Block on a Rigid Horizontal Surface

In this section, we explore the basic principles of dry friction by looking at a problem involving a block on a rigid horizontal surface. Consider the block of weight W, and let's examine the response of the block to the horizontal force P.

At P = 0, the body is completely at rest with the normal force equal to the weight and no friction. As P is gradually increased, the friction force F of equal magnitude is generated to keep the block in equilibrium. Once P becomes equal to the maximum friction force F_s, the block reaches an unstable equilibrium state. This means that it is on the verge of sliding. If P is increased beyond this value, the block will begin to slip in an accelerated motion, and the friction force actually drops to a value known as kinetic friction. If the value of P is then reduced to that equal to the kinetic friction, the block will continue to slide at a constant speed (i.e., acceleration = 0). The relationship between the applied force and friction is captured by the curve shown.

The maximum static friction force and kinetic friction force are both related to the normal force at the surface of contact through the relations

	(1)
	(2)

where μ_s and μ_k are the coefficients of static and kinetic friction, respectively. In general, the values of friction coefficients are dictated by the block and support surface materials. Some typical values for μ_s can be found in this Table.

The response of block to force P is captured in the following four cases.

Case 1: P < F_s, h < h_max. Block remains at rest and the equilibrium equations indicate that

There is no relationship between the friction and the normal force in this case.

Case 2: P = F_s, h < h_max. Block is about to slide to the right. This impending motion is detected since P is now equal to the maximum friction force. That is

Case 3: P > F_s, h < h_max. Block will slide to the right with an acceleration as the applied force is greater than the maximum resistance or friction force at the surface of contact. Once the block begins to slide, it is possible to maintain a uniform (zero acceleration) motion by reducing the value of P to that equal to the kinetic friction equation .

Case 4: Combination of P and h causing the block to tip over. In this case, the position of the normal force is clearly defined to be at the right edge of the block as that is the only place where the block is in contact with the horizontal surface. To determine whether tipping is a possibility or not, we solve for the value of h_max using the moment equilibrium as

If h = h_max and P < F_s, then the block is on the verge of tipping over with no possibility of sliding.

If b greater than or equals to h h_max and P < F_s, then the block will tip over with the right edge remaining stationary relative to the horizontal surface.

If h = h_max and P = F_s, then both tipping and sliding are impending. There is an equal chance that the block will either tip over or begin to slide.