Friction conditions; sliding or overturning of block

Conditions of friction

 No friction      No motion        Impending         Motion
 (P = 0)          (P < F_max)      motion            (P > F_max)
   W                W                W                W
   |                |                |                |
 -----[ ]-----   P->[ ]F<--      P->[ ]F=μsN<-     P->[ ]F=μkN<-
 N = W, F = 0    F = P           F = F_max         F < F_max
 at rest         at rest         about to slide    sliding, accelerates

• No friction: no applied tangential force (P = 0). Friction force F = 0, only the normal reaction N = W acts.
• No motion: P is applied but is small. The body stays at rest and F = P, so F is less than the maximum value, F < μₛN.
• Impending motion: P is increased until the body is just about to slide. F reaches its maximum (limiting) value F_max = μₛN.
• Motion: if P exceeds F_max the body slides. The friction force falls to the kinetic value F = μₖN, which is slightly less than F_max.

Sliding or overturning of a block

A block of weight W, base width b and height h is pulled by a horizontal force P applied at height a above the base. The coefficient of static friction is μ.

          P
      ----->+------+          a = height of P
            |      |          b = width of base
            |  W   |
   ---------+------+---> 
            F      N

• Sliding starts when P reaches the limiting friction: P_slide = μ W.
• Overturning (tipping) about the toe starts when the overturning moment equals the restoring moment, with the normal reaction acting at the toe: P a = W (b/2), so P_tip = W b / (2a).

Decision: compare the two values of P.

• If μW < Wb/(2a), that is μ < b/(2a), the block slides first.
• If Wb/(2a) < μW, the block overturns first.
• At equal values both are about to occur together.

Example: W = 500 N, b = 0.6 m, a = 1.0 m, μ = 0.4. P_slide = 200 N. P_tip = 500 × 0.6/(2 × 1.0) = 150 N. Since 150 N < 200 N, the block overturns before it slides.