Friction - recap
Friction is the resistance encountered by one body moving over another.
As seen previously when two solid surfaces are placed together, contact will occur only at asperities. Frictional forces and wear originate due to the interlocking of asperities because, in order for the surfaces to move relative to each other, asperities must deform and/or fracture, and adhesive forces must be overcome. In general, the greater the proportion of the surface that is in asperity contact, the greater the frictional force.
These diagrams show that the true area of contact is far less than the apparent area of contact (asperities circled).
Box on a slope:
For the above block to be able to slide to the right, the applied horizontal force must be greater than the frictional force, F.
Now, \[F \le {\rm{ }}\mu N\]
where N is the normal load and μ is the coefficient of friction. It is a common observation that frictional force needed to initiate motion is greater than that needed to maintain it, i.e., μstatic > μdynamic. The higher the value of μ, the steeper the slope can be for the box to remain stationary:
When the object is stationary:
Resolving parallel to the normal force, N, we have:
N = W cos θ
Resolving parallel to the frictional force, F, we have:
F = W cos (90° - θ) = W sin θ
Hence,
\[\frac{F}{N} = \tan \theta \]
and so the limiting angle at which the object remains on the slope, θcrit, therefore determines the coefficient of friction, μ:
μ = tan θcrit
[An analogous problem arises when leaning a uniform ladder against a smooth vertical wall where the bottom of the ladder is in contact with rough ground.]