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chap6 - Vectors Vector Operations Components Inclined...

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Vectors 2-D Force & Motion Problems Trig Applications Relative Velocities Free Body Diagrams Vector Operations Components Inclined Planes Equilibrium
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Vector Addition Tip to tail method Parallelogram method 8 N 4 N 3 N Suppose 3 forces act on an object at the same time. F net is not 15 N because these forces aren’t working together. But they’re not completely opposing each either. So how do find F net ? The answer is to add the vectors ... not their magnitudes, but the vectors themselves. There are two basic ways to add vectors w/ pictures:
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Tip to Tail Method in-line examples Place the tail of one vector at the tip of the other. The vector sum (also called the resultant ) is shown in red. It starts where the black vector began and goes to the tip of the blue one. In these cases, the vector sum represents the net force. You can only add or subtract magnitudes when the vectors are in-line! 16 N 20 N 4 N 20 N 16 N 12 N 9 N 9 N 12 N 21 N
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Tip to Tail – 2 Vectors 5 m 2 m To add the red and blue displacement vectors first note: Vectors can only be added if they are of the same quantity—in this case, displacement. The magnitude of the resultant must be less than 7 m (5 + 2 = 7) and greater than 3 m (5 - 2 = 3). 5 m 2 m blue + red Interpretation: Walking 5 m in the direction of the blue vector and then 2 m in the direction of the red one is equivalent to walking in the direction of the black vector. The distance walked this way is the black vector’s magnitude. Place the vectors tip to tail and draw a vector from the tail of the first to the tip of the second.
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Commutative Property blue + red red + blue As with scalars quantities and ordinary numbers, the order of addition is irrelevant with vectors. Note that the resultant (black vector) is the same magnitude and direction in each case. (We’ll learn how to find the resultant’s magnitude soon.)
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Tip to Tail – 3 Vectors We can add 3 or more vectors by placing them tip to tail in any order, so long as they are of the same type (force, velocity, displacement, etc.). blue + green + red
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Parallelogram Method This time we’ll add red & blue by placing the tails together and drawing a parallelogram with dotted lines. The resultant’s tail is at the same point as the other tails. It’s tip is at the intersection of the dotted lines. Note: Opposite sides of a parallelogram are congruent.
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Comparison of Methods red + blue Tip to tail method Parallelogram method The resultant has the same magnitude and direction regardless of the method used.
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Opposite of a Vector v - v If v is 17 m/s up and to the right, then - v is 17 m/s down and to the left. The directions are opposite; the magnitudes are the same.
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Scalar Multiplication x -2 x 3 x Scalar multiplication means multiplying a vector by a real number, such as 8.6. The result is a parallel vector of a different length. If the scalar is positive, the direction doesn’t change. If it’s negative, the direction is exactly opposite.
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