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Chapter 3

Chapter 3. Vectors. Vectors and Scalars , Addition of vectors Subtraction of vectors. Physics deals with many quantities that have both Magnitude Direction VECTORS !!!!!. y. . x. r. Scalar. A scalar quantity is a quantity that has magnitude only and has no direction in space.

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Chapter 3

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  1. Chapter 3 Vectors

  2. Vectors and Scalars, • Addition of vectors • Subtraction of vectors

  3. Physics deals with many quantities that have both • Magnitude • Direction • VECTORS !!!!! y  x r

  4. Scalar • A scalar quantity is a quantity that has magnitude only and has no direction in space • Examples of Scalar Quantities: • Length • Area • Volume • Time • Mass

  5. Vector • A vector quantity is a quantity that has both magnitude and a direction in space • Examples of Vector Quantities: • Displacement • Velocity • Acceleration • Force

  6. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Blue and greenvectors have same direction but different magnitude. Blue and purplevectors have same magnitude and direction so they are equal. Blue and orange vectors have same magnitude but different direction. Two vectors are equal if they have the same direction and magnitude (length).

  7. Examples A = 20 m/s at 35° NE B = 120 lb at 60° SE C = 5.8 mph/s west

  8. Example • The directionof the vector is 55° North of East • Themagnitudeof the vector is 2.3.

  9. Try Again • Direction: • Magnitude: 43° East of South 3

  10. Try Again It is also possible to describe this vector's direction as 47 South of East. Why?

  11. Vector Addition vectors may be added graphically or analytically Triangle (Head-to-Tail) Method 1. Draw the first vector with the proper length and orientation. 2. Draw the second vector with the proper length and orientation originating from the head of the first vector. 3. The resultant vector is the vector originating at the tail of the first vector and terminating at the head of the second vector. 4. Measure the length and orientation angle of the resultant.

  12. Adding vectors in same direction: Example:Travel 8 km East on day 1, 6 km East on day 2. Displacement = 8 km + 6 km = 14 km East Example:Travel 8 km East on day 1, 6 km West on day 2. Displacement = 8 km - 6 km = 2 km East “Resultant” = Displacement

  13. Adding more than two vectors graphically

  14. Subtraction of Vectors -graphically

  15. Parallelogram (Tail-to-Tail) Method 1. Draw both vectors with proper length and orientation originating from the same point. 2. Complete a parallelogram using the two vectors as two of the sides. 3. Draw the resultant vector as the diagonal originating from the tails. 4. Measure the length and angle of the resultant vector.

  16. Components of Force: y x

  17. Resolving a Vector Into Components +y The horizontal, or x-component, of A is found by Ax = A cos q. A Ay q Ax The vertical, or y-component, of A is found by Ay = A sin q +x By the Pythagorean Theorem, Ax2 + Ay2 = A2 Every vector can be resolved using these formulas, such that A is the magnitude of A, and q is the angle the vector makes with the x-axis.

  18. Analytical Method of Vector Addition 1.Find the x- and y-components of each vector. Ax = A cosq Ay = A sin q By = B sin q Bx = B cosq Cx = C cosq Cy = C sin q Rx Ry 2.Sum the x-components. This is the x-component of the resultant. 3.Sum the y-components. This is the y-component of the resultant. 4.Use the Pythagorean Theorem to find the magnitude of the resultant vector. Rx2 + Ry2 = R2

  19. 5. Find the reference angle by taking the inverse tangent of the absolute value of the y-component divided by the x-component. q = Tan-1Ry/Rx 6. Use the “signs” of Rx and Ry to determine the quadrant. NW NE (-,+) (+,+) (-,-) (-,+) SW SE

  20. 25 m Step 1 • Sketch the given vector with the tail located at the origin of an x-y coordinate system. (Ex. 25 m at an angle of 36º) 36º

  21. 25 m Step 2 • Draw a line segment from the tip of the vector perpendicular to the x-axis 36º Notice, you now have a right triangle with a known hypotenuse and known angle measurements

  22. 25 m Step 3 • Replace the perpendicular sides of the right triangle with vectors drawn tip – to - tail

  23. 25 m Step 4 • Use sine and cosine functions to find the horizontal and vertical components of the given vector. Ry 36º Rx Cos(36) = Rx/25 Rx = 25cos(36) Rx = 20.2 m Sin(36) = Ry/25 Ry= 25sin(36) Ry = 14.7 m

  24. Example: 6 N 5 N 135° 45° 30° Ry R R = (0.09)2 + (6.74)2 = 6.74 N Rx  = arctan 6.74/0.09 = 89.2°

  25. Problem 1

  26. Problem 2

  27. Problem 3

  28. Problem 4

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