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Vectors
All vector quantities have magnitude (size) and
direction.
Vectors in physics that we will use:
 Force, displacement, velocity, and
acceleration.
A vector quantity is graphically represented by a
vector.
An example is a force vector.
•
F
 The label for the vector indicates the type of
vector.
Every vector has 4 parts:
 • is the point of application sometimes
referred to as the point of concurrency.
 F is the label symbolizing the type of vector.
 The arrowhead indicates the direction of the
vector.
 The length of the vector indicates the
magnitude of the vector.
Free-Body Diagrams (Force Diagrams)
A free-body diagram sometimes called a force
diagram shows all the forces that are exerted on
a single object.
 Usually a point is used to represent the
object.
Examples of force diagrams:
T
•
Fw
T
FN
•
Fw
•
Fw
Consistency is important!
 Fw represents the weight of an object.
 T represents the tension in a rope or string
and FN the normal (perpendicular) force.
Guidelines for Free-Body Diagrams
Only forces are to appear in diagrams.
All the forces exerted on the object but none of
those objects exerting those forces are included.
Each force is represented by a force vector.
 The tail end of the arrow is placed at the
point of application of the force.
 The direction of the arrow indicates the
direction of the force.
Forces in Free-Body Diagrams
The following is a list of common forces found in
force diagrams.
Normal Force (FN)
 For each surface in contact with an object,
there is a normal force which is
perpendicular to the surfaces.
Frictional Force (Ff)
 For each surface in contact with an object,
there is a normal force which is
perpendicular to the surfaces.
When a surface is described as frictionless,
then Ff = 0 and is not included in the
diagram.
Gravitational Force (Fw)
 Be careful!
Even when an object is said to be in a state
of weightlessness, it still has weight.
Tension (T or FT)
 For each string or rope, there is a tension.
If the string or rope is slack, then T = 0.
Air Resistance (FR)
 Occurs when an object moves relative to the
air, i.e. a parachute.
 If an object is a small dense object moving
very quickly, this is usually ignored.
Net Force
 The net or resultant force is the vector sum
of all the forces acting on the object.
 The net or resultant force is the vector sum
of all the forces acting on the object.
 The net force is never drawn in the diagram!
 It is the net force that determines the motion
of an object.
More New Vocabulary
Forces are measured in units of newtons (N).
 In dynamics, specifically Newton’s 2nd law,
we will more formally define a newton.
Right now, consider 1 N ≈ 1/5 pound.
A resultant force is the one force that combines
the effects of all the forces acting on an object.
The direction in which a force acts is sometimes
referred to as bearing.
Sometimes instead of using N, E, S, and W to
designate the direction, an angular measure is
given.
000°
270°
090°
180°
You start at 000° and go
clockwise.
A Typical Force Problem
Two soccer players kick a soccer ball at exactly
the same time. One player exerts a force of 55 N
north and the second player exerts a force of
75 N east. Mathematically determine the
magnitude and direction of the resultant.
F2
F1 FR
•
θ
F2
F1
•
θ
FR
The force diagram can be drawn using the
parallelogram method (left) or the triangle method
(right).
They give the same results and sometimes one
may be favored more than the other depending
on the application.
To solve the problem mathematically, we use the
pythagorean theorem, c2 = a2 + b2, which must be
written in terms of forces.
To determine the direction (bearing), we use one
of the trigonometric functions which must be
written in terms forces.
FR = (F12 + F22)1/2
FR = ((55 N)2 + (75 N)2)1/2 = 93 N
θ=
sin-1
F1
55 N
= sin-1
= 36°
FR
93 N
FR = 93 N, 36° N of E
Graphical Solution
We will not solve the problem graphically but
rather just outline the steps.
 Choose a scale of ? cm = ? N, where the ?’s
are chosen to conveniently draw the force
(vector) diagram.
 The resultant vector can be measured in cm
from using either diagram.
 The resultant vector must have the same
point of application as the other vectors.
 According to your scale, you convert the
number of cm to the number of N (if drawing
a force diagram).
 The angle is measured with a protractor.
Velocity Vectors
A motorboat travels at 7.5 m/s W. The motorboat
heads straight across a river 120 m wide.
(a) If the river flows downstream at a rate of
3.2 m/s S, determine the resultant velocity of
the boat.
VB
• θ
VR Vr
VB = 7.5 m/s W
Vr = 3.2 m/s S
VR = (VB2 + Vr2)1/2
VR = ((7.5 m/s)2 + (3.2 m/s)2)1/2 = 8.2 m/s
θ = sin-1 Vr = sin-1 3.2 m/s = 23°
VR
8.2 m/s
VR = 8.2 m/s, 23° S of W or 8.2 m/s, 247°
(b) How long does it take the boat to reach the
opposite side of the river?
Δs = 120 m VB = 7.5 m/s W
vave
Δt =
Δs
=
Δt
120 m
7.5 m/s
= 16 s
(c) How far downstream is the boat when it
reaches the other side of the river?
vave = Δs
Δt
Δs = 3.2 m/s × 16 s = 51 m
The Resultant and Equilibrant
One force of 17 N at a bearing of 090° acts
concurrently with a force of 11 N at a bearing of
180°.
(a) Determine the magnitude and bearing of the
resultant.
•
F2
F1
θ
FR
F1 = 17 N
F2 = 11 N
FR = (F12 + F22)1/2
FR = ((17 N)2 + (11 N2)1/2 = 20.0 N
θ = cos-1 F1 = cos-1 17 N
FR
20. N
FR = 20.0 N, 122°
= 32°
(b) Determine the magnitude and bearing of the
equilibrant.
Feq
•
F2
F1
θ
Feq = 20.0 N, 302°
FR
The equilibrant is equal in magnitude and
opposite in direction (180°) to the resultant.
Fnet = FR + Feq = 0 N
Resolution of Forces
Harry pushes a lawn mower along a level ground.
The handle of the lawn mower makes an angle of
20.0° with the ground as Harry pushes with a
force of 78.0 N.
(a) Determine the useful component of the force.
FH
• θ
FR FV
FR = 78.0 N
θ = 20.0°
.
cos θ =
FH
FR
FH = FR × cos θ = 78.0 N × 0.940 = 73.3 N
(b) Determine the wasted component of the force.
sin θ =
FV
FR
FV = FR × sin θ = 78.0 N × 0.940 = 26.7 N
The force diagram shows the lawn mower being
pushed from right to left. It could also be shown
being pushed from left to right.
 The results would be the same.
 If a particular direction was given, it would
have to appear that way in the problem.
The wasted component would be FV because that
component of the force is trying to push the lawn
mower straight into the ground.
Wrap Up Questions
Can the magnitude of a vector have a negative
value? Justify your answer.
Remember that vectors are measurable
quantities that have a direction as well as a
magnitude. Such quantities as forces,
displacements, velocities, and accelerations can
not have a magnitude below zero but can point in
a negative direction.
Two forces, F1 = 12.0 N and F2 = 4.0 N are
concurrent (acting on the same point).
(a) What is the minimum resultant?
The minimum resultant would be 8.0 N if the
angle between them was 180°.
(b) What is the maximum resultant?
The maximum resultant would be 16.0 N if the
angle between them was 0°.