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
- 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
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.
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
- Yes. Yes. Yes.
- Vertical component =
6 grid marks. 1 grid mark = 1 m/s.
Vertical component = 6 m/s.
Horizontal component = 8 m/s
- Vertical component =
30 m/s, horizontal component = 40 m/s.
- The resultant is
shown on the diagram.
It’s length and direction are measured on
the graph.
- 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.
- The horizontal
components are 5, -3 and –1 which add up to
1.
- 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:
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