Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Transcription

1 1.1 Vector Alger Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples re temperture, density nd mss. In the following, lowercse (usully Greek) letters, e.g.,,, will e used to represent sclrs Vectors The concept of the vector is used to descrie physicl quntities which hve oth mgnitude nd direction ssocited with them. Exmples re force, velocity, displcement nd ccelertion. Geometriclly, vector is represented y n rrow; the rrow defines the direction of the vector nd the mgnitude of the vector is represented y the length of the rrow, Fig Anlyticlly, vectors will e represented y lowercse old-fce Ltin letters, e.g., r, q. The mgnitude (or length) of vector is denoted y or. It is sclr nd must e non-negtive. Any vector whose length is 1 is clled unit vector; unit vectors will usully e denoted y e. c () () Figure 1.1.1: () vector; () ddition of vectors Vector Alger The opertions of ddition, sutrction nd multipliction fmilir in the lger of numers (or sclrs) cn e extended to n lger of vectors. Solid Mechnics Prt III 3

2 The following definitions nd properties fundmentlly define the vector: 1. Sum of Vectors: The ddition of vectors nd is vector c formed y plcing the initil point of on the terminl point of nd then joining the initil point of to the terminl point of. The sum is written c. This definition is clled the prllelogrm lw for vector ddition ecuse, in geometricl interprettion of vector ddition, c is the digonl of prllelogrm formed y the two vectors nd, Fig The following properties hold for vector ddition: commuttive lw c c ssocitive lw. The Negtive Vector: For ech vector there exists negtive vector. This vector hs direction opposite to tht of vector ut hs the sme mgnitude; it is denoted y. A geometricl interprettion of the negtive vector is shown in Fig Sutrction of Vectors nd the Zero Vector: The sutrction of two vectors nd is defined y ( ), Fig If then is defined s the zero vector (or null vector) nd is represented y the symol o. It hs zero mgnitude nd unspecified direction. A proper vector is ny vector other thn the null vector. Thus the following properties hold: o o 4. Sclr Multipliction: The product of vector y sclr is vector with mgnitude times the mgnitude of nd with direction the sme s or opposite to tht of, ccording s is positive or negtive. If 0, is the null vector. The following properties hold for sclr multipliction: distriutive lw, over ddition of sclrs distriutive lw, over ddition of vectors ( ) ( ) ssocitive lw for sclr multipliction () () Figure 1.1.: () negtive of vector; () sutrction of vectors Solid Mechnics Prt III 4

3 Note tht when two vectors nd re equl, they hve the sme direction nd mgnitude, regrdless of the position of their initil points. Thus in Fig A prticulr position in spce is not ssigned here to vector it just hs mgnitude nd direction. Such vectors re clled free, to distinguish them from certin specil vectors to which prticulr position in spce is ctully ssigned. Figure 1.1.3: equl vectors The vector s something with mgnitude nd direction nd defined y the ove rules is n element of one cse of the mthemticl structure, the vector spce. The vector spce is discussed in the next section, The Dot Product The dot product of two vectors nd (lso clled the sclr product) is denoted y. It is sclr defined y cos. (1.1.1) here is the ngle etween the vectors when their initil points coincide nd is restricted to the rnge 0, Fig Figure 1.1.4: the dot product An importnt property of the dot product is tht if for two (proper) vectors nd, the reltion 0, then nd re perpendiculr. The two vectors re sid to e orthogonl. Also, cos(0), so tht the length of vector is. Another importnt property is tht the projection of vector u long the direction of unit vector e is given y u e. This cn e interpreted geometriclly s in Fig Solid Mechnics Prt III 5

4 u e u u e u cos Figure 1.1.5: the projection of vector long the direction of unit vector It follows tht ny vector u cn e decomposed into component prllel to (unit) vector e nd nother component perpendiculr to e, ccording to u ee u u ee u (1.1.) The dot product possesses the following properties (which cn e proved using the ove definition) { Prolem 6}: (1) (commuttive) () c c (distriutive) (3) (4) 0; nd 0 if nd only if o The Cross Product The cross product of two vectors nd (lso clled the vector product) is denoted y. It is vector with mgnitude sin (1.1.3) with defined s for the dot product. It cn e seen from the figure tht the mgnitude of is equivlent to the re of the prllelogrm determined y the two vectors nd. Figure 1.1.6: the mgnitude of the cross product The direction of this new vector is perpendiculr to oth nd. Whether points up or down is determined from the fct tht the three vectors, nd form right hnded system. This mens tht if the thum of the right hnd is pointed in the Solid Mechnics Prt III 6

5 direction of, nd the open hnd is directed in the direction of, then the curling of the fingers of the right hnd so tht it closes should move the fingers through the ngle, 0, ringing them to. Some exmples re shown in Fig Figure 1.1.7: exmples of the cross product The cross product possesses the following properties (which cn e proved using the ove definition): (1) (not commuttive) () c c (distriutive) (3) (4) o o re prllel ( linerly dependent ) if nd only if nd The Triple Sclr Product The triple sclr product, or ox product, of three vectors u, v, w is defined y u v w v wu w u v Triple Sclr Product (1.1.4) Its importnce lies in the fct tht, if the three vectors form right-hnded trid, then the volume V of prllelepiped spnned y the three vectors is equl to the ox product. To see this, let e e unit vector in the direction of of w on u v is h w e, nd w u v w u v e u v, Fig Then the projection u v h V (1.1.5) Solid Mechnics Prt III 7

6 w u v e h Figure 1.1.8: the triple sclr product Note: if the three vectors do not form right hnded trid, then the triple sclr product yields the negtive of the volume. For exmple, using the vectors ove, wv u. V Vectors nd Points Vectors re ojects which hve mgnitude nd direction, ut they do not hve ny specific loction in spce. On the other hnd, point hs certin position in spce, nd the only chrcteristic tht distinguishes one point from nother is its position. Points cnnot e dded together like vectors. On the other hnd, vector v cn e dded to point p to give new point q, q v p, Fig Similrly, the difference etween two points gives vector, q p v. Note tht the notion of point s defined here is slightly different to the fmilir point in spce with xes nd origin the concept of origin is not necessry for these points nd their simple opertions with vectors. v q p Figure 1.1.9: dding vectors to points Prolems 1. Which of the following re sclrs nd which re vectors? (i) weight (ii) specific het (iii) momentum (iv) energy (v) volume. Find the mgnitude of the sum of three unit vectors drwn from common vertex of cue long three of its sides. Solid Mechnics Prt III 8

7 3. Consider two non-colliner (not prllel) vectors nd. Show tht vector r lying in the sme plne s these vectors cn e written in the form r p q, where p nd q re sclrs. [Note: one sys tht ll the vectors r in the plne re specified y the se vectors nd.] 4. Show tht the dot product of two vectors u nd v cn e interpreted s the mgnitude of u times the component of v in the direction of u. 5. The work done y force, represented y vector F, in moving n oject given distnce is the product of the component of force in the given direction times the distnce moved. If the vector s represents the direction nd mgnitude (distnce) the oject is moved, show tht the work done is equivlent to F s. 6. Prove tht the dot product is commuttive,. [Note: this is equivlent to sying, for exmple, tht the work done in prolem 5 is lso equl to the component of s in the direction of the force, times the mgnitude of the force.] 7. Sketch if nd re s shown elow. 8. Show tht. 9. Suppose tht rigid ody rottes out n xis O with ngulr speed, s shown elow. Consider point p in the ody with position vector r. Show tht the velocity v of p is given y v ωr, where ω is the vector with mgnitude nd whose direction is tht in which right-hnded screw would dvnce under the rottion. [Note: let s e the rc-length trced out y the prticle s it rottes through n ngle on circle of rdius r, then v v r (since s r, ds / dt r( d / dt) ).] ω O r r p v 10. Show, geometriclly, tht the dot nd cross in the triple sclr product cn e interchnged: c c. 11. Show tht the triple vector product c lies in the plne spnned y the vectors nd. Solid Mechnics Prt III 9