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Sect. 3-4: Analytic Method of Addition

Sect. 3-4: Analytic Method of Addition. Resolution of vectors into components : YOU MUST KNOW & UNDERSTAND TRIGONOMETERY TO UNDERSTAND THIS!!!!. Vector Components.

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Sect. 3-4: Analytic Method of Addition

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  1. Sect. 3-4: Analytic Method of Addition Resolution of vectors into components: YOU MUST KNOW & UNDERSTAND TRIGONOMETERY TO UNDERSTAND THIS!!!!

  2. Vector Components • Any vector can be expressed as the sum of two other vectors, called its components. Usually, the other vectors are chosen so that they are perpendicular to each other. • Consider vector V in a plane(say the xy plane) • Vcan be expressed in terms of theComponentsVx ,Vy • Finding the ComponentsVx& Vyis Equivalentto finding 2 mutually perpendicular vectors which, when added(with vector addition)will giveV. • That is, findVx& Vysuch that V  Vx + Vy (Vx|| x axis, Vy|| y axis) Finding Components “Resolving into components”

  3. Mathematically, a componentis a projectionof a vector along an axis • Any vector can be completely described by its components • It is useful to use rectangular components • These are the projections of the vector along the x- and y-axes

  4. When Vis resolved into components:Vx& Vy V  Vx + Vy(Vx|| x axis, Vy|| y axis) By the parallelogram method, the vector sum is:V = V1 + V2In 3 dimensions, we also need a component Vz.

  5. Brief Trig Review • Adding vectors in 2 & 3 dimensions using components requires TRIGONOMETRY FUNCTIONS • HOPEFULLY, A REVIEW!! • See also Appendix A!! • Given any angleθ, we can construct a right triangle: Hypotenuse h, Adjacent side  a, Opposite side  o h o a

  6. Define trig functions in terms of h, a, o: = (opposite side)/(hypotenuse) = (adjacent side)/(hypotenuse) = (opposite side)/(adjacent side) [Pythagorean theorem]

  7. Signs of sine, cosine, tangent • Trig identity: tan(θ) = sin(θ)/cos(θ)

  8. Trig Functions to Find Vector Components We can & will use all of this to add vectors analytically! [Pythagorean theorem]

  9. Example V = displacement 500 m, 30º N of E

  10. Using Components to Add Two Vectors • Consider 2 vectors, V1 & V2. We want V = V1 + V2 • Note:The components of each vector are really one-dimensional vectors, so they can be added arithmetically.

  11. We want the vector sum V = V1 + V2 “Recipe” (for adding 2 vectors using trig & components) 1.Sketcha diagram toroughlyadd the vectors graphically. Choose x & y axes. 2.Resolve each vector into x& y components using sines & cosines. That is, find V1x, V1y, V2x, V2y. (V1x = V1cos θ1, etc.) 4. Add the components in each direction. (Vx = V1x + V2x, etc.) 5.Find the length & direction of V, using:

  12. Example 3-2 A rural mail carrier leaves the post office & drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?

  13. A plane trip involves 3 legs, with 2 stopovers: 1) Due east for 620 km, 2) Southeast (45°) for 440 km, 3)53° south of west, for 550 km.Calculate the plane’s total displacement. Example 3-3

  14. Sect. 3-5: Unit Vectors • Convenient to express vector V in terms of it’s componentsVx,Vy, Vz& UNIT VECTORS along x,y,z axes UNIT VECTOR a dimensionless vector, length = 1 • Define unit vectors along x,y,z axes: i alongx; j along y; k along z |i| = |j| = |k| = 1 • Vector V. Components Vx,Vy, Vz:

  15. Simple Example • Position vector r in x-y plane. Components x, y: r  xi + y j Figure 

  16. Vector Addition Using Unit Vectors • Suppose we want to add two vectors V1 & V2in x-y plane: V = V1 + V2 “Recipe” 1. Find x & y components of V1 & V2(using trig!) V1 = V1xi + V1yj V2 = V2xi + V2yj 2. x component of V: Vx = V1x + V2x y component of V: Vy = V1y + V2y 3. So V= V1 + V2 = (V1x+ V2x)i + (V1y+ V2y)j

  17. Example 3-4 Rural mail carrier again. Drives 22.0 km in North. Then 60.0° south of east for 47.0 km. Displacement?

  18. Another Analytic Method • Laws of Sines & Law of Cosines from trig. • Appendix A-9, p A-4, arbitrary triangle: • Law of Cosines: c2 = a2 + b2 - 2 a b cos(γ) • Law of Sines: sin(α)/a = sin(β)/b = sin(γ)/c β c a α γ b

  19. Given A, B, γ B A β C A • Add 2 vectors: C = A + B • Law of Cosines: C2 = A2 + B2 -2 A B cos(γ) Gives length of resultant C. • Law of Sines: sin(α)/A = sin(γ)/C, or sin(α) = A sin(γ)/C Gives angle α α γ B

  20. Sect. 3-6: Kinematics in 2 Dimensions

  21. Sect. 3-6: Vector Kinematics • Motion in 2 & 3 dimensions. Interested in displacement, velocity, acceleration. All are vectors. For 2- & 3-dimensional kinematics, everything is the same as in 1-dimensional motion, but we must now use full vector notation. • Position of an object (particle) is described by its position vector, r. • Displacementof the object is thechange in its position: Particle Path Displacement takes time Δt

  22. Velocity:Average Velocity =ratio of displacement Δrtime interval Δtfor the displacement: • vavg • vavgis in the direction of Δr. vavgindependent of the path taken because Δr is also. In the limit as Δt & Δr 0 vavg Instantaneous Velocity v. • Direction of v at any point in a particle’s path is along a line tangent to the path at that point & in motion direction. The magnitude of the instantaneous velocity vector v is the speed, which is a scalar quantity.

  23. Acceleration:Average Acceleration = ratio of the change in the instantaneous velocity vector divided by time during which the change occurs. aavg aavgis in the direction of In the limit as Δt &  0 aavgInstantaneous Velocity a. Various changes in particle motion may cause acceleration: 1) Magnitude velocity vector may change. 2) Direction of velocity vector may change. even if magnitude remains constant. 3) Both may change simultaneously.

  24. Using unit vectors:

  25. Kinematic Equations for 2-Dimensional Motion • In the special case when the 2-dimensional motion has a constant acceleration a, kinematic equations can be developed that describe the motion • Can show that these equations (next page) are similar to those of 1-dimensional kinematics. • Motion in 2 dimensions can be treated as 2 independentmotions in each of the 2 perpendicular directions associated with the x & y axes • Motion in y direction does not affect motion in x direction & motion in x direction does not affect the motion in y direction. • Results are shown on next page.

  26. Generalization of 1-dimensional equations for constant acceleration to 2 dimensions. These are valid for constant acceleration ONLY!

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