14  Friction Force

Author

Theresa Honein

Published

January 1, 2026

Most theories of friction arise from studies by the French scientist Charles Augustin de Coulomb (1736–1806).

14.1 Introduction

Consider a block on a rough horizontal surface pushed with force \({\bf P} = P{\bf E}_x\).

Upon increasing \(P\), we observe:

  • For small \(P\), the block remains at rest.
  • Beyond a critical value \(P = P^*\), the block starts to move.
  • Once in motion, a constant value \(P = P^{**}\) is required to maintain constant speed.
  • Both \(P^*\) and \(P^{**}\) are proportional to the normal force \(\lnorm{\bf N}\rnorm\).

14.2 Formalism: Coulomb’s Theory of Friction

We distinguish two types of friction: static and kinetic.

14.2.1 Static Friction

When the relative velocity is zero (\({\bf v}_{rel} = {\bf 0}\)), the friction force \({\bf F}_f\) is unknown and lies along the contact surface. We must solve for both its magnitude and direction. We say the static friction force points in the direction that impedes motion.

TipThink!

Question: Consider a block on a rough inclined surface that is not moving. Draw the FBD and write the equation of motion. Do you have as many equations as unknowns?

Yes. The friction and normal forces have unknown magnitudes, but the block’s acceleration is known (it matches the surface acceleration). There are as many equations as unknowns.

The static friction criterion: \[\begin{align} \lnorm{\bf F}_f\rnorm \leq \mu_s\lnorm{\bf N}\rnorm, \end{align}\] where \(\mu_s\) is the static friction coefficient.

14.2.2 Kinetic Friction

When two bodies are in relative motion (\({\bf v}_{rel} \neq {\bf 0}\)), the friction force is known: \[\begin{align} {\bf F}_f = \mu_k\lnorm{\bf N}\rnorm\lp-\frac{{\bf v}_{rel}}{\lnorm{\bf v}_{rel}\rnorm}\rp, \end{align}\] where \(\mu_k\) is the kinetic friction coefficient.

Remarks:

  • \(\mu_k\) is also written \(\mu_d\) (dynamic friction coefficient).
  • Kinetic friction is a prescription, not a constraint force.
  • The magnitude of kinetic friction does not depend on speed.

14.2.3 Example: Crate Pushed with Incremental Force

  1. Kinematics: \({\bf r} = x{\bf E}_x+y_0{\bf E}_y\), \({\bf v} = \dot{x}{\bf E}_x\), \({\bf a} = \ddot{x}{\bf E}_x\).

  2. FBD with forces \({\bf P} = P{\bf E}_x\), \({\bf W} = -mg{\bf E}_y\), \({\bf N} = N{\bf E}_y\), \({\bf F}_f = F_f(-{\bf E}_x)\).

    When kinetic: \({\bf F}_f = -\mu_k\lnorm{\bf N}\rnorm\,\text{sgn}(\dot{x}){\bf E}_x\).

  3. BoLM: \[\begin{align} P{\bf E}_x-mg{\bf E}_y+N{\bf E}_y-F_f{\bf E}_x = m\ddot{x}{\bf E}_x. \end{align}\]

  4. Initially at rest (static friction). Setting \(\ddot{x}=0\): \[\begin{align} P = F_f, \quad N = mg. \end{align}\] The block stays at rest while \(P \leq \mu_s mg\). The maximum static value is \(P^* = \mu_s mg\).

Once sliding, \({\bf F}_f = -\mu_k mg{\bf E}_x\) and: \[\begin{align} \ddot{x} = \frac{P}{m}-\mu_k g. \end{align}\] For constant speed, \(P^{**} = \mu_k mg\).

14.2.4 Example: A Box on a Ramp

(MKB problem 03/002.)

When it is unknown whether the block is in motion:

  1. Assume sticking: \({\bf v}_{rel} = {\bf 0}\).
  2. Calculate the static friction force.
  3. Check the static friction criterion.
    • If satisfied: the block sticks. Analysis complete.
    • If not satisfied: the block slides. Apply kinetic friction and resolve.

14.3 Relative Velocity

A common misconception: if an object is moving, the friction acting on it must be kinetic. Counter-example: a crate on a moving truck. If the crate moves with the truck (same velocity), the friction between them is static. Only if the crate slides on the truck bed is the friction kinetic.

Similar examples include a person in an elevator and a person riding a bus.

14.3.1 Example: Nonzero Relative Velocity

  • Box held fixed on a moving conveyor belt: The belt moves, so \({\bf v}_{rel} = -{\bf v}_c \neq {\bf 0}\) — friction is kinetic. The box is stationary in space but moving relative to the belt.

  • Person standing in an accelerating bus: Friction is static. The impending motion of the person (relative to the bus) is toward the rear, so the static friction force points toward the front.

  • Car rolling without slipping: Friction is static and points in the direction of motion, opposing the tendency of the tire to slip backward.

14.4 Normal and Friction Force Prescription

See this video.

  • For a particle moving on a space curve: \({\bf N} = N_n{\bf e}_n+N_b{\bf e}_b\) and \({\bf F}_f = F_f{\bf e}_t\).
  • For a particle moving on a surface with normal \({\bf n}\) and tangent directions \({\bf t}_1\), \({\bf t}_2\): \({\bf N} = N{\bf n}\) and \({\bf F}_f = F_{f1}{\bf t}_1+F_{f2}{\bf t}_2\).

14.5 Summary

  • Static friction: \(\|\mathbf{F}_f\|\le\mu_s\|\mathbf{N}\|\).
  • Kinetic (Coulomb) friction: \(\mathbf{F}_f = -\mu_k\|\mathbf{N}\|\dfrac{\mathbf{v}_{\mathrm{rel}}}{\|\mathbf{v}_{\mathrm{rel}}\|}\).

Strategy: assume sticking; check the static criterion. If violated, resolve with kinetic friction.

14.6 Lecture Videos

14.7 Exercises

The following problems are from Set 10 – Friction.

1. [MKB 03-002] Consider taking \(\mathbf{E}_x\) along and \(\mathbf{E}_y\) perpendicular to the incline. Assume sticking first; check the static criterion. (ans. (a) \(a=1.118\) m/s\(^2\) down incline; (b) \(a=0\); (c) \(a=2.04\) m/s\(^2\) up incline)

2. [MKB 03-030] Apply the 4 steps twice: first for the combined system, then for mass \(m_2\) alone. (ans. \(0.0577(m_1+m_2)g \le P \le 0.745(m_1+m_2)g\))

3. [MKB 03-047] Compare the Serret–Frenet basis here with Problem 03-043 from Set 07. (ans. \(N=156.2\) lb, \(a=27.7\) ft/sec\(^2\))

4. [03-049] Set up a cylindrical polar coordinate system with origin at the base of the cylinder axis; apply the 4 steps to the small object \(A\). (ans. \(\omega=\sqrt{\mu_s g/r}\))

5. [03-050] This determines the maximum lateral acceleration before slipping. (ans. \(a_n=0.818g\), \(F=2460\) lb)

6. [03-071] Polar coordinate system with origin at the centre of the disk. (ans. \(N=\frac{\mu_s g}{4\pi^2}\left(\frac{1}{r\alpha}-1\right)\))