Vectors

1
Vectors and Scalars
LO Understand how to resolve vectors
graphically and via calculation
BAT Define Scalar and vector quantities giving
examples (D) Resolve two vectors graphically
(C) Resolve two vectors via calculation (A/B)
2
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

3
Vectors

• A vector quantity is a quantity that has both
  magnitude and a direction in space

• Examples of Vector Quantities
• Displacement
• Velocity
• Acceleration
• Force

4
Vector Diagrams

• Vector diagrams are shown using an arrow
• The length of the arrow represents its magnitude
• The direction of the arrow shows its direction

5
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
6
The Parallelogram 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

7
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
  Pythagoras Theorem on the triangle abc

12 N
a
d
?
13 N
5 N
5
b
c
12

• Resultant displacement is 13 N 67º with the 5 N
  force

8
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
c
d
7.07 N
5

• Now find the resultant of the 10 N and 7.07 N
  forces

5 N
90º
45º
?

• The 2 forces are in a straight line (45º 135º
  180º) and in opposite directions

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
9
Recap

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

10
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
11
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
y25 N
30º
x43.3 N

• We can see that it would be more efficient to
  pull the table with a horizontal force of 50 N

12
Calculating the Magnitude of the Perpendicular
Components

• If a vector of magnitude v and makes an angle ?
  with the horizontal then the magnitude of the
  components are
• x v Cos ?
• y v Sin ?

v
yv Sin ?
y
?
xv Cos ?
x

• Proof

13
Problem Calculating the magnitude of
perpendicular components

• A force of 15 N acts on a box as shown. What is
  the horizontal
• component of the force?

Solution
15 N
12.99 N
Vertical Component
60º
Horizontal Component
7.5 N
14

• 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
its 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
15
Homework

• Read text book pages 90-96 and complete the
  question worksheet.
• Attempt all the questions!
• Show your workings!
• Remember units!
• Due Friday 14th September!