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Kinematics & Dynamics in 2 & 3 Dimensions; Vectors First, a review of some Math Topics in Ch. 1. Then, some Physics Topics in Ch. 4! Vectors: Some Topics in Ch. 1, Section 7 General Discussion. Vector A quantity with magnitude & direction. Scalar A quantity with magnitude only. • Here, we’ll mainly deal with Displacement & Velocity. But, our discussion is valid for any vector! • The Ch. 1 vector review has a lot of math! It requires a detailed knowledge of trigonometry! Problem Solving • A diagram or sketch is helpful & vital! • I don’t see how it is possible to solve a vector problem without a diagram! Coordinate Systems Rectangular (Cartesian) Coordinates • “Standard” coordinate axes. • A point in the plane is (x,y) • If its convenient, we could reverse + & - - ,+ +,+ -,- +,- A “Standard Set” of xy Coordinate Axes Vector & Scalar Quantities • Vector Quantity with magnitude & direction. • Scalar Quantity with magnitude only. Equality of Two Vectors •Consider 2 vectors, A & B A = B means A & B have the same magnitude & direction. Vector Addition, Graphical Method • Addition of Scalars: We use “Normal” arithmetic! • Addition of Vectors: Not so simple! • Vectors in the same direction: – We can also use simple arithmetic • Example 1: Suppose we travel 8 km East on day 1 & 6 km East on day 2. Displacement = 8 km + 6 km = 14 km East • Example 2: Suppose we travel 8 km East on day 1 & 6 km West on day 2. Displacement = 8 km - 6 km = 2 km East “Resultant” = Displacement Adding Vectors in the Same Direction: Graphical Method of Vector Addition • For 2 vectors NOT along the gsame line, adding is more complicated: • Example: D1 = 10 km East D2 = 5 km North. What is the resultant (final) displacement? • 2 Methods of Vector Addition: –Graphical (2 methods of this also!) –Analytical (TRIGONOMETRY) Graphical Method of Adding Vectors “Recipe” • Draw the first vector. • Draw the second vector starting at the tip of the first vector • Continue to draw vectors “tip-to-tail” • The sum is drawn from the tail of the first vector to the tip of the last vector Example: • Example: 2 vectors NOT along the same line. Figure! D1 = 10 km E, D2 = 5 km N. Resultant = DR = D1 + D2 = ? • In this special case ONLY, D1 is perpendicular to D2. • So, we can use the Pythagorean Theorem. DR = 11.2 km Note! DR < D1 + D2 (scalar addition) D1 = 10 km E, D2 = 5 km N. Resultant = DR = D1 + D2 = ? DR = 11.2 km Note! DR < D1 + D2 The Graphical Method of Addition • Plot the vectors to scale, as in the figure. • Then measure DR & θ. Results in DR = 11.2 km, θ = 27º N of E • This example illustrates general rules of graphical addition, which is also called the “Tail to Tip” Method. • Consider R = A + B (See figure!). Graphical Addition Recipe 1. Draw A & B to scale. 2. Place the tail of B at the tip of A 3. Draw an arrow from the tail of A to the tip of B 4. This arrow is the Resultant R Measure its length & the angle with the x-axis. Order Isn’t Important! Adding vectors in the opposite order gives the same result: In the example in the figure, DR = D1 + D2 = D2 + D1 Graphical Method of Vector Addition • Adding (3 or more) vectors: V = V1 + V2 + V3 • Even if the vectors are not at right angles, they can be added graphically using the tail-to-tip method. Parallelogram Method • A 2nd Graphical Method of Adding Vectors (equivalent to the tail-to-tip method, of course!) V = V1 + V2 1. Draw V1 & V2 to scale from a common origin. 2. Construct a parallelogram using V1 & V2 as 2 of the 4 sides. 3. Resultant V = Diagonal of the Parallelogram from a Common Origin (measure length & the angle it makes with the x axis) See Figure Next Page! Parallelogram Method A common error! Mathematically, we can move vectors around (preserving their magnitudes & directions) Subtraction of Vectors • First, Define The Negative of a Vector: - V vector with the same magnitude (size) as V but with opposite direction. Math: V + (- V) 0 • Then add the negative vector. • For 2 vectors, V1 & V2: Subtracting Vectors To subtract one vector from another, add the first vector to the negative of the 2nd vector, as in the figure below: Multiplication by a Scalar • A vector V can be multiplied by a scalar c V' = cV V' vector with magnitude cV & same direction as V. • If c is negative, the resultant is in the opposite direction. Example • Consider a 2 part car trip: • Displacement A = 20 km due North. • Displacement B = 35 km 60º West of North. • Find (graphically) resultant displacement vector R (magnitude & direction). R = A + B. See figure below. Use ruler & protractor to find the length of R & the angle β. Answers: Length = 48.2 km β = 38.9º