Friction
Chapter 8: Engineering Mechanics Statics, R. C. Hibbeler, 12th Ed
CHARACTERISTICS OF DRY FRICTION
• Friction is a force that resists the movement of two contacting surfaces that slide relative
to one another
• This force always acts tangent to the surface at the points of contact and is directed so as
to oppose the possible or existing motion between the surfaces
• Dry friction (also known as Coulomb Friction) occurs between the contacting surfaces of bodies
when there is no lubricating fluid
THEORY OF DRY FRICTION
• The theory of dry friction can be explained by considering the effects caused by pulling
horizontally on a block of uniform weight W which is resting on a rough horizontal
surface that is nonrigid or deformable
• As shown on the free-body diagram of the block, the floor exerts an uneven distribution
of both normal force ∆Nn and frictional force ∆Fn along the contacting surface
• For equilibrium, the normal forces must act upward to balance the block’s weight W, and
the frictional forces act to the left to prevent the applied force P from moving the block to
the right
THEORY OF DRY FRICTION
• Close examination of the contacting surfaces between the floor and block reveals how
these frictional and normal forces develop
• It can be seen that many microscopic irregularities exist between the two surfaces and, as
a result, reactive forces ∆Rn are developed at each point of contact
• As shown, each reactive force contributes both a frictional component ∆Fn and a normal
component ∆Nn
THEORY OF DRY FRICTION
Equilibrium:
• The effect of the distributed normal and frictional loadings is indicated by their resultants N and F on
the free-body diagram
• Notice that N acts a distance ‘x’ to the right of the line of action of W
• This location, which coincides with the centroid or geometric center of the normal force distribution is
necessary in order to balance the “tipping effect” caused by P
• For example, if P is applied at a height ‘h’ from the surface, then moment equilibrium about point O is
satisfied if: Wx = Ph, or x = Ph/W
THEORY OF DRY FRICTION
Impending Motion:
• As P is slowly increased, F correspondingly increases
until it attains a certain maximum value Fs called the
limiting static frictional force
• When this value is reached, the block is in unstable
equilibrium since any further increase in P will cause the
block to move
• Experimentally, it has been determined that this limiting
static frictional force Fs is directly proportional to the
resultant normal force N
• Expressed mathematically, Fs = µsN , where the constant
of proportionality, µs, is called the coefficient of static
friction
THEORY OF DRY FRICTION
Impending Motion:
• Thus, when the block is on the verge of sliding, the
normal force N and frictional force Fs combine to create a
resultant Rs
• The angle Φs that Rs makes with N is called the angle of
static friction
• From the figure:
THEORY OF DRY FRICTION
Typical values of µs:
Materials
Coefficient of Station Friction,
S
Metal on metal
0.150.20
Masonry on masonry
0.600.70
Wood on wood
0.250.50
Metal on masonry
0.300.70
Metal on wood
0.200.60
Rubber on concrete
0.500.90
THEORY OF DRY FRICTION
Motion:
• If the magnitude of P acting on the block is increased so
that it becomes slightly greater than Fs, the frictional force
at the contacting surface will drop to a smaller value Fk,
called the kinetic frictional force
• The block will begin to slide with increasing speed
• Experiments with sliding blocks indicate that the
magnitude of the kinetic friction force is directly
proportional to the magnitude of the resultant normal
force, expressed mathematically as: Fk = µkN
• Here the constant of proportionality, µk, is called the
coefficient of kinetic friction
THEORY OF DRY FRICTION
Motion:
• As shown, in this case, the resultant force at the surface
of contact, Rk, has a line of action defined by Φk
• This angle is referred to as the angle of kinetic friction,
where:
THEORY OF DRY FRICTION
Variation of Frictional Force F versus Applied Load P:
1 F is a static frictional force if equilibrium is maintained
2 F is a limiting static frictional force Fs when it reaches a maximum value needed to
maintain equilibrium
3 F is termed a kinetic frictional force Fk when sliding occurs at the contacting surface
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Theory of Dry Friction
– Summary:
• If slip is impending, the magnitude of the friction & the angle of friction
are given by:
f  S N
tan S  S
• If the surfaces are sliding relative to each other, the magnitude of the
friction force & the angle of friction are given by:
f  k N
tan k  k
• Otherwise, the friction force must be determined from the equilibrium
equations
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Theory of Dry Friction
• The sequence of decisions in evaluating the friction force &
angle of friction is summarized in the Fig:
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Example 1 Determining the Friction Force
The arrangement in Fig. exerts
a horizontal force on the
stationary 180-N crate. The
coefficient of static friction
between the crate & the ramp
is S = 0.4.
(a) If the rope exerts a 90-N force on the crate,
what is the friction force exerted on the crate by
the ramp?
(b) What is the largest force the rope can exert on
the crate without causing
it to slide up the ramp?
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Example 1 Determining the Friction Force
Strategy
(a) The crate is not sliding on the ramp & we don’t
know whether slip is impending, so we must
determine the friction force by using the
equilibrium equations.
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Example 1 Determining the Friction Force
Strategy
(b) We want to determine the value of the force
exerted by the rope that causes the crate to be
on the verge of slipping up the ramp. When slip
is impending, the magnitude of the friction force
is f = SN & the friction force opposes the
impending slip. We can use the equilibrium
equations to determine the force exerted by the
rope.
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Example 1 Determining the Friction Force
Solution
(a) Draw the free-body diagram of the crate,
showing the force T exerted by the rope, the
weight W of the crate & the normal force N &
friction force f exerted by the ramp:
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Example 1 Determining the Friction Force
Solution
We can choose the direction of f arbitrarily & our
solution will indicate the actual direction of the
friction force.
By aligning the coordinate system with the ramp as
shown, we obtain the equilibrium equation:
Σ Fx = f + T cos 20°  W cos 20° = 0
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Example 1 Determining the Friction Force
Solution
Solving for the friction force, we obtain:
f = T cos 20° + W sin 20°
= (90 N) cos 20° + (180 N) sin 20°
= 23.0 N
The minus sign indicates that the direction of the
friction force on the crate is down the ramp.
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Example 1 Determining the Friction Force
Solution
(b) In this case the friction force f = SN & it
opposes the impending slip.
To simplify our solution for T, we align the
coordinate system as shown:
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Example 1 Determining the Friction Force
Solution
The equilibrium equations:
Σ Fx = T  N sin 20°  SN cos 20° = 0
Σ Fy = N cos 20°  SN sin 20°  W= 0
Solving the 2nd equilibrium equation for N, we
W
obtain:
N
cos 20  S sin 20
180 N

cos 20  0.4 sin 20
 224 N
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Example 1 Determining the Friction Force
Solution
Then, from the 1st equilibrium equation, the force T
is:
T = N (sin 20° + S cos 20°) = 0
= (224 N) (sin 20° + 0.4 cos 20°)
= 161 N
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Example 1 Determining the Friction Force
Critical Thinking
• When you use the equilibrium equations to
determine a friction force, often you will not know
its direction before-hand:
– Depending on the value of the force T exerted
on the crate by the rope, the friction force
exerted on the crate by the ramp can point
either up or down the ramp
– In drawing the free-body diagram of the crate
in (a), we arbitrarily assumed that the friction
force pointed up the 23ramp
Example 1 Determining the Friction Force
Critical Thinking
– The negative value obtained from the equilibrium equations, f =
23.0 N, tells us that the force is in the opposite direction, down
the ramp
• In contrast, when you use the equation f = SN, the friction force
must point in the correct direction on the free-body diagram
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Example 1 Determining the Friction Force
Critical Thinking
– In drawing the free-body diagram in (b), we wanted to
determine the largest force T that would not cause the crate to
slide up the ramp, so we assumed that the slip of the crate up
the ramp was impending
– This told us that the friction force, resisting the impending slip,
pointed down the ramp
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Example 2 Determining the Friction Force
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Example 2 Determining the Friction Force
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