Summary Friction of Rigid Bodies - Engineering Mechanics - Civil Engineering

Table of Contents
1. Types of Friction
2. Types of Dry Friction
3. Comparison: Static vs Kinetic Friction
4. Laws of Coulomb's (Dry) Friction
5. Limiting Force of Friction
6. Coefficient of Friction
7. Angle of Friction
8. Cone of Friction
9. Angle of Repose
10. Simple Example: Block on an Inclined Plane (impending motion)
11. Remarks, Limitations and Practical Points
12. Applications and Examples in Engineering
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A. FRICTION

In many problems of statics and dynamics we assume the forces between contacting bodies act normal to the surfaces. That assumption is valid only for smooth surfaces. Real surfaces have microscopic irregularities (asperities). When two surfaces are pressed together and there is a tendency for relative motion, these asperities interlock and resist motion. The resistance that arises at the contacting surfaces is called friction.

Types of Friction

Friction is classified according to the nature of the interacting media and the type of motion:

• Dry friction (or solid friction) - occurs between unlubricated solid surfaces in contact when there is sliding or a tendency to slide.
• Fluid friction - occurs within fluids (liquids or gases) when adjacent layers move with different velocities; includes viscous effects and drag.
• Internal friction (or hysteresis) - occurs inside solid materials when subjected to cyclic loading; energy is dissipated during repeated deformation and recovery.

Types of Dry Friction

Dry friction is further classified by the relative motion between the contacting surfaces:

• Static friction - the frictional force acting when two surfaces are at rest with respect to each other but there is a tendency to move. It prevents the start of motion up to a limiting value.
• Kinetic (or dynamic) friction - the frictional force acting when two surfaces slide relative to each other. It usually has a nearly constant value (for a given pair of materials and contact conditions) and is generally less than the limiting static friction.
• Rolling resistance - a much smaller retarding force when a body rolls over a surface; arises from deformation of the rolling body and/or the surface.

Comparison: Static vs Kinetic Friction

Laws of Coulomb's (Dry) Friction

The commonly used empirical laws of dry friction (Coulomb's laws) are:

• The direction of the frictional force on a body is opposite to the direction of the tendency of relative motion between the contacting surfaces, tangent to the surface at the point of contact.
• The frictional force is approximately independent of the apparent area of contact for rigid bodies under normal engineering conditions.
• The frictional force depends on the nature of the surfaces in contact and the normal reaction between them.
• The limiting frictional force is proportional to the normal reaction: Fs(max) = μs N, where Fs(max) is the limiting static friction and N is the normal reaction.
• When one body is just on the verge of sliding over another, the frictional force has its maximum value (limiting static friction).
• Limiting static friction is generally greater than kinetic friction for the same pair of surfaces.
• Kinetic friction is approximately independent of the relative sliding velocity over a broad low-to-moderate speed range (with exceptions in some materials and lubricated contacts).

Limiting Force of Friction

The limiting static frictional force is the maximum value of static friction that can act at a contact before sliding begins. If the applied tangential force exceeds this limit, relative motion commences and kinetic friction applies.

For a block on a surface the limiting static friction Fs satisfies

where μ_s is the coefficient of static friction and N is the normal reaction (normal force) at the contact.

Coefficient of Friction

The coefficient of friction is a dimensionless quantity that characterises the interaction between two surfaces:

• Coefficient of static friction, μs = limiting static friction ÷ normal reaction = Fs(max) ÷ N.
• Coefficient of kinetic friction, μk = kinetic friction ÷ normal reaction = Fk ÷ N.
• Both μs and μk depend on material pair and surface condition (smooth, rough, lubricated, dry, contaminated, etc.).
• Units: dimensionless (no units).

Angle of Friction

The angle of friction, denoted by φ, is the angle between the resultant of the normal reaction and the frictional force and the normal reaction itself at the point of contact.

If the frictional force has magnitude F and the normal reaction is N, then the resultant R makes an angle φ with N and

tan φ = F / N

When the body is on the verge of sliding, F = Fs(max) and tan φ = μs.

Cone of Friction

The cone of friction at a contact point is a right circular cone whose axis is along the normal reaction and whose semi‐vertical angle is the angle of friction φ. If the resultant contact force R lies inside this cone, the contact can remain without sliding; if R lies on the surface of the cone the contact is at impending sliding; if R lies outside, sliding occurs.

Angle of Repose

The angle of repose is the maximum angle of inclination of a plane on which a body placed on it remains at rest under the action of gravity and friction alone.

For a rigid body on the verge of sliding on an inclined plane, the angle of repose θ satisfies

tan θ = μs

Hence, for impending motion, the angle of repose θ equals the angle of friction φ: θ = φ.

Simple Example: Block on an Inclined Plane (impending motion)

Consider a block of weight W on a plane inclined at angle α to the horizontal. Let the block be just about to slide down. The equilibrium of forces gives:

The component of weight down the plane is W sin α.

The normal reaction is N = W cos α.

At impending motion, limiting friction Fs(max) = W sin α.

Using Fs(max) = μs N we get

W sin α = μ_s W cos α

Therefore

tan α = μ_s

So the limiting angle α at which the block will just start to slide equals the angle of repose φ where tan φ = μs.

Remarks, Limitations and Practical Points

• Coefficients μ_s and μ_k are determined experimentally; typical values depend on material pairs (e.g., rubber on concrete is large, polished steel on steel is much smaller, lubricated contacts have very low μ).
• Friction is an empirical phenomenon; Coulomb's laws are approximations that are valid for many engineering contexts but have exceptions (dependency on speed, temperature, surface contamination, adhesion, real contact area).
• Rolling resistance is affected by deformation and is not accurately modelled by μ_N; it is often expressed as a moment or an effective resistance force proportional to N but depending on geometry and material properties.
• Frictional forces convert mechanical work into heat; in machine design, bearings, brakes and clutches exploit or minimise friction depending on the application.

Applications and Examples in Engineering

• Design of ramp angles and slopes using angle of repose.
• Determining minimum friction required to prevent sliding of retaining elements, foundations, or stacked materials.
• Brake and clutch design using controlled frictional contact.
• Analysis of belt drives, frictional contact in bolted joints and contact mechanics.
• Calculation of safe traction for vehicles and foundations using appropriate coefficients of friction.