Chapter 4 Vectors

1. Chapter 4 Vectors The Cardinal Directions

2. Vectors • An arrow-tipped line segment used to represent different quantities. • Length represents magnitude. • Arrow head represents direction.

3. Vector Addition in 1 - Dimension • When vectors point in the same direction we add them just as we would add any two numbers.

4. Vector Addition in 1 - Dimension • When vectors point in opposite directions we subtract them just as we would with any two numbers.

6. Vector Addition in 2-Dimensions • Vectors in 2-dim are added by placing the tail of one to the head of another.

7. Remember This?

8. Addition of Several Vectors • The order of addition is not important. • R is called the resultant.

9. Independence of Vector Quantities • Perpendicular vectors can be treated independently of each other.

11. Analytical Method of Vector Addition • The sum of any two vectors can be determined using trigonometry.

12. Adding Perpendicular Vectors

13. Angle θ is = • 25 deg • 14 deg • 35 deg • 45 deg

14. Angle θ is = • 25 deg b) 14 deg c) 35 deg d) 45 deg

15. Vector Components • We can take two vectors and replace them with a single vector that has the same effect. This is vector addition. • We can start with a single vector and think of it as a resultant of two perpendicular vectors called components. • This process is called vector resolution.

16. Example

17. Example 2

18. Problem Solving Strategy • In resolving vectors choose the most convenient axis according to the specifics of the problem. • Choose the axis that simplifies the solution. • Axis may be up-down, left-right, east-west or north-south. • Be sure to specify the positive direction for each.

19. Adding Vectors at Any Angle • Vector resolution is the method used. • Resolve all vectors into x and y components. • Add all x’s and all y’s together. • Use xtot and ytot to create a right triangle. • Use Pythagorean formula to calculate resultant and trig to find angle.

22. R is = ? • 15 N • 12 N • 20 N • 11N

23. R is = ? • 15 N • 12 N • 20 N • 11N

24. Θ is = ? • 53 deg • 35 deg • 25 deg • 45 deg

25. Θ is = ? • 53 deg • 35 deg • 25 deg • 45 deg

26. Applications of Vectors Vectors can be used to represent: • displacement • velocity • acceleration • force

27. Equilibrium • When the net force is zero, the object is in equilibrium. • When the vector sum of the forces is not zero, a force can be applied that will produce equilibrium. This force is called the equilibrant. • It is equal in magnitude but opposite in direction to the resultant.

28. 3 Forces in Equilibrium: • produce a net force. • produce a triangle for a vector diagram. • are called an equilibrant. • produce an acceleration.

29. 3 Forces in Equilibrium: • produce a net force. • produce a triangle for a vector diagram. • are called an equilibrant. • produce an acceleration.

30. Gravitational Force and Inclined Planes • Gravitational force always points towards center of Earth. • This is weight. • Choose one axis parallel to the plane and the other perpendicular to it.

31. Formulas • R2 = A2 + B2 – 2AB cos Θ • Ax = A cos Θ • AY = A sin Θ • A = Ax + AY