Vectors and Scalars

Scalars 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

Vectors A vector quantity is a quantity that has both magnitude and a direction in space Examples of Vector Quantities: Displacement Velocity (speed in a given direction) Acceleration (rate of which something speeds up or slows down in a direction) Force

Vector Diagrams Vector diagrams are shown using an arrow Because these two vectors are the same length but are in opposite directions we have u and –u. Vector diagrams are shown using an arrow The length of the arrow represents its magnitude The direction of the arrow shows its direction u v -u Because these two vectors (u and v) are going in the same direction and have the same magnitude (length) then the vectors are the same u=v.

Resultant of Two Vectors The resultant is the sum or the combined effect of two vector quantities Vectors in the same direction: 6 N 4 N = 10 N 6 m = 10 m 4 m Vectors in opposite directions: 6 m s-1 10 m s-1 = 4 m s-1 6 N 10 N = 4 N

The Parallelogram Law The Triangle Law When two vectors are joined tail to tail Complete the parallelogram The resultant is found by drawing the diagonal The Triangle Law When two vectors are joined head to tail Draw the resultant vector by completing the triangle

Adding Vectors “Tip to Tail” method Suppose that we have the following two vectors Graphically represent u+v. v v u u u+v Resultant vector

Graphically represent 2u+3v

Represent v-u Is that the same thing as v+(-u)? v u v-u -u

Sketch u+v, 2(u+v), u-v u v

Problem: Resultant of 2 Vectors Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? Solution: Complete the parallelogram (rectangle) The diagonal of the parallelogram ac represents the resultant force The magnitude of the resultant is found using Pythagorean Theorem on the triangle abc 12 N a d θ 5 N 13 N 5 b c 12 Resultant displacement is 13 N 67º with the 5 N force. Our text book often refers to direction based off of due north going clockwise.

Problem: Resultant of 3 Vectors Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. Solution: Find the resultant of the two 5 N forces first (do right angles first) 5 d c 7.07 N Now find the resultant of the 10 N and 7.07 N forces 5 N 5 90º The 2 forces are in a straight line (45º + 135º = 180º) and in opposite directions 45º θ a 2.93 N b 5 N 135º So, Resultant = 10 N – 7.07 N = 2.93 N in the direction of the 10 N force 10 N

Recap What is a scalar quantity? Give 2 examples What is a vector quantity? How are vectors represented? What is the resultant of 2 vector quantities? What is the triangle law? What is the parallelogram law?

Resolving a Vector Into Perpendicular Components When resolving a vector into components we are doing the opposite to finding the resultant We usually resolve a vector into components that are perpendicular to each other Here a vector v is resolved into an x component and a y component v y x

Practical Applications Here we see a table being pulled by a force of 50 N at a 30º angle to the horizontal When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N 50 N y=25 N 30º x=43.3 N We can see that it would be more efficient to pull the table with a horizontal force of 50 N

Calculating the Magnitude of the Perpendicular Components If a vector of magnitude v and makes an angle θ with the horizontal then the magnitude of the components are: x = v Cos θ y = v Sin θ v y=v Sin θ y θ x=v Cos θ x Proof:

Problem: Calculating the magnitude of perpendicular components 2002 HL Sample Paper Section B Q5 (a) A force of 15 N acts on a box as shown. What is the horizontal component of the force? Solution: 12.99 N 15 N Component Vertical 60º Horizontal Component 7.5 N

2003 HL Section B Q6 Solution: A person in a wheelchair is moving up a ramp at constant speed. Their total weight is 900 N. The ramp makes an angle of 10º with the horizontal. Calculate the force required to keep the wheelchair moving at constant speed up the ramp. (You may ignore the effects of friction). Solution: If the wheelchair is moving at constant speed (no acceleration), then the force that moves it up the ramp must be the same as the component of it’s weight parallel to the ramp. Complete the parallelogram. Component of weight parallel to ramp: 156.28 N 10º 80º 10º Component of weight perpendicular to ramp: 886.33 N 900 N

Summary If a vector of magnitude v has two perpendicular components x and y, and v makes and angle θ with the x component then the magnitude of the components are: x= v Cos θ y= v Sin θ v y=v Sin θ y θ x=v Cosθ