When exactly is friction limiting? I have seen so many problems that assume
\[ f = \mu N \] but many that don't.
If the body is moving relative to the surface, then is friction limiting? What about in rolling motion? Thanks, everyone.

Actually, for real materials, there are two different values of the coefficient of friction: the coefficient of static friction and the coefficient of kinetic friction. Basically, when you pull a body tangentially across another surface with increasing force, you get a graph like this for the frictional force:

Graph

What happens is that the frictional force keeps increasing with the applied force (the body is still at rest), until suddenly, the applied force breaks the interaction forces between the two surfaces, causing it to start moving. The value of friction at this instant is called the limiting friction.

While moving relative to each other, the two surfaces experience a frictional force less than this limiting frictional force that is required to start the motion in the first place. This is explained by interactions at the microscopic level.

Note that the frictional force in the kinetic case is constant: when a body moves relative to a surface, it experiences a friction force equal to \(\mu_k N\).

Hence, we use two coefficients of friction, one for the static case \(\mu_s\) and one for the kinetic case \(\mu_k\). The confusion usually arises because many problems assume \(\mu_s = \mu_k\) without explicitly stating it.

In general, friction is a complicated interplay of molecular forces between the surface and the moving object, as well as dissipation as the object itself deforms throughout the roll. Whenever you can approximate the friction as some coefficient times the normal force, it is most likely an approximation made by the person posing the problem so that the problem is approachable. For example, rolling a cylinder made of clay will dissipate a lot of energy to the clay as it reshapes during the roll, but a hard steel cylinder will lose less due to this effect, and more due to interactions with the surface.

@Josh Silverman
–
Yeah, what Josh Silverman said. The expression for the frictional force is empirically derived and is hence invalid in several cases. You might want to go through these couple of websites; they clearly explain the role of friction in rolling (see what I did there? :P)

@Raj Magesh
–
Oh, and in general, I can't think of any reason why frictional force in the pure rolling case should be equal to limiting friction, since the point of contact is essentially static.

If a body is in pure rolling with no external force being applied, frictional force is of course going to be zero. However, if a body is being pushed, assume some frictional force \(f \) to be acting in an arbitrary direction, write translational and rotational dynamics equations and solve them to obtain the required value of frictional force, which is not necessarily limiting friction. If you get a value of friction greater than the limiting friction, then you know that the body cannot be in pure rolling, since friction isn't sufficient to support it.

The coefficient of rolling friction is completely different from what we are talking about: the friction that is necessary to start rolling motion is NOT given by \(f = \mu_r N\). This friction is not for an ideal rigid body; rather, it describes the net resistive forces present that try to prevent rolling motion from occurring, like deformations at the contact surface. Rolling friction is generally ignored in most physics problems.

When two surfaces are in contact then due to pressure they two surface makes bonds with each other called as cold-welds. Untill these cold welds are intact the friction is statics and when these are broken then the friction becomes kinetic.
limiting friction is a static friction when these cold welds comes to verge of breaking. Or it can be said that limiting friction is maximum value of static friction.

As in, friction is not always given by \[ f = \mu N .\]
where N is the normal force acting. This is the maximum (limiting) value of friction, The frictional force can take any value lesser than this. But often I've seen solutions involving friction considering this maximum as the friction.
That's what got me confused.

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TopNewestActually, for real materials, there are two different values of the coefficient of friction: the coefficient of static friction and the coefficient of kinetic friction. Basically, when you pull a body tangentially across another surface with increasing force, you get a graph like this for the frictional force:

Graph

What happens is that the frictional force keeps increasing with the applied force (the body is still at rest), until suddenly, the applied force breaks the interaction forces between the two surfaces, causing it to start moving. The value of friction at this instant is called the limiting friction.

While moving relative to each other, the two surfaces experience a frictional force less than this limiting frictional force that is required to start the motion in the first place. This is explained by interactions at the microscopic level.

Note that the frictional force in the kinetic case is constant: when a body moves relative to a surface, it experiences a friction force equal to \(\mu_k N\).

Hence, we use two coefficients of friction, one for the static case \(\mu_s\) and one for the kinetic case \(\mu_k\). The confusion usually arises because many problems assume \(\mu_s = \mu_k\) without explicitly stating it.

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So if the body is rolling on a surface, we can always say \[ f = \mu_r N \] if \[\mu_r\] is the coefficient of rolling friction?

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It really depends on the situation.

In general, friction is a complicated interplay of molecular forces between the surface and the moving object, as well as dissipation as the object itself deforms throughout the roll. Whenever you can approximate the friction as some coefficient times the normal force, it is most likely an approximation made by the person posing the problem so that the problem is approachable. For example, rolling a cylinder made of clay will dissipate a lot of energy to the clay as it reshapes during the roll, but a hard steel cylinder will lose less due to this effect, and more due to interactions with the surface.

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If a body is in pure rolling with no external force being applied, frictional force is of course going to be zero. However, if a body is being pushed, assume some frictional force \(f \) to be acting in an arbitrary direction, write translational and rotational dynamics equations and solve them to obtain the required value of frictional force, which is not necessarily limiting friction. If you get a value of friction greater than the limiting friction, then you know that the body cannot be in pure rolling, since friction isn't sufficient to support it.

The coefficient of rolling friction is completely different from what we are talking about: the friction that is necessary to start rolling motion is NOT given by \(f = \mu_r N\). This friction is not for an ideal rigid body; rather, it describes the net resistive forces present that try to prevent rolling motion from occurring, like deformations at the contact surface. Rolling friction is generally ignored in most physics problems.

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isquite confusing.Log in to reply

When two surfaces are in contact then due to pressure they two surface makes bonds with each other called as cold-welds. Untill these cold welds are intact the friction is statics and when these are broken then the friction becomes kinetic.

limiting friction is a static friction when these cold welds comes to verge of breaking. Or it can be said that limiting friction is maximum value of static friction.

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@Ameya Daigavane can you say more about what you mean by friction being limiting?

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As in, friction is not always given by \[ f = \mu N .\] where N is the normal force acting. This is the maximum (limiting) value of friction, The frictional force can take any value lesser than this. But often I've seen solutions involving friction considering this maximum as the friction. That's what got me confused.

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