Module 2

Lesson 1.7
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. Concepts
. Examples
. Review
. Answers
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Lesson 1.5
Lesson 1.6
Lesson 1.8
Lab 1.5
Lab 1.6
Lab 1.7
Lab 1.8
Project 2
Class Test 2

Module 1

Module 3

Module 4

LESSON 1.7 - COMPONENTS OF VECTORS

Objectives
On completion of this lesson, you should be able to:
(For vectors acting in the same plane - or in two dimensions):
Explain: How a vector can be regarded as consisting of a number of component vectors.
Describe the use of components to determine the resultant of a number of vectors acting on a point..
Calculate: The resultant of two or more vectors acting on the same point in the same plane by combining their components in directions at right angles to each other.

ACTIVITY
Two people lift a weight supported at the center of a rope by pulling at each end of the rope.
(a) If each person holds the rope at roughly the same height above the floor, see how increasing the distance between the points at which the people pull on the rope increases the amount of force they need to apply in order to lift the weight.
(b) With two people pulling at opposite ends of the rope, lower the ends of the rope until it becomes impossible to lift the weight.

Components of vectors
The effect of a vector must very often be determined in a direction that differs from the direction of the vector. Under these circumstances, a portion of the vector has influence in that direction. This portion is known as the component of the vector in that direction.

Resolving vectors into components
In dealing with vectors in two dimensions, it is common to use x and y axes as references for the direction of a vector. We often refer to the direction of a vector with reference to the vertical or with reference to the horizontal. We are often interested in the components of a particular vector in these directions and refer to the vertical and horizontal components of a vector. The two mutually perpendicular components can be added to produce the initial vector.

Example 1.7.1 Resolving a vector into components
The arrow on the diagram below represents a force vector with a magnitude of 7.07 N.
What are the magnitudes of the vertical and horizontal components of this force?

Solution
From the diagram below, we can see that the force of 7.07 N acting at 45º to the horizontal could be the resultant of two mutually perpendicular vectors, each with a magnitude of 5 N .

Example 1.7.2 Resultant of 3 vectors from components
Determine the resultant of the vectors shown on the diagram below. Find the vertical and horizontal components of the vectors. Combine the vertical components into a single vertical vector. Repeat this for the horizontal components and determine the resultant of these two mutually perpendicular vectors.

Solution
The vertical component of Vector 1 is 4 and the vertical component of Vector 2
is –3. These combine to give a net vertical component of 1.

The horizontal component of Vector 1 is 4 and the horizontal component of Vector 2
is –1. These combine to give a net horizontal component of 3.

The green arrow in the diagram below represents the resultant of two mutually perpendicular vectors: A horizontal vector with a magnitude of 3 and a vertical vector with a magnitude of 1.

The green arrow represents the resultant of Vector 1 plus Vector 2.

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Review Questions

1. If a force acts at 25º to the vertical, can a portion of the force act in the vertical direction?
   Is this the vertical component of the force?
   Does the force acting at 25º to the vertical also have a horizontal component?
   
   
   
   
   
   Question 2
   The diagram above represents the velocity of an object moving at 10 m/s in a direction 36.87º to the horizontal. Use the diagram to measure
   a) The vertical component of its velocity and:
   b) The horizontal component of its velocity.

Question 3

1. 
   If a cannonball is fired at an angle of 36.87º to the horizontal with a velocity of 50 m/s, use the vector diagram below to estimate
   a) The vertical component of its velocity? and
   b) The horizontal component of its velocity?

Question 4
Use Diagram 4.2 to determine the resultant of the vectors acting on point A in Diagram 4.1 by moving the vectors such that Vector 2 starts at the end of Vector 1 etc.

1. 
   
   
   
   
   Question 5
   What is the sum of the vertical components of the forces shown in Diagram 5.1 below?
   
   
   
   
   
   Question 6
   What is the sum of the horizontal components of the forces shown in Diagram 5.1 above?
   
   
   Question 7
   Determine the vertical and horizontal components of the 4 forces shown acting on point A in Diagram 7.1 below.
   Use Diagram 7.2 to indicate the sum of the vertical components and the sum of the horizontal components and to estimate the resultant of the 4 forces.

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MINI LAB
Two people lift a weight supported at the center of a rope by pulling at each end of the rope.
(a) If each person holds the rope at roughly the same height above the floor, see how increasing the distance between the points at which the people pull on the rope increases the amount of force they need to apply in order to lift the weight.
(b) With two people pulling at opposite ends of the rope, lower the ends of the rope until it becomes impossible to lift the weight.

You will need:
2 volunteers
Rope: ~8 meters (25ft) long
A heavy object with a handle or opening through which the rope can be threaded.
(A 1-gallon milk container or oil can filled with water could be used.)

Procedure:
Thread the rope through the handle of the container
With the 2 volunteers standing roughly 4 meters apart - each person 2 meters from the heavy object, ask the volunteers to lift the weight by pulling the rope in a direction away from each other.
Increase the distance between the volunteers by 2 meters and repeat the lifting procedure.
Keep increasing the distance between the volunteers and ask them to explain why it becomes more difficult to lift the object as their distance apart increases.

(b) With the volunteers holding each end of the rope, as them to lower the point from which they pull the rope to lift the object.
Ask them to explain why it becomes more difficult to lift the object as the point at which they are pulling on the rope gets nearer to the floor.

Question:
Why is it impossible to lift the weight when the ends of the rope are at the same height above the floor as the point at which the rope acts on the weight?

HANDS-ON HOMEWORK
Select one or more of the recommended activities for Lesson 1.8, collect the items needed and test the procedure before demonstrating the activity during the next theory lesson.

Lesson 1.7 Components of Vectors

1. Yes. Yes. Yes.
2. Vertical component = 6 grid marks. 1 grid mark = 1 m/s.
   Vertical component = 6 m/s.
   Horizontal component = 8 m/s
3. Vertical component = 30 m/s, horizontal component = 40 m/s.
4. The resultant is shown on the diagram.
   It’s length and direction are measured on the graph.
   
5. The blue vector has a vertical component of 2, the red vector has a vertical component of 1 and the green vector has a vertical component of –2. The sum of the vertical components is thus 1.
6. The horizontal components are 5, -3 and –1 which add up to 1.
7. The vertical components are: 2, 2, -2 and –4. These add up to –2. The horizontal components are 3, -1, -5 an 4. These add up to 1.
   We can estimate the resultant by measuring its length and direction the diagram below: