1 / 27

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.

nealm
Download Presentation

Sect. 3-4: Analytic Method of Addition

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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!

More Related