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Friction
∗
Sunil Kumar Singh
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 2.0†
Abstract
Friction is indispensable for our world.
Friction force (also referred simply as friction) plays the role of a dampener to the motion of a body that
takes place in contact with another body. All motions involved in our daily life take place in contact with
another body or medium. The force of opposition to the motion of a body in relation to another rigid body
is called friction and in relation to uid (liquid or gas) medium is called drag.
Friction is a huge disadvantage to us. A good part of the energy used in this world goes waste to counteract
this force. On the other hand, this force is also responsible for our existence and the very existence of life
on the planet. Imagine its absence. How would have we stopped once in motion? Everything would have
been perpetually in motion with no control. Could this life have evolved without friction in the rst place?
Friction is simply indispensable to us.
1 Genesis of friction
When bodies come in contact with each other, large numbers of atoms of the two surfaces come close to
each other, aecting a temporary joint (referred as cold welds in technical parlance) i.e. atoms from the two
surfaces become part of one body mass. These joints result from electromagnetic force operating between
atoms, when brought suciently close. The temporary joints between two surfaces inhibit relative movement
between two surfaces.
In general two ordinary surfaces are uneven at microscopic level. The surface is actually formed of small
hills and valleys. All points across the surfaces do not come in contact. Still, there are large numbers of such
contact points, forming temporary joints.
∗ Version 1.14: Dec 13, 2008 12:38 am -0600
† http://creativecommons.org/licenses/by/2.0/
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Surfaces in contact
Figure 1: The surface is actually formed of small hills and valleys.
The weight of the body plays important role in determining friction.
When the overlying body has
greater mass, it applies greater force at the interface. More points come in contact or become suciently
close to form joints.
Further, all contact points are not cold welded or joints.
Due to the weight of the
overlying body, more of the contact points become temporary joints or become stronger joints, requiring
larger external force to initiate motion. Thus, friction depends on (i) numbers of points in contact (nature
of the surfaces in contact) and (ii) normal force at the contact surface which presses them to come closure.
Generally, a smooth polished surface is known to oer smaller friction with respect to a rough surface.
When the surface is smooth, then there are more contact points, but corresponding force per point is smaller.
In this case, the weight of the block, pressing against the surface beneath, is distributed across larger numbers
of contact points. The net result is that there are greater numbers of contact points, but fewer welded joints
opposing motion. Thus, a smooth surface oers smaller friction in comparison to rough surface.
However, if the surfaces are genuinely smooth to perfection and brought together, then there are much
greater numbers of contact points, which are already suciently close and produce still larger numbers of
weld sites. In such case, two bodies become almost inseparable and require a much greater external force
to separate two bodies. If the joint is done between very smooth surfaces in the absence of air i.e. vacuum,
then the cold welding, at contact sites covering larger contact area, makes the two pieces as one and the
bodies are mechanically inseparable.
2 Static friction
Static friction comes into play when a body tends to move over another body (surface), but the body has
yet not been initiated into motion. When we apply a small force to initiate motion of a block, lying on a
horizontal surface as shown in the gure; the temporary joints may not yield to the external force. This
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means that the force is not sucient enough to break o the welds, which are formed at contact points. In
this case, the external force (
F ||
) parallel to the contact surface is equal to the static friction,
fs
, i.e.
f s = F || = F
A block on plane surface
Figure 2: A block is pulled by a horizontal force.
Note that friction force comes into being even before motion is initiated.
If we increase the external
force in the direction parallel to the contact surface, then friction force increases so that two forces are equal
in magnitude and opposite to each other. It is important to understand that friction responds to the net
external force parallel to the contact surface to match the magnitude. In this sense, static friction force is
a "self adjusting" force against net external force parallel to the contact force (F||) or the component of
external forces along contact surface.
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Friction and applied force
Figure 3: Friction is equal to external force parallel to the contact force.
Self adjusting force means that if a 2 N force parallel to contact surface fails to initiate motion, then
static friction is 2 N; if a 4 N force parallel to contact surface fails to initiate motion, then static friction is
4 N and so on.
Example 1
Problem :
A force of 10 N is applied on a block of 10 kg. The force makes an angle 60
◦
with
the horizontal. If the block fails to move, then nd static friction between two surfaces.
Solution :
Here, the block is not initiated into motion by the external force. Now, we know
that static friction is equal to the component of net external force parallel to contact surface :
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A block on a plane surface
Figure 4: The block is pulled by an external force of 10 N.
fs = F || = F cos600 = 10 x
1
2
= 5N
We must realize following characteristics of the static friction from the discussion so far :
•
So long the body does not start moving, the static friction is equal to the component of net external
force in the direction parallel to contact surface.
•
So long the body does not start moving, the static friction is a self adjusting force i.e. it varies with
net external force in the direction parallel to contact surface.
Also, we must realize that friction force (
(
f ||
fs
) and component of external force parallel to contact surface
) are parallel to each other and may not be concurrent in reality. However, we treat them concurrent,
while analyzing - for the simple reason that we are considering translational motion. In this situation, the
forces can be considered concurrent.
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A block on a plane surface
Figure 5: The forces are not concurrent, but are considered concurrent for translational motion.
note: In this course we shall follow a convention to denote various friction forces. According to
this convention, "
FF
" denotes friction in general, whose nature is not known; "
adjusting static friction ; "
FS
fs
" denotes limiting or maximum static friction and "
" denotes self
FK
" denotes
kinetic friction.
2.1 Nature of force system
If the body remains at rest under the action of an external force, then the forces on the body are balanced.
This means that the forces on the body, including friction, should complete a closed force polygon so that
the net force is zero. Let us consider the case of a block of mass m, lying on a horizontal surface. There
are four forces (i) weight of the block,mg, (ii) normal force, N, (iii) external force (F) and (iv) friction force
(
fs
).
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A block resting on a plane surface
Figure 6: The forces form a closed vector polygon.
Till the body starts moving, the external force and static friction adjust themselves to complete the closed
polygon.
3 Limiting (maximum) static friction
If we continue to increase the component of external force parallel to the contact surface, then for a particular
magnitude, the weld joints at the contact points are broken o and the block starts moving. For any particular
pair of surfaces, there is a limiting or maximum static friction (
Fs
).
This limiting friction force between two surfaces can not be specied on the basis of the constitution of
the material of bodies alone (such as the iron, lead etc), but its value depends on two additional factors (i)
the relative smoothness or roughness of the two surfaces in contact and (ii) normal force between the two
surfaces. Normal force is responsible to increase numbers of joints between the surfaces and as such aects
friction between surfaces.
In order to understand this aspect of friction, let us consider that the mass of
block is 10 kg and acceleration due to gravity is 10
m / s2
. From force analysis, we observe that normal
force on the body is equal to the weight of the block :
N = mg = 10 X 10 = 100
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Newton
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A block resting on a plane surface
Figure 7: The free body diagram superimposed on body systems.
Now, let us put another block of mass of 10 kg over the block on the surface. As a result, the normal
force increases to 200 N as shown in the gure below.
Two blocks resting on a plane surface
Figure 8: Greater normal force means more joints between surfaces.
As the block is now pressing the underlying surface with greater force, more of the contact points are
converted into weld sites. Plainly speaking, the atoms at contact points have moved closure, increasing the
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inter-atomic forces. For the same pair of surfaces, we would, then, need a bigger maximum force (
Fs
) to
initiate motion as shown in the gure below.
Friction .vs. external force plot
(a)
(b)
Figure 9: Greater normal force extends maximum static friction.
Experimentally, it is found that maximum friction force is proportional to normal force on the block.
Fs ∝ N
⇒ Fs = µs N
where
µs
is called coecient of static friction, specic to a given pair of the surfaces in contact. It depends
on the nature of the surfaces in contact. Importantly, it neither depends on the normal force unlike friction
We should also note that coecient of friction, µs , being
Fs
is a ratio of two forces and is, therefore, dimensionless. Further, coecient of static friction is
N
dened only for the maximum static friction, when body is about to move and not for friction before this
(force) nor it depends on the area in contact.
equal to
stage. What it means that we can not estimate self adjusting static friction, using above relation. The self
adjusting friction, as we can remember, is equal to the component of external force parallel to the contact
surface.
Here, we distinguish maximum static friction force by denoting it with the symbol friction before motion is denoted by fs
Fs
, whereas static
.
We summarize following characterizing aspects of static (
fs
) and maximum static friction (
Fs
) :
1. Static friction applies in the direction opposite to the component of net external force parallel to contact
surface. This, as a matter of fact, is the criteria of deciding direction of friction.
2. Self adjusting static friction (
fs
) is equal to the component of net external force parallel to contact
surface. It ranges from 0 to a maximum value
Fs
. Static friction is a self adjusting friction force. It is
important to emphasize that static friction, unlike maximum (limiting) static friction, is not given by
the expression of coecient of friction. Note the relation for static friction :
fs = F ||
(and
fs 6= µs N
)
3. Maximum static friction is proportional to normal force applied on the body and is a single value
(unlike static friction) quantity.
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4. The relation between maximum (limiting) static friction and normal force is a scalar relation. This
relation connects forces, which are mutually perpendicular to each other.
Example 2
Problem :
A force "F" is applied on a block of mass "m", making an angle "θ " with the
horizontal as shown in the gure. If the block just starts to move with the applied force, nd (i)
maximum static friction and (ii) static friction coecient for the two surfaces in contact.
Block on a horizontal plane
Figure 10: The block is pulled by an external force.
Solution :
(i) The maximum static friction is equal to the force parallel to contact surface to
initiate the motion. Thus,
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Block on a horizontal plane
Figure 11: The block is pulled by an external force.
Fs = F cosθ
(ii) Coecient of static friction is ratio of normal force and friction.
We, therefore, need to
know the normal force on the block. Now, force analysis in y-direction results in following relation
for normal force,
P
Fy ⇒ N + F sinθ − mg = 0
⇒ N = mg − F sinθ
Hence,
⇒ µs =
Fs
N
=
F cosθ
mg − F sinθ
4 Kinetic friction
If the external force parallel to contact surface exceeds maximum static friction, then the body starts moving
over underlying surface. It does not mean, however, that friction between surfaces disappears. New contact
points between surfaces come in contact, some of which are momentarily joined and then broken on continuous
basis. The friction force is dropped slightly (almost instantly); but remains constant - independent of the
velocity of the body.
We denote kinetic friction force as
Fk
and corresponding friction coecient as
is related to normal force as :
Fk ∝ N
⇒ Fk = µk N
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µk
. The kinetic friction
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The coecient of kinetic friction is generally independent of the velocity of the body and is practically
considered constant.
4.1 Motion over a rough surface
Once the external force along the contact surface exceeds limiting or maximum static friction, the body
starts moving on the surface. The nature of motion, subsequent to initiation, depends on the external force.
There are two possibilities :
1:Uniform motion :
The external force along the contact surface exactly equals to kinetic friction during the motion. In this
case, the body is subjected to a balanced external force system (including friction) having zero net force. As
such, the body will continue moving with whatever initial velocity is given to the body. What it means that
we apply external force,
F > Fs
, momentarily to impart initial velocity to the body and then subsequently
maintain uniform velocity by external force, F, such that
F = Fk
.
A typical friction force - time plot for the situation is shown in the gure.
Here, we have considered
that external force along the contact surface is increased slowly till it equals maximum static friction, then
external force is adjusted simply to maintain velocity of the body.
Freiction .vs. time plot
Figure 12
The maximum static friction coecient,
µs
, and kinetic friction coecient,
µk
, are usually close values
for a pair of given surfaces and may be taken to be equal as an approximation.
2: Accelerated motion :
If we maintain external force along the contact surface in excess of kinetic friction, then body undergoes
accelerated motion.
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Accelerated motion
Figure 13
The free body diagram is shown here in the gure. Let a be the acceleration towards right.
Free body diagram
Figure 14
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P
Fx = F − Fk = ma
⇒ a =
F − Fk
m
Example 3
Problem :
A block of 10 kg is moving with a speed 10 m/s along a straight line on a horizontal
surface at a particular instant. If the coecient of kinetic friction between block and the surface is
0.5, then nd (i) how long does it travel before coming to a stop and (ii) the time taken to come to
a stop. Assume g = 10
Solution :
m / s2
.
Kinetic friction is a constant force that acts opposite to the relative velocity of
the block with respect to the surface on which it moves. This constant force results in constant
deceleration, whose magnitude is given by :
⇒ a =
Fk
m
=
µk N
m
Now the free body diagram of the block as superimposed on the body diagram is shown in the
gure.
Block moving on a horizontal plane
Figure 15: Friction decelerates the block to a stop.
As there is no motion in y-direction,
N = mg
In x-direction, the magnitude of acceleration is :
⇒ a =
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µk N
m
=
µk mg
m
= µk g = 0.5 X 10 = 5 m / s 2
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As the deceleration is constant, we can employ equation of motion to determine the displacement
in x-direction.
v 2 = u 2 + 2 ax
Here, v = 0, u = 10 m/s and a = -5
m/s2
.
⇒ 0 = 102 − 2 x 5x
⇒ x =
100
10
= 10 m
If t is time taken, then using the relation v = u + at, we have :
⇒
⇒
0 = 10 5t
t = 2s
5 Laws of friction
The properties of friction as explained above are summarized in three laws of friction as :
1: If a force fails to initiate the motion of a body in contact with another surface, then static friction is
equal to the component of net external force parallel to the surface of the contact surface.
fs = F ||
2: The static friction has a maximum value (Fs), which is given by :
Fs = µs N
where
µs
is the coecient of static friction and "N" is the magnitude of normal force. The maximum
static friction (
Fs
) and normal force (N) are mutually perpendicular to each other . The coecient of static
friction depends on the nature of the surfaces in contact.
3: If the body begins to move along the surface, then friction force, called kinetic friction, is reduced
slightly, which is given by :
Fk = µk N
where "
µk
" is coecient of kinetic friction and depends on the nature of surfaces in contact. Experi-
mentally, it is found that
µk < µs
.
6 Area of contact and friction
We observe that area of contact does not appear anywhere in our consideration of friction.
Though, we
might generally believe that a greater contact area should oer greater friction and would be dicult to
move.
In order to understand the absence of area in our consideration, let us consider the dierent shapes of
bodies of equal mass in contact with a given surface.
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Shapes of bodies and friction
Figure 16: Irrespective of the dierence in contact areas, the friction in all three cases is same.
Irrespective of the dierence in contact areas, the friction in all three cases is same.
If we recall, friction results from the temporary joints formed between the contact surfaces. Thus, friction
depends on (i) the numbers of contact points and (ii) the force inducing joints at these contact points.
The actual contact area may be much less - only up to 40 % of the total surface area in the case of
ordinary plane surface. This means that friction does not depend on the total area, but only the part of
the area which is actually in contact with the other surface. This fact is incorporated in our consideration
through "coecient of friction" which represents the mutual characteristic of two surfaces not of one
surface. It largely accounts for the numbers of contact points between two surfaces and hence is characteristic
of a pair of surfaces in contact.
On the other hand, formation of joints at the contact points depend on the distribution of normal force
at contact points. Normal force, in turn, is distributed across the surface. If the density of contact points
throughout is uniform, we can say that formation of joints depends on the normal force per unit area. Normal
force is equal to the product of normal force per unit area and area as given here :
N =
N
A
A
It is clear that by considering normal force we have implicitly accounted for the area of contact. In the
nutshell, we can say in a very general way that (i) coecient of friction largely accounts for the contact
points between a given pair of surfaces and (ii) normal force accounts for the distribution of force at contact
points across the whole area.
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