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Chapter 2
Mechanical Equilibrium
I. Force (2.1)
A. force– is a push or pull
1. A force is needed to change an
object’s state of motion
2. State of motion may be one of two
things
a. At rest
b. Moving uniformly
along a straight-line path.
B. Net force
1. Usually more than one force is acting on an
object
2. combination of all forces acting on object
is called net force.
3. The net force on an
object changes its
motion
4. Can add or subtract to get resultant net force
5. If forces acting on object equal zero then we say
the net force acting on the object = 0
6. Scientific units for force are Newtons (N)
C. Tension and Weight
1. Tension is a “stretching force”
2. When you hang an object from a spring scale the
there are two forces acting on object.
a. Force of gravity pulling down (also called
weight)
b. Tension force pulling
upward
c. Two forces are equal and
opposite in direction and add
to zero (net force = 0)
D. Force Vectors
1. Forces can be represented by arrows
a. length of arrow represents amount
(magnitude) of force
b. Direction of arrow represents
direction of force
c. Refer to arrow as a vector
(represents both magnitude and
direction of force
2. Vector quantity- needs both magnitude and
direction to complete description (i.e. force, velocity,
momentum)
3. Scalar quantity- can be described by magnitude
only and has no direction (i.e. temperature, speed,
distance)
Scaler or Vector quantity?
Time interval?
scaler
Mass?
scaler
Speed?
scaler
Velocity?
vector
Temperature?
scaler
Displacement?
vector
Acceleration?
vector
Arrow-tipped line segment
Length represents magnitude
Arrow points in specified direction of vector
Vectors are equal if: magnitude and directions are
the same
Vectors are not equal if: have different magnitude
or direction
or
Are these vectors equal or not?
NO
YES
NO
NO
YES
II. Mechanical Equilibrium (2.2)
A. Mechanical equilibrium- a state wherein no
physical changes occur (state of steadiness)
1. When net force equals zero, object is
said to be in mechanical equilibrium
a. Known as equilibrium rule
b. Can express rule mathematically as
F  0
1). ∑ symbol stands for “the sum of”
2). F stands for “forces”
2N
2N
-2 N
-2 N
F  (2  2)  0
III. Support Force (2.3)
A. support force- the upward force that
balances the weight of an object on a surface
1. The upward force balances the weight
of an object
2. Support force often called normal
force
Support force
weight
B. For an object at rest on a horizontal surface, the
support force must equal the objects weight.
1. Upward force is positive (+) and the
downward force is negative (-).
2. Two forces add mathematically to zero
F  0
2N
2N
-2 N
-2 N
F  (2  2)  0
IV. Equilibrium of Moving Objects (2.4)
A. Equilibrium can exist in both objects at rest
and objects moving at constant speed in a
straight-line path.
1. Equilibrium means
state of no change
2. Sum of forces equal
zero
F  0
B. Objects at rest are said to be in static equilibrium
C. Objects moving at constant speed in a straight-line
path are said to be in dynamic equilibrium
Static equilibrium
dynamic equilibrium
V. Vectors (2.5)
A. Parallel vectors
1. Add vectors if in same direction
2. Subtract vectors if in opposite
direction
3. The sum of two or more vectors is
called the resultant vector.
+
=
+
B. Parallel vectors– simple to add or subtract
add
subtract
Let’s say you are taking a trip to Hawaii. The distance
to Hawaii is 4100km and you travel at 900km/hr.
How long should it take you to reach Hawaii?
Let’s do the math.
(4100h)
(4100km )
(900km/h)
=
(900)
= 4.56 hours
It should take you the same amount of time to
return….. Right? Does it? Why not?
Remember, we can use vectors to describe things such
as velocity. Vectors tell us direction and magnitude
Let’s look at the velocity vectors that might describe the
airplane’s velocity and the wind’s velocity
Airplane vector to Hawaii =
Wind vector =
subtract vectors =
What is the difference in speed?
What about the direction?
Airplane vector from Hawaii =
Wind vector =
Add together =
B. Non-parallel vectors
1. Construct a parallelogram to determine
resultant vector
2. The diagonal of the parallelogram shows
the resultant
a. Perpendicular vectors
R
At one point the lunar lander was traveling
15ft/sec forward while traveling down at
4ft/sec. What was the direction they were
actually moving?
Combining vectors that are NOT parallel
•Result of adding two vectors called the resultant
•Resultant of two perpendicular vectors is the
diagonal of the rectangle with the two vectors as
sides
Resultant
vector
•Use simple three step technique to find resultant of
a pair of vectors that are at right angles to each
other.
First– draw two vectors with their tails
touching.
Second-draw a parallel projection of each vector
with dashed lines to form a rectangle
Third-draw the diagonal from the point where the
two tails are touching
resultant
b. Perpendicular vectors with equal sides (special
case)
1). For a square the length of diagonal is 2
or 1.414
2). Resultant = 1.414 x one of
2 sides
1
Resultant =
1
2
c. Parallelogram (not perpendicular)
•Construct parallelogram
•Construct with two vectors as sides
•Resultant is the diagonal
resultant
C. Applying the Parallelogram Rule- as angle
increases, tension increases.
Pythagorean Theorum- can be used if vectors
added are at right angles
R  A B
2
2
R
A
90°
B
2
or
R
A B
2
2
SOHCAHTOA
• sin, cos, and tan are functions of the angle of a triangle
compared to the lengths of the sides of a triangle
• If you know the distances of the triangle sides, you can
determine the inside angles.
• If you know the angle and one side, you can calculate the
length of the other side of the triangle.
• Remember the following acronym: SOHCAHTOA
• These will give you the formula
depending on which side or
angle you need to calculate
 O 
sin   

 H 
 A
cos  

H 
O
tan   

 A
Components calculated using following formulas
 adjacent   Ax 
   
Ax  A cos  ; therefore, cos   
 hypotenuse   A 
 opposite   Ay 

  
Ay  A sin  ; therefore, sin   
 hypotenuse   A 
When angles larger than 90°, sign of one or more
components may be negative
• Remember what the sides of a triangle are called
Hypotenuse
Opposite
θ
Adjacent
Example:
sin 25° =
Length of Opposite side
Length of Hypotenuse
sin 25° =
Length of Opposite side
100 meters
length of Opposite side = 38.3 meters
100 meters
θ
Adjacent
Opposite = ?
Graphical Addition of Vectors
•Simple method for combining vectors to get
vector
•Use ruler to measure length of vector
•Use protractor to measure angle
55°
resultant