Chapter Two Notes:
Mechanical Equilibrium
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A force is a push or a pull:
◦ A force is necessary to cause a change in the state
of motion of an object.
◦ Net Force– The net force is the sum of all forces
acting on an object.
 You pull horizontally on an object with a force of 10
pounds.
 A friend pulls on the same object with a force of 5
pounds
 If the friend pulls in the same direction, the net force is
15 pounds.
 If the friend pulls in the opposite direction, the net
force is 5 pounds in the direction you are pulling.
 Hold an object in your hand, and the net force is zero.
The earth pulls down on the object, while you push up
to hold it there.
◦ Tension and Weight–
 Suspend an object from a rope, and the force the rope
exerts on the object to keep it from falling is called
tension. The pull of earth on the object is called
weight.
 The net force on the object is zero…
◦ Force Vectors Forces are represented by arrows, in the direction of
the force. The length of the arrow represents the
amount (magnitude) of the force.
 A vector is the name we give to these arrows. A
vector quantity is a quantity that needs both a
magnitude and a direction to be complete.
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A scalar quantity, on the other hand can be described by only a
magnitude, and has no direction.
We will come back to vectors in chapter 5.
Mechanical Equilibrium is a state where no physical changes
occur. In other words, the net force is zero.
Equilibrium Rule - You can express the equilibrium rule
mathematically!
ΣF = 0
The symbol Σ stands for “sum of” and F stands for “Force”. (This
represents the fact that all the forces acting on an object add
vectorially to zero.)
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If you have an object at rest on a desk, one force
is gravity pulling down on the object. Since the
object is in equilibrium, the net force must be
zero, and there must be another force acting on
the object to produce this state. That force must
be opposite the force of gravity, which is acting
down.
That force must be upward, and must be
provided by the desk. We call this the support
force.
For an object at rest on a horizontal surface, the
support force must equal the object’s weight.
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When an object
isn’t moving it’s in
equilibrium. The forces on it add up to zero.
But the state of rest is only one for of
equilibrium. An object moving in a straight
line path at a constant speed is also in a state
of equilibrium.
Once in motion, if there is no net force to
change the state of motion, it’s in equilibrium.
Object at rest are said to be in static
equilibrium; objects moving at constant speed
in a straight-line path are said to be in dynamic
equilibrium.
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There are several ways to find the sum of two or
more vectors.
 (Head-Tail, Components, Parallelogram Method)
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Addition of vectors by Head to Tail method
(Graphical Method)
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Head to Tail method or graphical method is one
of the easiest method used to find the resultant
vector of two of more than two vectors.
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(not in our book!)
DETAILS OF METHOD
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Consider two vectors and
directions shown below:
acting in the
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In order to get their resultant vector by
head to tail method we must follow the
following steps:
 STEP # 1
Choose a suitable scale for the vectors so
that they can be plotted on the paper.
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STEP # 2
Draw representative line
of vector . Draw
representative line
of vector
such that the tail
of
coincides with the head of vector .
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STEP # 3
Join 'O' and 'B'.
represents resultant vector of given
vectors and i.e.
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STEP # 4
Measure the length of line segment and multiply
it with the scale chosen initially to get the
magnitude of resultant vector
.
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STEP # 5
The direction of the resultant vector is directed
from the tail of vector
to the head of vector
.
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-----------------////---------------- NEXT METHOD:
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(The Books method)
Parallelogram Method to Calculate Resultant
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Before tackling the parallelogram method for solving
resultant vectors, you should be comfortable with the
following topics:
SOHCAHTOA (basic sine, cosine, tangent )
law of cosines
law of sines
the following properties of parallelograms
◦ opposite sides of parallelograms are congruent
◦ opposite angles of parallelograms are congruent
To best understand how the parallelogram method
works, lets examine the two vectors below. The
vectors have magnitudes of 17 and 28 and the angle
between them is 66°. Our goal is to use the
parallelogram method to determine the magnitude of
the resultant.
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Step 1) Draw a parallelogram based on the two vectors that
you already have. These vectors will be two sides of the
parallelogram (not the opposite sides since they have the
angle between them)
Step 2) We now have a parallelogram and know two angles
(opposite angles of parallelograms are congruent). We can
also figure out the other pair of angles since the other pair
are congruent and all four angles must add up to 360. ???
Determine other Angles
Step 4) Draw the parallelograms diagonal. This diagonal is the
resultant vector
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Step 1) Draw a parallelogram based on the two vectors that
you already have. These vectors will be two sides of the
parallelogram (not the opposite sides since they have the
angle between them)
Step 2) We now have a parallelogram and know two angles
(opposite angles of parallelograms are congruent). We can
also figure out the other pair of angles since the other pair
are congruent and all four angles must add up to 360. ???
Determine other Angles
Step 4) Draw the parallelograms diagonal. This diagonal is the
resultant vector
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Step 1) Draw a parallelogram based on the two vectors that
you already have. These vectors will be two sides of the
parallelogram (not the opposite sides since they have the
angle between them)
Step 2) We now have a parallelogram and know two angles
(opposite angles of parallelograms are congruent). We can
also figure out the other pair of angles since the other pair
are congruent and all four angles must add up to 360. ???
Step 3) Determine other Angles
Step 4) Draw the parallelograms diagonal. This diagonal is the
resultant vector
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Step 1) Draw a parallelogram based on the two vectors that
you already have. These vectors will be two sides of the
parallelogram (not the opposite sides since they have the
angle between them)
Step 2) We now have a parallelogram and know two angles
(opposite angles of parallelograms are congruent). We can
also figure out the other pair of angles since the other pair
are congruent and all four angles must add up to 360. ???
Step 3) Determine other Angles
Step 4) Draw the parallelograms diagonal. This diagonal is the
resultant vector
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To Find the Length of Diagonal
Use the law of cosines to determine the
length of the resultant