Statics · Static and kinetic friction
Statics · Static and kinetic friction
When the surface joins the calculation
So far all contact surfaces were smooth: supports transmitted exactly the forces their valency allowed. Real surfaces are rough — they additionally transmit a force along the contact surface. Using the crate on the loading ramp, this module develops how large that force can become, why it turns into an inequality rather than an equation, and what that means for the ladder against the hall wall.
To the exercises →Solve this chapter's task to keep reading.
The crate on the ramp
On the right stands a crate of weight on the loading ramp of the hall. The ramp is inclined by , and a rope holds the crate parallel to the ramp with the force .
Resolving into the directions normal to the ramp and along the ramp gives and the downslope force . Without a rope and without a rough surface, these 273.62 N would accelerate the crate down the slope.
But it does not — at least not immediately. The contact surface transmits a force along the ramp. This tangential component is exactly what this chapter is about.
A roller support transmits exactly one force, a pinned support exactly two — the valency is a property of the construction. The rough surface is different: how much it transmits tangentially depends on how heavily it is loaded. It has no fixed valency, but a limit.
Static and sliding friction
The textbook separates two cases that are both called “friction” in everyday speech. If the body stays at rest, one speaks of static friction; if it moves relative to the surface, of sliding friction. The difference is not merely linguistic — the two forces are of fundamentally different kinds:
- The static friction force is a reaction force. For statically determinate systems it can be determined from the equilibrium conditions — exactly like a support reaction.
- The sliding friction force is an applied force. Its magnitude is prescribed and does not depend on what equilibrium demands.
From this follows a rule often overlooked in free-body diagrams: because is a reaction force, its sense of direction in the free-body diagram may be assumed arbitrarily. If a negative value comes out in the end, it simply points the other way — just as with a support force.
What does the static friction force depend on as long as the body remains at rest?
Coulomb's laws of friction
The magnitude of the static friction force cannot exceed a limiting static friction force . A body sticks to another one if the condition of sticking is fulfilled:
Note the inequality sign. is not the value of the static friction force but its upper limit; which value actually takes is still decided by equilibrium. Only in the limiting case — just before the body slips — does .
If the body moves, an equation applies instead of the inequality, Coulomb's law of friction . The coefficient of static friction and the coefficient of sliding friction belong to material pairings and are determined experimentally; usually is somewhat smaller than .
On the ramp, , so . The downslope force is 273.62 N — static friction alone is therefore not enough. Without a rope the crate slips.
In the size of the contact area does not occur — a wide and a narrow block of equal weight stick equally well. The reason: at twice the area the contact pressure halves, so the product stays the same. That holds only approximately, but well enough for statics.
A crate stands on level ground and nobody pushes. How large is ?
Angle of friction and friction wedge
The normal force and the static friction force can be combined into a resultant force . Its direction is given by the angle relative to the normal:
In the limiting case this becomes . The textbook calls this limiting angle the angle of friction ; it is a direct measure of the coefficient of static friction. On the ramp, .
Plotting to both sides of the normal in the plane problem, a friction wedge. As long as lies inside the wedge, , and the body remains at rest. That is the same statement as the condition of sticking — only as a drawing instead of an inequality.
On the inclined plane without a rope, coincides with the vertical, and is exactly the angle of inclination . The crate therefore holds by itself exactly when is.
Hands-on task: Set the ramp so flat that the crate just barely holds without a rope.
Once you have hit the limiting angle, check here.
In the plane problem the shape is a wedge, because can only tilt to two sides. In space the resultant can tilt in any direction — there a friction cone arises, a cone of revolution about the normal with the apex angle . The condition stays the same: if lies inside, the body sticks.
Problem types and solution path
Static friction problems almost always ask one of two questions: Does the system hold under the given load? or Within what limits must a force lie for it to hold? The same route answers both: isolate the body, write down equilibrium, apply the condition of sticking.
The sense of direction of may be assumed arbitrarily. The inequality then yields two limiting cases — which is why the result is a range and not a single number.
Hands-on task: Set the rope force so that the crate is just barely not pulled up the slope.
Once you have hit the upper limit, check here.
sliding friction
If the crate slips, the character of the problem changes. The inequality becomes the equation , and is thereby known from the outset — an applied force, no longer an unknown.
In return a new question arises: the direction. The friction force is always directed against the (relative) velocity . It therefore follows the motion, not the loading — and reverses when the crate is pulled up the slope instead of down.
“Relative” is to be taken literally: if not the body but the surface moves — a workpiece on a running belt, say — then what counts is the velocity of the body with respect to the surface, and the friction force drives it instead of braking it.
How does the calculation differ as soon as the body slips?
Systems with several contact points
It gets interesting when a body touches at several places. The ladder against the hall wall is the classic case: it stands at on rough ground () and leans against a smooth wall. A person of weight stands at of the ladder length.
Because the wall is smooth, it transmits only a normal force — horizontally. At the ground this horizontal force has to be taken up by static friction. The ladder therefore slips as soon as the wall pushes harder than the ground can hold.
Hands-on task: Move the person up until the ladder just barely holds.
Once you have hit the highest point, check here.
In the moment balance, appears on both sides and drops out. Whether the ladder is two or nine metres long is therefore irrelevant for slipping — only the Angle and the relative position of the person count. That is why ladders carry a stated inclination angle and not a length.
The ladder is set more steeply . What follows from that?