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Transcript
Analytic Method of Vector Addition Analytic Method of Addition • First: Discuss the resolution of vectors into components: YOU MUST KNOW & UNDERSTAND TRIGONOMETERY TO UNDERSTAND THIS!!!! Vector Components • Consider a vector V in a plane (say, the xy plane) • We can express V in terms of Components Vx , Vy • Finding components Vx & Vy is equivalent to finding 2 mutually perpendicular vectors which, when added (with vector addition) will give V. • That is, find Vx & Vy such that V Vx + Vy (Vx || x axis, Vy || y axis) Finding components “Resolving into components” • Mathematically, a component is a projection of 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 xand y-axes When V is resolved into components: Vx & V y V Vx + Vy (Vx || x axis, Vy || y axis) By the parallelogram method, the vector sum is: V = Vx + Vy In 3 dimensions, we also need a component Vz Brief Trig Review • Adding vectors in 2 & 3 dimensions using components requires TRIGONOMETRY FUNCTIONS HOPEFULLY, A REVIEW!! See also Appendix A!! • Given any angle θ, we can construct a right triangle: Hypotenuse h Adjacent side a Opposite side o h a o • Define the trigonometry functions in terms of h, a, o: = (opposite side)/(hypotenuse) = (adjacent side)/(hypotenuse) = (opposite side)/(adjacent side) [Pythagorean theorem] Signs of Sine, Cosine, Tangent Trig identity: tan(θ) = sin(θ)/cos(θ) Using Trig Functions to Find Vector Components We can & we will use all of this to Add Vectors Analytically! [Pythagorean theorem] Example V = Displacement 500 m, 30º N of E Unit Vectors • Its convenient to express vector A in terms of it’s components Ax, Ay, Az & UNIT VECTORS along x,y,z axes UNIT VECTOR a dimensionless vector, length = 1 • Define unit vectors along the x,y,z axes: i along x; j along y; k along z |i| = |j| = |k| = 1 Figure • Example: Vector A in the x-y plane. Components Ax, Ay: A Axi + Ayj Figure Simple Example • Position vector r in the x-y plane. Components x, y: r xi + y j Figure Vector Addition Using Unit Vectors • Suppose we want to add two vectors V1 & V2 in the 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 Using Components to Add Two Vectors • Consider 2 vectors, V1 & V2. We want V = V1 + V2 • Note: The components of each vector are really onedimensional vectors, so they can be added arithmetically. We want the vector sum V = V1 + V2 “Recipe” (for adding 2 vectors using trig & components) 1. Sketch a diagram to roughly add 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: Example 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? 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? Solution NOTE!!!! Example 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. 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. Solution Another Analytic Method • Laws of Sines & Law of Cosines from trig. • 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 • 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