURES 20 - 29 INTRODUCTION to CLASSICAL MECHANICS Prof
    LECTURES 20 - 29 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford HT 2017 1 OUTLINE : INTRODUCTION TO MECHANICS LECTURES 20-29 20.1 The hyperbolic orbit 20.2 Hyperbolic orbit : the distance of closest approach 20.3 Hyperbolic orbit: the angle of deflection, φ 20.3.1 Method 1 : using impulse 20.3.2 Method 2 : using hyperbola orbit parameters 20.4 Hyperbolic orbit : Rutherford scattering 21.1 NII for system of particles - translation motion 21.1.1 Kinetic energy and the CM 21.2 NII for system of particles - rotational motion 21.2.1 Angular momentum and the CM 21.3 Introduction to Moment of Inertia 21.3.1 Extend the example : J not parallel to ! 21.3.2 Moment of inertia : mass not distributed in a plane 21.3.3 Generalize for rigid bodies 22.1 Moment of inertia tensor 2 22.1.1 Rotation about a principal axis 22.2 Moment of inertia & energy of rotation 22.3 Calculation of moments of inertia 22.3.1 MoI of a thin rectangular plate 22.3.2 MoI of a thin disk perpendicular to plane of disk 22.3.3 MoI of a solid sphere 23.1 Parallel axis theorem 23.1.1 Example : compound pendulum 23.2 Perpendicular axis theorem 23.2.1 Perpendicular axis theorem : example 23.3 Example 1 : solid ball rolling down slope 23.4 Example 2 : where to hit a ball with a cricket bat 23.5 Example 3 : an aircraft landing 24.1 Lagrangian mechanics : Introduction 24.2 Introductory example : the energy method for the E of M 24.3 Becoming familiar with the jargon 24.3.1 Generalised coordinates 24.3.2 Degrees of Freedom 24.3.3 Constraints 3 24.3.4 Configuration
    [Show full text]
  • Action Principle for Hydrodynamics and Thermodynamics, Including General, Rotational Flows
    th December 15, 2014 Action Principle for Hydrodynamics and Thermodynamics, including general, rotational flows C. Fronsdal Depart. of Physics and Astronomy, University of California Los Angeles 90095-1547 USA ABSTRACT This paper presents an action principle for hydrodynamics and thermodynamics that includes general, rotational flows, thus responding to a challenge that is more that 100 years old. It has been lifted to the relativistic context and it can be used to provide a suitable source of rotating matter for Einstein’s equation, finally overcoming another challenge of long standing, the problem presented by the Bianchi identity. The new theory is a combination of Eulerian and Lagrangian hydrodynamics, with an extension to thermodynamics. In the first place it is an action principle for adia- batic systems, including the usual conservation laws as well as the Gibbsean variational principle. But it also provides a framework within which dissipation can be introduced in the usual way, by adding a viscosity term to the momentum equation, one of the Euler-Lagrange equations. The rate of the resulting energy dissipation is determined by the equations of motion. It is an ideal framework for the description of quasi-static processes. It is a major development of the Navier-Stokes-Fourier approach, the princi- pal advantage being a hamiltonian structure with a natural concept of energy as a first integral of the motion. Two velocity fields are needed, one the gradient of a scalar potential, the other the time derivative of a vector field (vector potential). Variation of the scalar potential gives the equation of continuity and variation of the vector potential yields the momentum equation.
    [Show full text]
  • 6. Non-Inertial Frames
    6. Non-Inertial Frames We stated, long ago, that inertial frames provide the setting for Newtonian mechanics. But what if you, one day, find yourself in a frame that is not inertial? For example, suppose that every 24 hours you happen to spin around an axis which is 2500 miles away. What would you feel? Or what if every year you spin around an axis 36 million miles away? Would that have any e↵ect on your everyday life? In this section we will discuss what Newton’s equations of motion look like in non- inertial frames. Just as there are many ways that an animal can be not a dog, so there are many ways in which a reference frame can be non-inertial. Here we will just consider one type: reference frames that rotate. We’ll start with some basic concepts. 6.1 Rotating Frames Let’s start with the inertial frame S drawn in the figure z=z with coordinate axes x, y and z.Ourgoalistounderstand the motion of particles as seen in a non-inertial frame S0, with axes x , y and z , which is rotating with respect to S. 0 0 0 y y We’ll denote the angle between the x-axis of S and the x0- axis of S as ✓.SinceS is rotating, we clearly have ✓ = ✓(t) x 0 0 θ and ✓˙ =0. 6 x Our first task is to find a way to describe the rotation of Figure 31: the axes. For this, we can use the angular velocity vector ! that we introduced in the last section to describe the motion of particles.
    [Show full text]
  • Chapter 6 the Equations of Fluid Motion
    Chapter 6 The equations of fluid motion In order to proceed further with our discussion of the circulation of the at- mosphere, and later the ocean, we must develop some of the underlying theory governing the motion of a fluid on the spinning Earth. A differen- tially heated, stratified fluid on a rotating planet cannot move in arbitrary paths. Indeed, there are strong constraints on its motion imparted by the angular momentum of the spinning Earth. These constraints are profoundly important in shaping the pattern of atmosphere and ocean circulation and their ability to transport properties around the globe. The laws governing the evolution of both fluids are the same and so our theoretical discussion willnotbespecifictoeitheratmosphereorocean,butcanandwillbeapplied to both. Because the properties of rotating fluids are often counter-intuitive and sometimes difficult to grasp, alongside our theoretical development we will describe and carry out laboratory experiments with a tank of water on a rotating table (Fig.6.1). Many of the laboratory experiments we make use of are simplified versions of ‘classics’ that have become cornerstones of geo- physical fluid dynamics. They are listed in Appendix 13.4. Furthermore we have chosen relatively simple experiments that, in the main, do nor require sophisticated apparatus. We encourage you to ‘have a go’ or view the atten- dant movie loops that record the experiments carried out in preparation of our text. We now begin a more formal development of the equations that govern the evolution of a fluid. A brief summary of the associated mathematical concepts, definitions and notation we employ can be found in an Appendix 13.2.
    [Show full text]
  • Chapter 13: the Conditions of Rotary Motion
    Chapter 13 Conditions of Rotary Motion KINESIOLOGY Scientific Basis of Human Motion, 11 th edition Hamilton, Weimar & Luttgens Presentation Created by TK Koesterer, Ph.D., ATC Humboldt State University Revised by Hamilton & Weimar REVISED FOR FYS by J. Wunderlich, Ph.D. Agenda 1. Eccentric Force àTorque (“Moment”) 2. Lever 3. Force Couple 4. Conservation of Angular Momentum 5. Centripetal and Centrifugal Forces ROTARY FORCE (“Eccentric Force”) § Force not in line with object’s center of gravity § Rotary and translatory motion can occur Fig 13.2 Torque (“Moment”) T = F x d F Moment T = F x d Arm F d = 0.3 cos (90-45) Fig 13.3 T = F x d Moment Arm Torque changed by changing length of moment d arm W T = F x d d W Fig 13.4 Sum of Torques (“Moments”) T = F x d Fig 13.8 Sum of Torques (“Moments”) T = F x d § Sum of torques = 0 • A balanced seesaw • Linear motion if equal parallel forces overcome resistance • Rowers Force Couple T = F x d § Effect of equal parallel forces acting in opposite direction Fig 13.6 & 13.7 LEVER § “A rigid bar that can rotate about a fixed point when a force is applied to overcome a resistance” § Used to: – Balance forces – Favor force production – Favor speed and range of motion – Change direction of applied force External Levers § Small force to overcome large resistance § Crowbar § Large Range Of Motion to overcome small resistance § Hitting golf ball § Balance force (load) § Seesaw Anatomical Levers § Nearly every bone is a lever § The joint is fulcrum § Contracting muscles are force § Don’t necessarily resemble
    [Show full text]
  • Chapter 8: Rotational Motion
    TODAY: Start Chapter 8 on Rotation Chapter 8: Rotational Motion Linear speed: distance traveled per unit of time. In rotational motion we have linear speed: depends where we (or an object) is located in the circle. If you ride near the outside of a merry-go-round, do you go faster or slower than if you ride near the middle? It depends on whether “faster” means -a faster linear speed (= speed), ie more distance covered per second, Or - a faster rotational speed (=angular speed, ω), i.e. more rotations or revolutions per second. • Linear speed of a rotating object is greater on the outside, further from the axis (center) Perimeter of a circle=2r •Rotational speed is the same for any point on the object – all parts make the same # of rotations in the same time interval. More on rotational vs tangential speed For motion in a circle, linear speed is often called tangential speed – The faster the ω, the faster the v in the same way v ~ ω. directly proportional to − ω doesn’t depend on where you are on the circle, but v does: v ~ r He’s got twice the linear speed than this guy. Same RPM (ω) for all these people, but different tangential speeds. Clicker Question A carnival has a Ferris wheel where the seats are located halfway between the center and outside rim. Compared with a Ferris wheel with seats on the outside rim, your angular speed while riding on this Ferris wheel would be A) more and your tangential speed less. B) the same and your tangential speed less.
    [Show full text]
  • The Equations of Motion in a Rotating Coordinate System
    The Equations of Motion in a Rotating Coordinate System Chapter 3 Since the earth is rotating about its axis and since it is convenient to adopt a frame of reference fixed in the earth, we need to study the equations of motion in a rotating coordinate system. Before proceeding to the formal derivation, we consider briefly two concepts which arise therein: Effective gravity and Coriolis force Effective Gravity g is everywhere normal to the earth’s surface Ω 2 R 2 R R Ω R g* g g* g effective gravity g effective gravity on an earth on a spherical earth with a slight equatorial bulge Effective Gravity If the earth were a perfect sphere and not rotating, the only gravitational component g* would be radial. Because the earth has a bulge and is rotating, the effective gravitational force g is the vector sum of the normal gravity to the mass distribution g*, together with a centrifugal force Ω2R, and this has no tangential component at the earth’s surface. gg*R= + Ω 2 The Coriolis force A line that rotates with the roundabout Ω A line at rest in an inertial system Apparent trajectory of the ball in a rotating coordinate system Mathematical derivation of the Coriolis force Representation of an arbitrary vector A(t) Ω A(t) = A (t)i + A (t)j + A (t)k k 1 2 3 k ´ A(t) = A1´(t)i ´ + A2´ (t)j ´ + A3´ (t)k´ inertial j reference system rotating j ´ reference system i ´ i The time derivative of an arbitrary vector A(t) The derivative of A(t) with respect to time d A dA dA dA a =+ijk12 + 3 dt dt dt dt the subscript “a” denotes the derivative in an inertial reference frame ddAdAi′ ′ In the rotating frame a1=++i′′A ..
    [Show full text]
  • Effects of the Earth's Rotation
    Effects of the Earth’s Rotation C. Chen General Physical Oceanography MAR 555 School for Marine Sciences and Technology Umass-Dartmouth 1 One of the most important physical processes controlling the temporal and spatial variations of biological variables (nutrients, phytoplankton, zooplankton, etc) is the oceanic circulation. Since the circulation exists on the earth, it must be affected by the earth’s rotation. Question: How is the oceanic circulation affected by the earth’s rotation? The Coriolis force! Question: What is the Coriolis force? How is it defined? What is the difference between centrifugal and Coriolis forces? 2 Definition: • The Coriolis force is an apparent force that occurs when the fluid moves on a rotating frame. • The centrifugal force is an apparent force when an object is on a rotation frame. Based on these definitions, we learn that • The centrifugal force can occur when an object is at rest on a rotating frame; •The Coriolis force occurs only when an object is moving relative to the rotating frame. 3 Centrifugal Force Consider a ball of mass m attached to a string spinning around a circle of radius r at a constant angular velocity ω. r ω ω Conditions: 1) The speed of the ball is constant, but its direction is continuously changing; 2) The string acts like a force to pull the ball toward the axis of rotation. 4 Let us assume that the velocity of the ball: V at t V + !V " V = !V V + !V at t + !t ! V = V!" ! V !" d V d" d" r = V , limit !t # 0, = V = V ($ ) !t !t dt dt dt r !V "! V d" V = % r, and = %, dt V Therefore, d V "! = $& 2r dt ω r To keep the ball on the circle track, there must exist an additional force, which has the same magnitude as the centripetal acceleration but in an opposite direction.
    [Show full text]
  • Confusion Around the Tidal Force and the Centrifugal Force
    Jein Institute for Fundamental Science・Research Note, 2015 Confusion around the tidal force and the centrifugal force Confusion around the tidal force and the centrifugal force Takuya Matsuda, NPO Einstein, 5-14, Yoshida-Honmachiu, Sakyo-ku, Kyoto, 606-8317 Hiromu Isaka, Shimadzu Corp., 1, Nishinokyo-Kuwabaracho, Nakagyoku, Kyoto, 604-8511 Henri M. J. Boffin, ESO, Alonso de Cordova 3107, Vitacur, Casilla 19001, Santiago de Cile, Chile We discuss the tidal force, whose notion is sometimes misunderstood in the public domain literature. We discuss the tidal force exerted by a secondary point mass on an extended primary body such as the Earth. The tidal force arises because the gravitational force exerted on the extended body by the secondary mass is not uniform across the primary. In the derivation of the tidal force, the non-uniformity of the gravity is essential, and inertial forces such as the centrifugal force are not needed. Nevertheless, it is often asserted that the tidal force can be explained by the centrifugal force. If we literally take into account the centrifugal force, it would mislead us. We therefore also discuss the proper treatment of the centrifugal force. １．Introduction 2．Confusion around the centrifugal force In the present paper, we discuss the tidal force. We distinguish the tidal As was described earlier, the Earth and the Moon revolve about the force and the tidal phenomenon for the sake of exactness. The tide is a common center of gravity in the inertial frame. Fig. 1 shows a wrong geophysical phenomenon produced by the tidal force. The surface of the picture of the sea surface affected by the tidal force due to the Moon.
    [Show full text]
  • Coriolis and Centrifugal Forces
    Coriolis and Centrifugal Forces We are now going to analyze a very simple physical situation from a less simple point of 1. A kid on a merry-go-round view. Let’s say we have a body which moves at A kid is sitting on the edge of a merry-go-round, constant velocity with respect to an inertial and he drops a ball. We will be interested in frame. No net force acts on it. Now suppose we what happens in the horizontal plane - that is, look at this body from a frame of reference which when the whole thing is observed from above. is accelerating. In general, we no longer observe Never mind the vertical motion of the ball. the body to be moving with constant velocity. It If we are attached to the earth, viewing the whole appears to accelerate, and from our point of view thing from above, the ball just goes sailing off in it looks as though there were a force acting on it. a straight line as shown in this top view: Such a force is called an “effective” or “ficti- tious” force. The acceleration due to such a force is caused solely by the motion of the observer. ball ends A common example of this effect is when you here jump up and down, or spin around on your fancy desk chair. Everything appears to bounce up and down or go around and around, as the case may be. In this section, we will be concerned with the kid ends kid and ball particular case in which we are observing the here start here uniformly-moving body from a frame of refer- ence which is rotating at a constant rate.
    [Show full text]
  • The Principle of Stationary Action1 D
    The principle of stationary action1 D. E. Soper2 University of Oregon Physics 611, Theoretical Mechanics 3 October 2012 1 Introduction We have considered a system of particles J with masses mJ in the presence of a potential 1 X X (e) V (x ;:::; x ; t) = V (jx − x j) + V (x ; t): (1) 1 N 2 JK J K J J JK J Here I use boldface for vectors in three dimensions and I set VKJ = VJK and VJJ = 0. For this system, the equations of motion are ~ mI x¨J = −rJ V (x1;:::; xN ; t): (2) We now reformulate the theory. Let us use the notation x(t) to stand for the entire collection of coordinates fx1(t);:::; xN (t)g at time t. Consider paths that the system might take between a fixed initial position x(ti) at time ti and a fixed final position x(tf ) at time tf . We use x to label the whole function describing the path, the function whose value at time t is x(t). We define the action ( ) Z tf X 1 S[x] = dt m x_ 2 − V (x(t); t) : (3) 2 J J ti J Thus the action is a function defined on the space of paths. We consider how the action changes if we change from one path x to an- other path x+δx that is infinitesimally different and has the same endpoints: δx(ti) = δx(tf ) = 0: (4) 1Copyright, 2012, D. E. Soper [email protected] 1 Define the variation δS[x] of the action to be the part of S[x+δx]−S[x] that is linear in δx, omitting contributions proportional to the square or higher powers of δx.
    [Show full text]