When a body is in motion, it has resistance because the body interacts with its surroundings. This resistance is a force of friction. Friction opposes relative motion between systems in contact but also allows us to move, a concept that becomes obvious if you try to walk on ice. Friction is a common yet complex force, and its behavior still not completely understood. Still, it is possible to understand the circumstances in which it behaves.
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The basic definition of friction is relatively simple to state.
Friction is a force that opposes relative motion between systems in contact.
There are several forms of friction. One of the simpler characteristics of sliding friction is that it is parallel to the contact surfaces between systems and is always in a direction that opposes motion or attempted motion of the systems relative to each other. If two systems are in contact and moving relative to one another, then the friction between them is called kinetic friction. For example, friction slows a hockey puck sliding on ice. When objects are stationary, static friction can act between them; the static friction is usually greater than the kinetic friction between two objects.
If two systems are in contact and stationary relative to one another, then the friction between them is called static friction. If two systems are in contact and moving relative to one another, then the friction between them is called kinetic friction.
Imagine, for example, trying to slide a heavy crate across a concrete floor&#;you might push very hard on the crate and not move it at all. This means that the static friction responds to what you do&#;it increases to be equal to and in the opposite direction of your push. If you finally push hard enough, the crate seems to slip suddenly and starts to move. Now static friction gives way to kinetic friction. Once in motion, it is easier to keep it in motion than it was to get it started, indicating that the kinetic frictional force is less than the static frictional force. If you add mass to the crate, say by placing a box on top of it, you need to push even harder to get it started and also to keep it moving. Furthermore, if you oiled the concrete you would find it easier to get the crate started and keep it going (as you might expect).
Figure \(\PageIndex{1}\) is a crude pictorial representation of how friction occurs at the interface between two objects. Close-up inspection of these surfaces shows them to be rough. Thus, when you push to get an object moving (in this case, a crate), you must raise the object until it can skip along with just the tips of the surface hitting, breaking off the points, or both. A considerable force can be resisted by friction with no apparent motion. The harder the surfaces are pushed together (such as if another box is placed on the crate), the more force is needed to move them. Part of the friction is due to adhesive forces between the surface molecules of the two objects, which explains the dependence of friction on the nature of the substances. For example, rubber-soled shoes slip less than those with leather soles. Adhesion varies with substances in contact and is a complicated aspect of surface physics. Once an object is moving, there are fewer points of contact (fewer molecules adhering), so less force is required to keep the object moving. At small but nonzero speeds, friction is nearly independent of speed.
Figure \(\PageIndex{1}\): Frictional forces, such as \(\vec{f}\), always oppose motion or attempted motion between objects in contact. Friction arises in part because of the roughness of the surfaces in contact, as seen in the expanded view. For the object to move, it must rise to where the peaks of the top surface can skip along the bottom surface. Thus, a force is required just to set the object in motion. Some of the peaks will be broken off, also requiring a force to maintain motion. Much of the friction is actually due to attractive forces between molecules making up the two objects, so that even perfectly smooth surfaces are not friction-free. (In fact, perfectly smooth, clean surfaces of similar materials would adhere, forming a bond called a &#;cold weld.&#;)The magnitude of the frictional force has two forms: one for static situations (static friction), the other for situations involving motion (kinetic friction). What follows is an approximate empirical (experimentally determined) model only. These equations for static and kinetic friction are not vector equations.
The magnitude of static friction fs is
\[F_{f_{s}} \leq \mu_{s} F_{N}, \label{6.1}\]
where \(\mu_{s}\) is the coefficient of static friction and N is the magnitude of the normal force.
The symbol &#; means less than or equal to, implying that static friction can have a maximum value of \(\mu_{s}\)N. Static friction is a responsive force that increases to be equal and opposite to whatever force is exerted, up to its maximum limit. Once the applied force exceeds fs (max), the object moves. Thus,
\[F_{f_{s}} (max) = \mu_{s} F_{N} \ldotp\]
The magnitude of kinetic friction fk is given by
\[F_{f_{k}} \leq \mu_{k} F_{N}, \label{6.2}\]
where \(\mu_{k}\) is the coefficient of kinetic friction.
A system in which \(F_{f_{k}} = \mu_{k} F_{N}\) is described as a system in which friction behaves simply. The transition from static friction to kinetic friction is illustrated in Figure \(\PageIndex{2}\)
Figure \(\PageIndex{2}\): (a) The force of friction \(\vec{f}\) between the block and the rough surface opposes the direction of the applied force \(\vec{F}\). The magnitude of the static friction balances that of the applied force. This is shown in the left side of the graph in (c). (b) At some point, the magnitude of the applied force is greater than the force of kinetic friction, and the block moves to the right. This is shown in the right side of the graph. (c) The graph of the frictional force versus the applied force; note that fs (max) > fk. This means that \(\mu_{s}\) > \(\mu_{k}\)As you can see in Table 6.1, the coefficients of kinetic friction are less than their static counterparts. The approximate values of \(\mu\) are stated to only one or two digits to indicate the approximate description of friction given by the preceding two equations.
Equation \ref{6.1} and Equation \ref{6.2} include the dependence of friction on materials and the normal force. The direction of friction is always opposite that of motion, parallel to the surface between objects, and perpendicular to the normal force. For example, if the crate you try to push (with a force parallel to the floor) has a mass of 100 kg, then the normal force is equal to its weight,
\[w = mg = (100\; kg)(9.80\; m/s^{2}) = 980\; N,\]
perpendicular to the floor. If the coefficient of static friction is 0.45, you would have to exert a force parallel to the floor greater than
\[f_{s} (max) = \mu_{s} N = (0.45)(980\; N) = 440\; N\]
to move the crate. Once there is motion, friction is less and the coefficient of kinetic friction might be 0.30, so that a force of only
\[f_{k} = \mu_{k} N = (0.30)(980\; N) = 290\; N\]
keeps it moving at a constant speed. If the floor is lubricated, both coefficients are considerably less than they would be without lubrication. Coefficient of friction is a unitless quantity with a magnitude usually between 0 and 1.0. The actual value depends on the two surfaces that are in contact.
Many people have experienced the slipperiness of walking on ice. However, many parts of the body, especially the joints, have much smaller coefficients of friction&#;often three or four times less than ice. A joint is formed by the ends of two bones, which are connected by thick tissues. The knee joint is formed by the lower leg bone (the tibia) and the thighbone (the femur). The hip is a ball (at the end of the femur) and socket (part of the pelvis) joint. The ends of the bones in the joint are covered by cartilage, which provides a smooth, almost-glassy surface. The joints also produce a fluid (synovial fluid) that reduces friction and wear. A damaged or arthritic joint can be replaced by an artificial joint (Figure \(\PageIndex{3}\)). These replacements can be made of metals (stainless steel or titanium) or plastic (polyethylene), also with very small coefficients of friction.
Figure \(\PageIndex{3}\): Artificial knee replacement is a procedure that has been performed for more than 20 years. These post-operative X-rays show a right knee joint replacement. (credit: Mike Baird)Natural lubricants include saliva produced in our mouths to aid in the swallowing process, and the slippery mucus found between organs in the body, allowing them to move freely past each other during heartbeats, during breathing, and when a person moves. Hospitals and doctor&#;s clinics commonly use artificial lubricants, such as gels, to reduce friction.
The equations given for static and kinetic friction are empirical laws that describe the behavior of the forces of friction. While these formulas are very useful for practical purposes, they do not have the status of mathematical statements that represent general principles (e.g., Newton&#;s second law). In fact, there are cases for which these equations are not even good approximations. For instance, neither formula is accurate for lubricated surfaces or for two surfaces siding across each other at high speeds. Unless specified, we will not be concerned with these exceptions.
A 20.0-kg crate is at rest on a floor as shown in Figure \(\PageIndex{4}\). The coefficient of static friction between the crate and floor is 0.700 and the coefficient of kinetic friction is 0.600. A horizontal force \(\vec{P}\) is applied to the crate. Find the force of friction if (a) \(\vec{P}\) = 20.0 N, (b) \(\vec{P}\) = 30.0 N, (c) \(\vec{P}\) = 120.0 N, and (d) \(\vec{P}\) = 180.0 N.
Figure \(\PageIndex{4}\): (a) A crate on a horizontal surface is pushed with a force \(\vec{P}\). (b) The forces on the crate. Here, \(\vec{f}\) may represent either the static or the kinetic frictional force.The free-body diagram of the crate is shown in Figure \(\PageIndex{4b}\). We apply Newton&#;s second law in the horizontal and vertical directions, including the friction force in opposition to the direction of motion of the box.
Newton&#;s second law gives
\[\sum F_{x} = ma_{x}\]
\[P - f = ma_{x}\]
\[\sum F_{y} = ma_{y}\]
\[N - w = 0 \ldotp\]
Here we are using the symbol f to represent the frictional force since we have not yet determined whether the crate is subject to station friction or kinetic friction. We do this whenever we are unsure what type of friction is acting. Now the weight of the crate is
\[w = (20.0\; kg)(9.80\; m/s^{2}) = 196\; N,\]
which is also equal to N. The maximum force of static friction is therefore (0.700)(196 N) = 137 N. As long as \(\vec{P}\) is less than 137 N, the force of static friction keeps the crate stationary and fs = \(\vec{P}\). Thus, (a) fs = 20.0 N, (b) fs = 30.0 N, and (c) fs = 120.0 N. (d) If \(\vec{P}\) = 180.0 N, the applied force is greater than the maximum force of static friction (137 N), so the crate can no longer remain at rest. Once the crate is in motion, kinetic friction acts. Then
\[f_{k} = \mu_{k} N = (0.600)(196\; N) = 118\; N,\]
and the acceleration is
\[a_{x} = \frac{\vec{P} - f_{k}}{m} = \frac{180.0\; N - 118\; N}{20.0\; kg} = 3.10\; m/s^{2} \ldotp\]
Significance
This example illustrates how we consider friction in a dynamics problem. Notice that static friction has a value that matches the applied force, until we reach the maximum value of static friction. Also, no motion can occur until the applied force equals the force of static friction, but the force of kinetic friction will then become smaller.
By the end of this section, you will be able to:
Friction is a force that is around us all the time that opposes relative motion between systems in contact but also allows us to move (which you have discovered if you have ever tried to walk on ice). While a common force, the behavior of friction is actually very complicated and is still not completely understood. We have to rely heavily on observations for whatever understandings we can gain. However, we can still deal with its more elementary general characteristics and understand the circumstances in which it behaves.
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Friction is a force that opposes relative motion between systems in contact.
One of the simpler characteristics of friction is that it is parallel to the contact surface between systems and always in a direction that opposes motion or attempted motion of the systems relative to each other. If two systems are in contact and moving relative to one another, then the friction between them is called kinetic friction. For example, friction slows a hockey puck sliding on ice. But when objects are stationary, static friction can act between them; the static friction is usually greater than the kinetic friction between the objects.
If two systems are in contact and moving relative to one another, then the friction between them is called kinetic friction.
Imagine, for example, trying to slide a heavy crate across a concrete floor&#;you may push harder and harder on the crate and not move it at all. This means that the static friction responds to what you do&#;it increases to be equal to and in the opposite direction of your push. But if you finally push hard enough, the crate seems to slip suddenly and starts to move. Once in motion it is easier to keep it in motion than it was to get it started, indicating that the kinetic friction force is less than the static friction force. If you add mass to the crate, say by placing a box on top of it, you need to push even harder to get it started and also to keep it moving. Furthermore, if you oiled the concrete you would find it to be easier to get the crate started and keep it going (as you might expect).
Figure 1 is a crude pictorial representation of how friction occurs at the interface between two objects. Close-up inspection of these surfaces shows them to be rough. So when you push to get an object moving (in this case, a crate), you must raise the object until it can skip along with just the tips of the surface hitting, break off the points, or do both. A considerable force can be resisted by friction with no apparent motion. The harder the surfaces are pushed together (such as if another box is placed on the crate), the more force is needed to move them. Part of the friction is due to adhesive forces between the surface molecules of the two objects, which explain the dependence of friction on the nature of the substances. Adhesion varies with substances in contact and is a complicated aspect of surface physics. Once an object is moving, there are fewer points of contact (fewer molecules adhering), so less force is required to keep the object moving. At small but nonzero speeds, friction is nearly independent of speed.
Frictional forces, such as f, always oppose motion or attempted motion between objects in contact. Friction arises in part because of the roughness of the surfaces in contact, as seen in the expanded view. In order for the object to move, it must rise to where the peaks can skip along the bottom surface. Thus a force is required just to set the object in motion. Some of the peaks will be broken off, also requiring a force to maintain motion. Much of the friction is actually due to attractive forces between molecules making up the two objects, so that even perfectly smooth surfaces are not friction-free. Such adhesive forces also depend on the substances the surfaces are made of, explaining, for example, why rubber-soled shoes slip less than those with leather soles.
The magnitude of the frictional force has two forms: one for static situations (static friction), the other for when there is motion (kinetic friction).
When there is no motion between the objects, the magnitude of static friction fs is fs &#; μsN, where μs is the coefficient of static friction and N is the magnitude of the normal force (the force perpendicular to the surface).
Magnitude of static friction fs is fs &#; μsN, where μs is the coefficient of static friction and N is the magnitude of the normal force.
The symbol &#; means less than or equal to, implying that static friction can have a minimum and a maximum value of μsN. Static friction is a responsive force that increases to be equal and opposite to whatever force is exerted, up to its maximum limit. Once the applied force exceeds fs(max), the object will move. Thus fs(max) = μsN.
Once an object is moving, the magnitude of kinetic friction fk is given by fk = μkN, where μk is the coefficient of kinetic friction. A system in which fk = μkN is described as a system in which friction behaves simply.
The magnitude of kinetic friction fk is given by fk=μkN, where μk is the coefficient of kinetic friction.
As seen in Table 1, the coefficients of kinetic friction are less than their static counterparts. That values of μ in Table 1 are stated to only one or, at most, two digits is an indication of the approximate description of friction given by the above two equations.
Table 1. Coefficients of Static and Kinetic Friction System Static friction μs Kinetic friction μk Rubber on dry concrete 1.0 0.7 Rubber on wet concrete 0.7 0.5 Wood on wood 0.5 0.3 Waxed wood on wet snow 0.14 0.1 Metal on wood 0.5 0.3 Steel on steel (dry) 0.6 0.3 Steel on steel (oiled) 0.05 0.03 Teflon on steel 0.04 0.04 Bone lubricated by synovial fluid 0.016 0.015 Shoes on wood 0.9 0.7 Shoes on ice 0.1 0.05 Ice on ice 0.1 0.03 Steel on ice 0.4 0.02The equations given earlier include the dependence of friction on materials and the normal force. The direction of friction is always opposite that of motion, parallel to the surface between objects, and perpendicular to the normal force. For example, if the crate you try to push (with a force parallel to the floor) has a mass of 100 kg, then the normal force would be equal to its weight, W = mg = (100 kg)(9.80 m/s2) = 980 N, perpendicular to the floor. If the coefficient of static friction is 0.45, you would have to exert a force parallel to the floor greater than fs(max) =μsN = (0.45)(980)N = 440N to move the crate. Once there is motion, friction is less and the coefficient of kinetic friction might be 0.30, so that a force of only 290 N fk = μkN = (0.30)(980N) = 290N would keep it moving at a constant speed. If the floor is lubricated, both coefficients are considerably less than they would be without lubrication. Coefficient of friction is a unit less quantity with a magnitude usually between 0 and 1.0. The coefficient of the friction depends on the two surfaces that are in contact.
Find a small plastic object (such as a food container) and slide it on a kitchen table by giving it a gentle tap. Now spray water on the table, simulating a light shower of rain. What happens now when you give the object the same-sized tap? Now add a few drops of (vegetable or olive) oil on the surface of the water and give the same tap. What happens now? This latter situation is particularly important for drivers to note, especially after a light rain shower. Why?
Many people have experienced the slipperiness of walking on ice. However, many parts of the body, especially the joints, have much smaller coefficients of friction&#;often three or four times less than ice. A joint is formed by the ends of two bones, which are connected by thick tissues. The knee joint is formed by the lower leg bone (the tibia) and the thighbone (the femur). The hip is a ball (at the end of the femur) and socket (part of the pelvis) joint. The ends of the bones in the joint are covered by cartilage, which provides a smooth, almost glassy surface. The joints also produce a fluid (synovial fluid) that reduces friction and wear. A damaged or arthritic joint can be replaced by an artificial joint (Figure 2). These replacements can be made of metals (stainless steel or titanium) or plastic (polyethylene), also with very small coefficients of friction.
Other natural lubricants include saliva produced in our mouths to aid in the swallowing process, and the slippery mucus found between organs in the body, allowing them to move freely past each other during heartbeats, during breathing, and when a person moves. Artificial lubricants are also common in hospitals and doctor&#;s clinics. For example, when ultrasonic imaging is carried out, the gel that couples the transducer to the skin also serves to to lubricate the surface between the transducer and the skin&#;thereby reducing the coefficient of friction between the two surfaces. This allows the transducer to mover freely over the skin.
A skier with a mass of 62 kg is sliding down a snowy slope. Find the coefficient of kinetic friction for the skier if friction is known to be 45.0 N.
The magnitude of kinetic friction was given in to be 45.0 N. Kinetic friction is related to the normal force N as fk = μkN; thus, the coefficient of kinetic friction can be found if we can find the normal force of the skier on a slope. The normal force is always perpendicular to the surface, and since there is no motion perpendicular to the surface, the normal force should equal the component of the skier&#;s weight perpendicular to the slope. (See the skier and free-body diagram in Figure 3.)
The motion of the skier and friction are parallel to the slope and so it is most convenient to project all forces onto a coordinate system where one axis is parallel to the slope and the other is perpendicular (axes shown to left of skier). N (the normal force) is perpendicular to the slope, and f (the friction) is parallel to the slope, but w (the skier&#;s weight) has components along both axes, namely w&#; and W//. N is equal in magnitude to w&#;, so there is no motion perpendicular to the slope. However, f is less than W// in magnitude, so there is acceleration down the slope (along the x-axis).
That is, N = w&#; = w cos 25º = mg cos 25º.
Substituting this into our expression for kinetic friction, we get fk = μkmg cos 25º, which can now be solved for the coefficient of kinetic friction μk.
Solving for μk gives [latex]\displaystyle\mu_k=\frac{f_k}{N}=\frac{f_k}{w\text{ cos }25^{\circ}}=\frac{f_k}{mg\text{ cos }25^{\circ}}\\[/latex]
Substituting known values on the right-hand side of the equation, [latex]\displaystyle\mu_k=\frac{45.0}{(62\text{ kg})(9.80\text{ m/s}^2)(0.906)}=0.082\\[/latex].
This result is a little smaller than the coefficient listed in Table 5.1 for waxed wood on snow, but it is still reasonable since values of the coefficients of friction can vary greatly. In situations like this, where an object of mass m slides down a slope that makes an angle θ with the horizontal, friction is given by fk = μk mg cos θ. All objects will slide down a slope with constant acceleration under these circumstances. Proof of this is left for this chapter&#;s Problems and Exercises.
An object will slide down an inclined plane at a constant velocity if the net force on the object is zero. We can use this fact to measure the coefficient of kinetic friction between two objects. As shown in Example 1, the kinetic friction on a slope fk = μk mg cos θ. The component of the weight down the slope is equal to mg sin θ (see the free-body diagram in Figure 3). These forces act in opposite directions, so when they have equal magnitude, the acceleration is zero. Writing these out:
fk = Fgx
μk mg cos θ = mg sin θ.
Solving for μk, we find that
[latex]\displaystyle\mu_{\text{k}}=\frac{mg\sin\theta}{mg\cos\theta}=\tan\theta\\[/latex]
Put a coin on a book and tilt it until the coin slides at a constant velocity down the book. You might need to tap the book lightly to get the coin to move. Measure the angle of tilt relative to the horizontal and find μk. Note that the coin will not start to slide at all until an angle greater than θ is attained, since the coefficient of static friction is larger than the coefficient of kinetic friction. Discuss how this may affect the value for μk and its uncertainty.
We have discussed that when an object rests on a horizontal surface, there is a normal force supporting it equal in magnitude to its weight. Furthermore, simple friction is always proportional to the normal force.
The simpler aspects of friction dealt with so far are its macroscopic (large-scale) characteristics. Great strides have been made in the atomic-scale explanation of friction during the past several decades. Researchers are finding that the atomic nature of friction seems to have several fundamental characteristics. These characteristics not only explain some of the simpler aspects of friction&#;they also hold the potential for the development of nearly friction-free environments that could save hundreds of billions of dollars in energy which is currently being converted (unnecessarily) to heat.
Figure 4 illustrates one macroscopic characteristic of friction that is explained by microscopic (small-scale) research. We have noted that friction is proportional to the normal force, but not to the area in contact, a somewhat counterintuitive notion. When two rough surfaces are in contact, the actual contact area is a tiny fraction of the total area since only high spots touch. When a greater normal force is exerted, the actual contact area increases, and it is found that the friction is proportional to this area.
But the atomic-scale view promises to explain far more than the simpler features of friction. The mechanism for how heat is generated is now being determined. In other words, why do surfaces get warmer when rubbed? Essentially, atoms are linked with one another to form lattices. When surfaces rub, the surface atoms adhere and cause atomic lattices to vibrate&#;essentially creating sound waves that penetrate the material. The sound waves diminish with distance and their energy is converted into heat. Chemical reactions that are related to frictional wear can also occur between atoms and molecules on the surfaces. Figure 5 shows how the tip of a probe drawn across another material is deformed by atomic-scale friction. The force needed to drag the tip can be measured and is found to be related to shear stress, which will be discussed later in this chapter. The variation in shear stress is remarkable (more than a factor of ) and difficult to predict theoretically, but shear stress is yielding a fundamental understanding of a large-scale phenomenon known since ancient times&#;friction.
Explore the forces at work when you try to push a filing cabinet. Create an applied force and see the resulting friction force and total force acting on the cabinet. Charts show the forces, position, velocity, and acceleration vs. time. Draw a free-body diagram of all the forces (including gravitational and normal forces).
Express your answers to problems in this section to the correct number of significant figures and proper units.
slope (one that makes an angle of 4º
with the horizontal) under the following road conditions. Assume that only half the weight of the car is supported by the two drive wheels and that the coefficient of static friction is involved&#;that is, the tires are not allowed to slip during the acceleration. (Ignore rolling.)
(a) On dry concrete; (b)
On wet concrete; (c)
On ice, assuming that
[latex]\mu _{\text{s}}=0.100\\[/latex], the same as for shoes on ice.
friction: a force that opposes relative motion or attempts at motion between systems in contact
kinetic friction: a force that opposes the motion of two systems that are in contact and moving relative to one another
static friction: a force that opposes the motion of two systems that are in contact and are not moving relative to one another
magnitude of static friction: [latex]{f}_{\text{s}}\le {\mu }_{\text{s}}N\\[/latex] , where [latex]{\mu }_{\text{s}}\\[/latex] is the coefficient of static friction and N is the magnitude of the normal force
magnitude of kinetic friction: [latex]{f}_{\text{k}}={\mu }_{\text{k}}N\\[/latex], where [latex]{\mu }_{\text{k}}\\[/latex] is the coefficient of kinetic friction
1. 5.00 N
4. (a) 588 N; (b) 1.96 m/s2
6. (a) 3.29 m/s2; (b) 3.52 m/s2; (c) 980 N, 945 N
10. 1.83 m/s2
14. (a) 4.20 m/s2; (b) 2.74 m/s2; (c) &#;0.195 m/s2
16. (a) 1.03 × 106 N; (b) 3.48 × 105 N
18. (a) 51.0 N; (b) 0.720 m/s2
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