LESSON 1.6 - VECTORS
Objectives
On completion of this lesson, you should be able to:
Define: Vectors,
Scalars.
Explain: What happens when two vectors
combine - or have a combined effect on an object.
Describe: Graphical methods used tp
represent and combine vectors.
Calculate: The resultant of two vectors
at right angles to each other by using the method of
completing the rectangle. Calculate the resultant of two
vectors that are not at right angles to each other by
completing the parallelogram..
CHOICE
OF ACTIVITIES
- Throw paper darts at
a target. Place a fan blowing across the path of
the darts and see how the darts need to be aimed
to hit the target.
- Use projectiles from
a Nerf gun instead of paper darts
- Attach a sail to a
model car and use a fan to sail the car across a
table. (Click
here for instructions)
Vector quantity: A
quantity that has magnitude and direction.
Vector: A line,
arrow or set of coordinates that represents a quantity
and its direction.
Resultant: The
combined effect of two or more vectors.
Displacement: When
the position of an object is changed, the displacement is
the shortest distance between its initial and final
positions.
Velocity:
Displacement (in a particular direction) in a specified
period of time.
Sin q : The Greek letter q is commonly used to denote an angle.
Sin q is a trigonometric value that that
in a right-angle triangle equals the length of the side
opposite to the angle q divided by the length of the
hypotenuse.
Cos q : The cosine of q is equal to the length of the side adjacent
to q divided by the length of the
hypotenuse.
Vectors and scalars
A scalar is a quantity that has magnitude only. Examples
of scalars are mass, time and energy.
Vectors have magnitude and direction. Common vectors are
force, displacement, velocity and acceleration.
Drawing vectors
Vectors are often represented graphically as a line with
an arrowhead indicating the direction of the quantity.
The length of the line indicates magnitude of the
quantity.
Vector addition
If two or more vectors influence an object or act on a
point, the vectors can be added to determine the net
effect or resultant of the vectors. This lesson deals
primarily with graphical methods of adding two vectors.
If the vectors are at right angles to each other, the
graphical method involves completing the rectangle. If
the vectors are at a different angle, the technique is
known as completing the parallelogram. Both methods
effectively involve moving the starting point of the
second vector to the end of the first.
Analytical method
Vectors can be added graphically or analytically.
Analytical addition usually requires the use of
trigonometric functions.
Example 1.6.1
Addition of 2 vectors by completing the rectangle
Two forces at right angles
to each other act on an object. Force 1 has a magnitude
of 4 N and Force 2 has a magnitude of 3 N. Determine the
resultant of the two forces.
Solution
The diagram below represents the two forces acting on the
object located at the point A.
The rectangle is completed in the second diagram by
moving the arrow representing Force 2 to start at the end
of Force 1. The magnitude of the resultant is indicated
by the length of the line connecting the starting point
of the arrow representing Force 1 to the end of the arrow
representing Force 2. The direction of this line
indicates the direction of the resultant.
Example 1.6.2
Addition of 2 vectors by completing the parallelogram
Two forces act on an
object with an angle of 53.13º between them. Force 1 has
a magnitude of 4 N and Force 2 has a magnitude of 5 N.
Determine the resultant of the two forces.
Solution
The diagram above
represents the two forces acting on the object.
The gray lines are used to
complete the parallelogram and the resultant is
determined by drawing the line as indicated. By measuring
the length of the green line, the resultant is estimated
to be ….. N.
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Review
Questions
- What is the
difference between speed and velocity?
- If an airplane is
flying in a crosswind that blows at 90º to the
plane’s direction through the air, how does
the wind affect the path of the plane relative to
the ground?
- On the diagram below,
one arrow represents the speed of the plane
relative to the air (it’s airspeed). The
second arrow represents the velocity of the wind
relative to the ground. Draw an arrow on the
diagram that represents the velocity of the plane
relative to the ground.
Qustion
4
Use the diagram below to determine the resultant
of 2 forces acting at right angles to each other.
In terms of the scale used, Force 1 has a
magnitude of 4 Newtons and has a length of 4
graduations on the scale. Force 2 has a magnitude
of 7 N and has a length of 7 graduations. Use a
ruler to measure the length of the resultant and
estimate its magnitude.
Question
5
Use the diagram below to determine the resultant
of 2 forces acting at less than 90º to each
other. Force 1 has a magnitude of 5 Newtons and
force 2 has a magnitude of 7 N. Complete the
parallelogram, use a ruler to measure the length
of the resultant and estimate the magnitude of
the resultant.
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MINI
LAB
CHOICE OF
ACTIVITIES
- Throw paper darts at
a target. Place a fan blowing across the path of
the darts and see how the darts need to be aimed
to hit the target.
- Use projectiles from
a Nerf gun instead of paper darts
- Attach a sail to a
model car and use a fan to sail the car across a
table.
EXPERIMENT
#1 PAPER DARTS
Purpose:
To illustrate the effect of a crosswind on the path of an
airplane.
Equipment :
A piece of paper and a fan
Activity
Aim paper darts at a target while a fan blows air across
the path of the dart. The darts need to be aimed in the
direction of the resultant of the two vectors: The path
of the dart through the air and the movement of the air
relative to the target.
Alternate
activity: Nerf darts
It may be less frustrating to use a projectile that is
easier to aim and is less affected by the fan.
EXPERIMENT
#2 SAILING
Purpose: To show how the forces on the sail and the keel
of a sailboat influence the speed and direction of a
sailboat.
Equipment :
A toy car fitted with a mast and a paper "sail".
A fan
Activity
Use a simple toy car that has no steering mechanism and
can move freely backwards and forwards.
Attach a thin dowel or strip of wood to the top of the
car to act as a mast. It will probably be necessary to
hold the mast in position with some string rigging.
Attach a piece of paper to the mast at an angle to the
direction that the car can move in. Use a piece of thin
string or cotton to hold the sail in position if
necessary.
Hold the fan so that the wind from the fan blows from the
side of the car. The sail should propel the car across
the wind from the fan.
Identify the force vectors acting on the car and explain
how it can be made to move upwind if the sail is
correctly angled.
HANDS-ON
HOMEWORK
Select one or more of the recommended
activities for Lesson 1.7, collect the items needed and test the
procedure before demonstrating the activity during the
next theory lesson.
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Lesson
1.6 Vectors
- Speed is a scalar
quantity. No direction needs to be specified.
Velocity is speed in a given direction. Velocity
is a vector quantity.
- The path that the
plane follows relative to the ground is a
combination of the air speed of the plane and the
wind speed.
-
4. On the diagram 55mm = 7
N. The length of the resultant is 63mm.
This is equivalent to 63/55 x 7 = 8 N.
We can measure the angle or calculate it to be tan-1
(4/7) = 29.7º.
5 On the diagram 58mm = 7
N. The length of the resultant is 90mm.
This is equivalent to 90/58 x 7 = 10.86 N.
We can measure the angle or calculate it to be tan-1
(2.7/7) = 21.1º.
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