AcademyApplying Force Models
Academy
Static and Kinetic Friction
Level 1 - Physics topic page in Applying Force Models.
Principle
Friction adjusts while surfaces stick and takes a fixed model value after sliding begins.
Notation
\(f_s\)
static friction magnitude
\(\mathrm{N}\)
\(f_k\)
kinetic friction magnitude
\(\mathrm{N}\)
\(\mu_s\)
coefficient of static friction
1
\(\mu_k\)
coefficient of kinetic friction
1
\(N\)
normal reaction magnitude
\(\mathrm{N}\)
Method
Derivation 1: Test whether sticking is possible
Static friction is not automatically at its maximum value. First find the friction required to keep the surfaces from slipping.
Assume no slipping
\[a_{\parallel}=0\]
Find required friction
\[\sum F_{\parallel}=0\Rightarrow f_{s,\mathrm{req}}\]
Compare with limit
\[f_{s,\mathrm{req}}\le\mu_sN\Rightarrow\text{sticking is possible}\]
Derivation 2: Switch model after slipping starts
If the required static friction is larger than the maximum available value, the surfaces slide and the kinetic-friction model replaces the static one.
Static maximum
\[f_{s,\max}=\mu_sN\]
Kinetic value
\[f_k=\mu_kN\]
Direction rule
\[\vec f\ \text{opposes relative or impending relative slipping}\]
The free-body diagram shows a horizontal push and a friction force parallel to the surface. The normal force must be found before either friction model can be evaluated.
Rules
These are the compact results from the method above.
Static range
\[0\le f_s\le \mu_sN\]
Static maximum
\[f_{s,\max}=\mu_sN\]
Kinetic friction
\[f_k=\mu_kN\]
Level normal
\[N=mg\]
Slope normal
\[N=mg\cos\theta\]
Examples
Question
An
\[8.0\,\mathrm{kg}\]
box is pushed horizontally by \[25\,\mathrm{N}\]
If \[\mu_s=0.40\]
does it move?Answer
On a level surface,
\[N=mg\]
The maximum static friction is \[f_{s,\max}=\mu_smg=0.40(8.0)(9.8)=31.4\,\mathrm{N}\]
Since \[25\,\mathrm{N}<31.4\,\mathrm{N}\]
it stays at rest and \[f_s=25\,\mathrm{N}\]
Checks
- Static friction is not automatically equal to \(\mu_sN\).
- Kinetic friction is used only after slipping begins.
- Friction is parallel to the contact surface.
- Find the normal force before calculating a friction limit.
- Friction opposes relative slipping, not necessarily motion relative to the ground.