FRICTION (Sections 8.1 - 8.2)

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FRICTION (Sections 8.1 - 8.2)
Sections Objective Students will be able to a)
Understand the characteristics of dry
friction. b) Draw a FBD including friction. c)
Solve problems involving friction.

• In-Class Activities
• Check homework, if any
• Reading quiz
• Applications
• Characteristics of dry friction
• Problems involving dry friction
• Concept quiz
• Group problem solving
• Attention quiz

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READING QUIZ
1. A friction force always acts _____ to the
contact surface. A) normal B) at 45 C)
parallel D) at the angle of static friction
2. If a block is stationary, then the friction
force acting on it is ________ . A) ? ?s N B)
?s N C) ? ?s N D) ?k N
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APPLICATIONS
In designing a brake system for a bicycle, car,
or any other vehicle, it is important to
understand the frictional forces involved.
For an applied force on the brake pads, how can
we determine the magnitude and direction of the
resulting friction force?
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APPLICATIONS (continued)
Consider pushing a box as shown here. How can you
determine if it will slide, tilt, or stay in
static equilibrium?
What physical factors affect the answer to this
question?
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CHARACTERISTICS OF DRY FRICTION (Section 8.1)
Friction is defined as a force of resistance
acting on a body which prevents or retards
slipping of the body relative to a second body.
Experiments show that frictional forces act
tangent (parallel) to the contacting surface in a
direction opposing the relative motion or
tendency for motion.
For the body shown in the figure to be in
equilibrium, the following must be true F P,
N W, and Wx Ph.
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CHARACTERISTICS OF FRICTION (continued)
To study the characteristics of the friction
force F, let us assume that tipping does not
occur (i.e., h is small or a is large). Then
we gradually increase the magnitude of the force
P. Typically, experiments show that the friction
force F varies with P, as shown in the left
figure above.
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FRICTION CHARACERISTICS (continued)
The maximum friction force is attained just
before the block begins to move (a situation that
is called impending motion). The value of the
force is found using Fs ?s N, where ?s is
called the coefficient of static friction. The
value of ?s depends on the materials in contact.
Once the block begins to move, the frictional
force typically drops and is given by Fk ?k N.
The value of ?k (coefficient of kinetic friction)
is less than ?s .
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DETERMING ?s EXPERIMENTALLY
A block with weight w is placed on an inclined
plane. The plane is slowly tilted until the block
just begins to slip.
Using these two equations, we get ?s (W sin
?s ) / (W cos ?s ) tan ?s This simple
experiment allows us to find the ?S between two
materials in contact.
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PROCEDURE FOR ANALYSIS (Section 8.2)
Steps for solving equilibrium problems involving
dry friction
1. Draw the necessary free body diagrams. Make
sure that you show the friction force in the
correct direction (it always opposes the motion
or impending motion).
2. Determine the number of unknowns. Do not
assume F ?S N unless the impending
motion condition is given.
3. Apply the equations of equilibrium and
appropriate frictional equations to solve for the
unknowns.
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IMPENDING TIPPING versus SLIPPING
For a given W and h, how can we determine if the
block will slide first or tip first? In this
case, we have four unknowns (F, N, x, and P) and
only three EofE.
Hence, we have to make an assumption to give us
another equation. Then we can solve for the
unknowns using the three EofE. Finally, we need
to check if our assumption was correct.
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IMPENDING TIPPING versus SLIPPING (continued)
Assume Slipping occurs Known F ?s N Solve
x, P, and N Check 0 ? x ? b/2
Or Assume Tipping occurs Known
x b/2 Solve P, N, and F Check F ? ?s
N
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EXAMPLE
Given A uniform ladder weighs 100 N. The
vertical wall is smooth (no friction). The
floor is rough and ?s 0.8. Find The minimum
force P needed to move ( tip or slide) the
ladder. Plan
a) Draw a FBD. b) Determine the unknowns. c) Make
any necessary friction assumptions. d) Apply EofE
(and friction equations, if appropriate ) to
solve for the unknowns. e) Check assumptions, if
required.
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EXAMPLE (continued)
1.2 m
There are four unknowns NA, FA, NB, and P. Let
us assume that the ladder will tip first. Hence,
NB 0
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EXAMPLE (continued)
Now check the assumption. Fmax ?s NA 0.8
100 N 80 N Is FA 75 N ? Fmax 80 N? Yes,
hence our assumption of tipping is correct.
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CONCEPT QUIZ
1. A 100 N box with wide base is pulled by a
force P and ?s 0.4. Which force orientation
requires the least force to begin sliding?
A) A B) B C) C
D) Can not be determined
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GROUP PROBLEM SOLVING
Given Drum weight 500 N, ?s 0.5, a 0.75
m and b 1m. Find The smallest magnitude of P
that will cause impending motion (tipping or
slipping) of the drum. Plan
a) Draw a FBD of the drum. b) Determine the
unknowns. c) Make friction assumptions, as
necessary. d) Apply EofE (and friction eqn. as
appropriate) to solve for the unknowns. e) Check
assumptions, as required.
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GROUP PROBLEM SOLVING (continued)
A FBD of the drum
There are four unknowns P, N, F and x. First,
lets assume the drum slips. Then the friction
equation is F ?s N 0.5 N.
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GROUP PROBLEM SOLVING (continued)
A FBD of the drum
? ? FX (4 / 5) P 0.5 N 0
? ? FY N (3 / 5) P 500 0 These
two equations give
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ATTENTION QUIZ
1. A 10 N block is in equilibrium. What is the
magnitude of the friction force between this
block and the surface? A) 0 N B) 1 N C) 2
N D) 3 N
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CENTER OF GRAVITY AND CENTROID (Chapter 9)
Sections Objective Students will a)
Understand the concepts of center of gravity,
center of mass, and centroid. b) Be able to
determine the location of these points for a
system of particles or a body.

• In-Class Activities
• Check homework, if any
• Reading quiz
• Applications
• Center of gravity, etc.
• Determine their location
• Concept quiz
• Group problem solving
• Attention quiz

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READING QUIZ
1. The _________ is the point defining the
geometric center of an object . A) center of
gravity B) center of
mass C) centroid D) none of the above
2. To study problems concerned with the motion
of matter under the influence of forces, i.e.,
dynamics, it is necessary to locate a point
called ________. A) center of gravity B)
center of mass C) centroid D) none of the
above
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APPLICATIONS
To design the structure for supporting a water
tank, we will need to know the weights of the
tank and water as well as the locations where the
resultant forces representing these distributed
loads are acting.
How can we determine these weights and their
locations?
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APPLICATIONS (continued)
One concern about a sport utility vehicle (SUVs)
is that it might tip over while taking a sharp
turn.
One of the important factors in determining its
stability is the SUVs center of mass.
Should it be higher or lower for making a SUV
more stable?
How do you determine its location?
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4N
CONCEPT OF CG and CM
3m
1m
The center of gravity (G) is a point which
locates the resultant weight of a system of
particles or body.
?
?

A
B
G
1 N
3 N
From the definition of a resultant force, the sum
of moments due to individual particle weight
about any point is the same as the moment due to
the resultant weight located at G. For the
figure above, try taking moments about A and B.
Also, note that the sum of moments due to the
individual particles weights about point G is
equal to zero.
Similarly, the center of mass is a point which
locates the resultant mass of a system of
particles or body. Generally, its location is the
same as that of G.
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CONCEPT OF CENTROID
The centroid C is a point which defines the
geometric center of an object.
The centroid coincides with the center of mass or
the center of gravity only if the material of the
body is homogenous (density or specific weight is
constant throughout the body).
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CG / CM FOR A SYSTEM OF PARTICLES (Section 9.1)
Consider a system of n particles as shown in the
figure. The net or the resultant weight is given
as WR ?W.
Similarly, we can sum moments about the x and
z-axes to find the coordinates of G.
By replacing the W with a M in these equations,
the coordinates of the center of mass can be
found.
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CG / CM / CENTROID OF A BODY (Section 9.2)
A rigid body can be considered as made up of an
infinite number of particles. Hence, using the
same principles as in the previous slide, we get
the coordinates of G by simply replacing the
discrete summation sign ( ? ) by the continuous
summation sign ( ? ) and W by dW.
Similarly, the coordinates of the center of mass
and the centroid of volume, area, or length can
be obtained by replacing W by m, V, A, or L,
respectively.
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STEPS FOR DETERMING AREA CENTROID
1. Choose an appropriate differential element dA
at a general point (x,y). Hint Generally, if y
is easily expressed in terms of x (e.g., y x2
1), use a vertical rectangular element. If the
converse is true, then use a horizontal
rectangular element.
2. Express dA in terms of the differentiating
element dx (or dy).
4. Express all the variables and integral limits
in the formula using either x or y depending on
whether the differential element is in terms of
dx or dy, respectively, and integrate.
Note Similar steps are used for determining CG,
CM, etc.. These steps will become clearer by
doing a few examples.
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EXAMPLE (continued)

4. x ( ?A x dA ) / ( ?A dA )
3
0 ? x ( 9 x2) d x 9 (x2)/2 (x4) /
4 3 0 ? x ( 9 x2) d x 9 x (x3) / 3
3 ( 9 ( 9 ) / 2 81 / 4 ) / ( 9 ( 3 )
( 27 / 3 ) ) 1.13 m
0


3
0

3
?A y dA ½ 0 ? ( 9 x2) ( 9 x2)
dx ?A dA 0 ? ( 9 x2) d x
3.60 m


y
3
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CONCEPT QUIZ
1. The steel plate with known weight and
non-uniform thickness and density is supported as
shown. Of the three parameters (CG, CM, and
centroid), which one is needed for determining
the support reactions? Are all three parameters
located at the same point? A) (center of
gravity, no)B) (center of gravity,
yes)C) (centroid, yes)D) (centroid, no)
2. When determining the centroid of the area
above, which type of differential area element
requires the least computational
work? A) Vertical B) Horizontal C) Polar D
) Any one of the above.
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GROUP PROBLEM SOLVING
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PROBLEM SOLVING (continued)
x 1.067 / 1.167 0.914 m
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ATTENTION QUIZ
1. If a vertical rectangular strip is chosen
as the differential element, then all the
variables, including the integral limit, should
be in terms of _____ . A) x B) y C) z D)
Any of the above.
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End of the Lecture
Let Learning Continue