1. Laws of Dry Coulomb Friction

Friction is the tangential resistive force developed at the contact interface between two physical surfaces that opposes relative motion or the tendency toward relative motion.

Fundamental Friction Relations

\[ F_{\text{max}} = \mu_s N \] \[ \tan \phi = \mu_s \quad (\text{Angle of Friction}) \] \[ \theta = \phi \quad (\text{Angle of Repose equals Angle of Friction}) \]

Where \( N \) is the normal reaction force perpendicular to the contact plane, \( \mu_s \) is the static coefficient of friction, and \( \phi \) is the limiting angle of friction.

Equilibrium of a Body on an Inclined Plane

For a block of weight \( W \) situated on a rough plane inclined at angle \( \alpha \) to the horizontal with coefficient of friction \( \mu = \tan\phi \):

  • Case 1: Minimum force \( P \) parallel to plane to prevent down-sliding:
    \( P_{\text{min}} = W \frac{\sin(\alpha - \phi)}{\cos\phi} \). If \( \alpha \leq \phi \), the body is self-locking and will never slide under its own weight.
  • Case 2: Minimum force \( P \) parallel to plane to pull body upward:
    \( P = W \frac{\sin(\alpha + \phi)}{\cos\phi} \).
  • Case 3: Minimum force in ANY direction to initiate upward motion:
    \( P_{\text{least}} = W \sin(\alpha + \phi) \) applied at angle \( \theta = \phi \) above the inclined surface.

2. Ladder Friction Analysis

A classic exam problem: A uniform ladder of length \( L \) and weight \( W \) rests against a rough vertical wall (friction coefficient \( \mu_w \)) and a rough floor (friction coefficient \( \mu_f \)).

Applying the 3 static equilibrium conditions:

  1. \( \sum F_x = 0 \implies N_w = F_f = \mu_f N_f \)
  2. \( \sum F_y = 0 \implies N_f + F_w = W \implies N_f + \mu_w N_w = W \)
  3. \( \sum M_A = 0 \) (Taking moments about the floor contact point \( A \)).

3. Rapid Zero-Force Member Inspection in Trusses

In competitive exams like BCECE LE and GATE, calculating forces in every member by writing simultaneous equations is too slow. Use these two visual zero-force inspection theorems:

Inspection Rule 1: Two Non-Collinear Members

If only two members meet at an unloaded joint and they are not collinear (do not lie on the same straight line), both members carry zero force: \( F_1 = 0 \) and \( F_2 = 0 \).

Inspection Rule 2: Three Members with Two Collinear

If three members meet at an unloaded joint and two of them are collinear, the third non-collinear member must carry zero force, while the two collinear members carry equal forces.