Transformation of kinematical quantities from rotating into static coordinate system

Transcription

1 Transformation of kinematical quantities from rotating into static coordinate sstem Dimitar G Stoanov Facult of Engineering and Pedagog in Sliven, Technical Universit of Sofia 59, Bourgasko Shaussee Blvd, 8800 Sliven, BULGARIA dgstoanov@abv.bg Abstract. In this article the problem for the using of uniforml rotating coordinate sstem is considered. The correct relations between the kinematical quantities characteriing the motion of a bod relative to uniforml rotating sstem and static coordinate sstem are obtained. PACS: q, 45.0.D-, Dd. Kewords: rotating reference frame, transformation. 1. Introduction In this article the problem of rotating coordinate sstem handling is discussed. It is not a new one, and is available in one form or another in all tetbooks of phsics at the undergraduate level. The common point of the bigger part of these books is the pointing out of ungrounded interrelations between the kinematical quantities in static and rotating coordinate sstems, and these interrelations have been deepl rooted in the consciousness of students and teachers. Nowadas such information is available in Wikipedia [1]. It is essentiall that this fact reduces the deepness of rationaliation and understanding on behalf of the students of wide circle of basic laws and effects in mechanics. As an eception is [], where an attempt is made for the derivation of the kinematical interrelations of coordinate sstems disposed in the common case and the kinematical interrelations of the motion of a bod also in common case. The approach is true and the final correct result should be got if in a later stage of the derivation a matri for the transformation of the coordinates has been applied. This is an eample how in order the present material to be simplified could be fallen in situation when the present material will be not quite adequate. Onl in [3] the correct interrelations are given, moreover combined with a proof in general case. The objective of this article is following [] up to a certain place and continuing after that in own wa the kinematical interrelations between the quantities in a static and uniforml rotating coordinate sstems to be derived correctl.. Relations between the kinematical quantities.1 A detail calculation Let the material point (small bod) M with mass m be moving in space. Let the inertial Cartesian coordinate sstem S to be static. The position of the material point in relation to the coordinate sstem S is determined b the radius-vector r r (1), where, and are the coordinates of the geometric point relative to S (see figure 1), in which the material point is: r = ( 1 )

2 Let the Cartesian coordinate sstem S, shown in figure 1, be rotating. The position of the r material point in relation to the coordinate sstem S is determined b the radius-vector (), where, and are the coordinates of the geometric point relative to S, in which the material point is: r = ( ) Suppose the coordinate sstem S is such that: - the origins of the coordinate sstems S and S coincide; - the ais of S coincides with the ais of S ; - at the initial moment t = 0 the aes OX ' and OX coincide; - S is rotating uniforml around the ais of S with angular velocit Ω. Figure 1. Mutual disposition of the coordinate sstems. A case with simple mutual disposition of the coordinate sstems is considered. This allows an intuitive check of the obtained results to be done. The calculations are given deliberatel in details in order to preclude an logical misunderstandings. The considered here special case of motion of the

3 rotating coordinate sstem S is sufficient the necessar relations between the kinematical quantities to be obtained as an illustration without an loss of generalit. At such formulation the coordinates of the geometric point relative to S are transformed to the coordinates of the same point relative to S b the equations []: = ) ), ( 3a ) = ) + ), ( 3b ) =. ( 3c ) Further we consider the motion of a material point which has radius-vector r r, velocit v and r r r r acceleration a relative to S and radius-vector, velocit v and acceleration a relative to S. Differentiating (3a), (3b) and (3c) with respect to time we obtain: v = v ). ) v ). ), ( 4a ) v = v ) + Ω. ) + v ). ), ( b ) v where v, 4 = v. ( 4 v and v are the components of the velocit v r of the material point relative to S. Similarl, differentiating (4a), (4b) and (4c) with respect to time for the components of the acceleration a r of the material point relative to S, we obtain: c ) a a = = a ). Ω.v ) a ) +. Ω.v ) + Ω a ) +. Ω.v ) + a ). Ω.v )..cos( Ω.t)..sin( Ω.t),..sin( Ω.t)..cos( Ω.t), (5a) (5b) a = a. (5c) Up to here all obtained results coincide with the results presented in []. Further on in [] a projection of the quantities from the right sides (concerning the rotating coordinate sstem S ) on the aes of the static coordinate sstem S is stated. To make the projection process more clear, we will use the matri for transformation of coordinates.. A presentation b the transformation matri In this part of the article the obtained et results for the velocit and the acceleration of the material point will be presented b the matri for transformation of coordinates. We define a matri (33) for transformation of coordinates R ), derived from (3а), (3b) and (3c): cos( θ ) sin( θ ) 0 ) R = sin( θ ) cos( θ ) 0. ( 6 ) Here θ is the angle between the aes OX and OX ' of the coordinate sstems S and S. The positive direction of the angle is counterclockwise (see figure1). The matri R ) acts upon vectors represented in S and as a result gives vectors from S.

4 Because of that ) det R = 1. ( 7 ) the magnitudes of vectors at transformation are not scaled (the length of the line segment is kept during the transition from one coordinate sstem to another). If the angle θ is a constant and is equal to ero R ) is the identit matri. If the angle θ is a constant and is not equal to ero R ) is not the identit matri. In this case S is turned relative to S, but is static. If θ = Ω. t the sstem S rotates with respect to S around ais with a constant angular velocit Ω. In this case R ) becomes cos( Ω.t ) sin( Ω.t ) 0 ) R = sin( Ω.t ) cos( Ω.t ) 0. ( 8 ) B using of (8) the components of the radius-vector (3а, 3b and 3c), the velocit (4a, 4b and 4c) and the acceleration (5a, 5b and 5c) can be presented in the following form ) = R. (9) v v v = v ) R. v. + Ω. v ( 10 ) a a a = a ) R. a. Ω.v +. Ω.v a.. ( 11 ) It eas to see, that the right side of the obtained equations (9), (10) and (11) are the products of the matri R ) b the column vectors of quantities given with respect to the sstem S. These equations can be written in a compact form using the properties of the vector product of vectors. For the goal a vector of the angular velocit Ω r of the coordinate sstem S relative to S is defined as in this considerable special case the sstems are of the following form Ω r 0 = 0 Ω (1 ) Then from (1) we derive: r ) r = R.r ( 13 )

5 r ) r r r v = R.( v + Ω ) ( 14 ) r r r r r r [ a +. Ω v + Ω ( Ω )] ( ) r ) a = R. 15 This is the final form of the kinematical interrelations between the static and the uniforml rotating coordinate sstem..3 Generaliation It is evident from the results obtained above the importance of the matri for transformation of coordinates R ). The obtained up to now results can illustrate the generaliation proved in [3]. This generaliation includes the following: Let s have vector A r r S and vector A S for which is valid the following link r ) r A = R.A The matri R ) is anti-smmetrical that allows the introduction of a vector of the instant angular velocit Ω r of rotation of S relative to S ( Ω r to be constant is not compulsor), which satisfies the condition [3] ) ) dr r ) ) dr r ) r v 1.A = R.R..A = R. Ω A dt dt (16) (17) Differentiating with respect to time (16) and using (17) we obtain [3] r r da ) da r r = R.( + Ω A ) dt dt (18) The rule (18) is valid for each rotating motion of the coordinate sstem S relative to S, i.e. with an arbitrar orientation in space, uniforml or non-uniforml in time. Differentiating (18) with respect to time and making certain transformations and using (17) we can obtain [3] r r r r d A ) d A dω r da r r r = R. A. ( A ). + + Ω + Ω Ω dt dt dt dt The comparison between (18) and (14) on one hand, and between (19) and (15) on the other hand shows the accurac of the obtained previousl and considered in details interrelations. This is an illustration of the importance of the transformation matri R ) as well as the possibilit for generaliations if this approach is used. 3. Discussion of results The transformation matri (6) is an important element at the kinematical interrelations (13), (14) and (15) because it is responsible for the correct direction of vectors in S. Unfortunatel, its significance is not appreciated properl up to now. According to the mathematics we can sum, subtract and multipl vectors, which are from one and the same vector space or are presented in one and the same coordinate sstem. This is the reason r ) r in [3] several times to be reminded that S and R.r S are two different vectors belonging to two different vector spaces. From (14) is easil obtained that (19)

6 r ) r r r r r r v = R.v + Ω v + Ω (0) r ) The transformation of v in (0) b R is necessar not onl for the uniforml rotating coordinate sstem but for the turned sstem S. If we multipl both sides of (15) b the mass of the material point and taking into account that according to the second law of Newton r r r m.a = F = Rˆ.F (1) where F r is the force acting on the material point we will obtain: r r r r r r r m.a = F -.m. Ω v m. Ω ( Ω ) () Thus, in the right side of () the Coriolis and centrifugal inertial forces. It is acting on the bod from the point of view of uniforml rotating coordinate sstem S, have appeared naturall and correctl. From () is evident that if there is no force F r, the material point will move relative to S with acceleration which value differs from ero. The reason is the presence of Coriolis and centrifugal inertial forces. Eactl these inertial forces make the uniforml rotating coordinate sstem S noninertial. 4. Conclusion In this article the geometric interrelations are precised and the common case of motion of a bod is considered. The conclusion is elementar and eas of access for students and therefore, it can be used in the teaching activit. The considered approach is valuable not onl with that it gives the right form of the kinematical interrelations but further developed naturall can give the epressions for the interrelations between the dnamical quantities for a single material point, a sstem of material points and a rigid bod in various coordinate sstems References [1] [] Kittel C, Knight W D and Ruderman M A 1973 Berkele Phsics Course, Mechanics vol 1 (New York: McGraw- Hill) pp [3] Arnold V I 1989 Mathematical methods of classical mechanics, Graduate Tets in Mathematics, v. 60, (New York: Springer-Verlag) pp 13-33