A Derivation of the Coriolis Force/Mass

Transcription

1 A Derivation of the Coriolis Force/Mass Introduction We live on a rotating planet and view the motions (and changes of motions) of things from this rotating (and hence accelerated) frame of reference. We re used to it, and it d be tough to change our point of view to an unaccelerated one, so we keep it. We pay a modest price, though we have to invent "apparent" forces if we want to use Newton s Second Law to help "explain" the accelerations that we observe. (That is, without these fictitious, apparent forces, the observed accelerations don t equal the sum of the real forces/mass acting on objects.) Moreover, for Newton s Second Law to be a useful quantitative relation (that is, if we want to use it to predict changes in velocity and hence help forecast the weather, for example), we have to express apparent forces quantitatively. That s our goal here. The Coriolis Force/Mass To get some insight into the Coriolis force/mass, we ll first adopt a virtually inertial (that is, virtually unaccelerated) frame of reference moving with the earth s center of mass. Within this frame of reference we establish a polar coordinate system in a plane normal to the earth s axis of rotation with its origin on the axis. This is convenient because at any particular latitude, the earth s surface lies at a nearly constant distance from the axis of rotation (and positions within the atmosphere and oceans depart from this by only small distances.) In any 2-D coordinate system such as polar coordinates, to specify completely the two-dimensional vector position of any object () requires two independent scalar pieces of information. In our polar coordinate system, can be represented as (, θ ), where is radial distance from the origin (in our case, perpendicular to the earth s axis of rotation), and θ is an angle measured from a reference line in the polar plane. On the earth, θ would be the longitude. If we define dimensionless unit vectors e and e θ in the two respective component directions, then the position vector can be written as: = e Page 1

2 or shown pictorially (looking down on the polar plane) as: θ Location of an object in the atmosphere or ocean at latitude ϕ and longitude θ. Notice that, although represents position in a two-dimensional plane, in a polar coordinate system has only one vector component, e. This is because the direction of the unit vector e depends on θ, and so e contains the information about the angle θ needed to specify fully in this coordinate system. That is, e represents the same information that (,θ) does, and so e θ is not explicitly needed to help construct in terms of its vector components. This is an important property of this coordinate system. In fact, the directions of both unit vectors e and e θ depend on θ, so that changes in θ imply that e and e θ change. Note, however, that changes in (the distance in the e, or radial, direction) don t affect the directions of e or e θ. Velocity in polar coordinates In this polar coordinate system, the velocity, V, of an object is the rate of change of the object s position with respect to time: d V = dt In the polar coordinate system, can change in either or both of the two coordinate directions--that is, it can change radially (in the direction) or tangentially (in the θ direction). The rate at which changes in the radial direction is just the rate at which lengthens or shortens, or the rate at which the object moves toward or away from the origin, d/dt. We give this radial component of the rate of change of position w/r/t time the symbol v : But can also change direction as well as length. Changes in s direction Page 2

3 d v dt are necessarily in the tangential direction (that is, it can t change direction by getting longer or shorter--it has to shift laterally ), and therefore this component of d/dt is in the tangential direction. Intuitively, we know that this tangential component should depend on how rapidly the angular coordinate of changes (that is, it should depend on dθ/dt the more rapidly θ changes, the larger this component of d/dt should be). In addition, though, for a given dθ/dt, the rate of change in tangentially is bigger for longer : 1 (t+ t) 2 (t+ t) 1 (t) θ 2 (t) Over a short time t, a two vectors 1 and 2 change direction by the same amount, θ. (Neither changes length.) The vector change in the longer vector is greater than the vector change in the shorter one. The vector changes in both 1 and 2 are in the tangential direction. Hence, the tangential component of d/dt, which we ll give the symbol v θ, should be involve both and dθ/dt. It can be shown that it is in fact the product of the two: dθ v θ dt In summary, we can write the velocity in vector-component form as: d d dθ V = v dt e + v θ e θ = ( )e dt + eθ dt Acceleration in polar coordinates epresenting the components of acceleration in polar coordinates gets more Page 3

4 complicated. For small The θ, final the result, magnitude though, of is e as θ follows: is approximately equal to the length of an arc between e θ1 and e θ2. (In the limit 2 as t becomes infinitesimally small, they become exactly equal.) dv The v radius θ dv v of the arc θ is v the θ a = a length of e θ1 or e θ2, which is 1 since e + a both are θ e θ = e eθ unit vectors, dt so the arc dtlength is just θ 1= θ. The direction of e θ is approximately in the same direction as e r (and The radial component of acceleration, a exactly the same in the limit as t becomes, comprises two parts. The infinitesimally small). Hence, e interpretation of the first part, dv θ θe r, and /dt, is probably intuitive. It accounts for changes in the rate at which the object approaches or moves away from the origin. The second term may not be so intuitive. It accounts for changes in the direction of the tangential component of velocity; these changes occur in the radial direction as the object moves tangentially: v θ e θ (t+ t) (v θ e θ ) In Lab 3 we defined field Velocity variables vectors as quantities (vector or scalar) that depend on position in space and on time. For scalar field variables, we defined the gradient of the vvariable θ e θ (t) as a vector pointed in the direction v θ e θ (t) in which thee scalar varies most rapidly in space, with a magnitude equal to the (t+ t) maximum rate at which the scalar varies with vposition. θ e θ (t+ t) We looked in Position vectors particular at how the gradient e (t) of a scalar can be represented in rectangular coordinates in terms of the partial derivatives of Close-up the scalar view of in each coordinate velocity vectors, direction. shifted to share a common base. The This diagram gradient shows of a scalar the position in rectangular and velocity coordinates V of an object turns at out two to slightly be different simply one times. application For simplicity, of an this entity shows called a case thewhere gradient the velocity operator is or purely del operator, tangential.. The The tangential gradient vector operator velocity is acomponent vector, but changes it looks direction rather odd (though not magnitude comparedin tothis other idealized vectorscase). withthe which vector we ve change worked in the because tangential it doesn t velocityhave component a clearly defined is the magnitude (negative) radial or even direction, a direction normal in the to the conventional tangential velocity sense. It component. is more like The a procedure larger the tangential or set of velocity instructions component, that can thebe faster carried it changes out, a intool the radial that acquires direction concrete (for a given meaning ). On the only other when hand, it is the applied fartheror the used. object That is from is, the origin acquires the clear more slowly meaning theonly tangential whenvelocity the operation vector changes it represents in the radial is carried direction out (for applied. a given tangential velocity). It can be shown that the rate at which changes in the radial direction due to changes in the direction of the tangential component of velocity Lab is v 3 showed 2 θ /. (Note one that way thisthat partthe of the del radial operator acceleration can be is applied. toward the In origin rectangular regardless of coordinates, the sign of v θ ). the del operator can be written as: The tangential component of acceleration, a θ, also comprises two parts. The first part, dv θ /dt, is probably intuitive. i It + accounts j + kfor changes in the rate at which the object moves along a circle of radius x (in y the positive z or negative tangential direction). We can associate this with changes in the speed of the tangential motion. Page 4

5 The second term may not be so intuitive. It accounts for changes in the direction of the radial component of velocity, which occur in the tangential direction when the object s motion has both tangential and radial components. Here s an idealized example where the velocity, which has both tangential and radial components, changes direction but not magnitude (i.e., speed), and neither v θ nor v change: Velocity vectors, broken into components V(t+ t) v θ e θ (t+ t) v θ e θ (t) V(t) (t+ t) v (t) e (t+ t) Position vectors v e (t) Because the direction of the velocity changes, it is therefore accelerated. The change in the velocity, V, has both a radial and tangential component (shown at right above). V s radial component arises from the change in direction of the tangential velocity component, as in the example on the previous page. Its tangential component comes from the change in direction of the radial velocity component, which occurs because the object is also moving tangentially. The left-hand diagram above shows this. Notice how the radial component of the velocity changes direction, rotating counterclockwise (in this example) but not changing length, as a consequence of the fact that the object is moving not only radially but also tangentially. This change in direction of the radial velocity component occurs in the negative tangential direction, in this case. It can be shown that the contribution to the tangential component of acceleration due to the change in direction of the radial component of velocity, is proportional to the product of the two velocity components and inversely proportional to the distance from the origin, v θ v /. V V(t) V(t+ t) Close-up view of velocity vectors, shifted to share a common base. Page 5