Unit 2 - Notes

MEC107 7 min read

Unit 2: Friction

1. Introduction to Friction

Friction is a fundamental concept in engineering mechanics defined as the opposing force that comes into play when two bodies are in contact and one body tends to move or actually moves over the surface of the other.

Nature of Force: It is a reactive force; it does not exist unless there is an active force trying to cause motion.
Cause of Friction: No surface is perfectly smooth. At a microscopic level, surfaces have irregularities (peaks and valleys). When two surfaces are placed together, these irregularities interlock. Additionally, molecular adhesion (van der Waals forces) between the surfaces contributes to the resistance against motion.
Significance: Friction is both a necessity (e.g., walking, braking, transmitting power via belts) and a nuisance (e.g., causing wear and tear, energy loss in bearings and engines).

2. Types of Friction

Friction is broadly classified based on the state of the contacting surfaces and the medium between them:

A. Dry Friction (Coulomb Friction)

Occurs when there is no lubricating fluid between two solid, contacting surfaces. This is the primary focus of rigid body mechanics.

B. Fluid Friction

Occurs when adjacent layers in a fluid (liquid or gas) move at different velocities. It depends heavily on the viscosity of the fluid and the relative velocity between layers.

C. Internal Friction

Occurs within solid materials subjected to cyclic loading. It is responsible for the dissipation of energy during elastic deformation (related to material damping).

3. Static and Dynamic (Kinetic) Friction

Within the realm of dry friction, the frictional force changes depending on whether the body is at rest or in motion.

A. Static Friction

Definition: The friction experienced by a body when it is at rest, but an external force is trying to move it.
Characteristics: It is a self-adjusting force. As the applied external force increases, the static frictional force increases equally to keep the body in equilibrium ( $F = P$ ).

B. Dynamic (Kinetic) Friction

Definition: The friction experienced by a body when it is actually in motion over another surface.
Characteristics: Its magnitude is relatively constant and is slightly less than the maximum static friction.
Sub-types of Dynamic Friction:
1. Sliding Friction: Experienced when one body slides over another (e.g., a block pushed across a floor).
2. Rolling Friction: Experienced when a body rolls over a surface (e.g., wheels, ball bearings). Rolling friction is generally much smaller than sliding friction.

4. Limiting Friction

Definition: The maximum value of static friction that occurs just before the body starts to slide over the surface is called Limiting Friction.
Mechanism: When the applied force $P$ gradually increases, the static friction $F$ increases to match it. Eventually, $F$ reaches a maximum limit beyond which it cannot increase. Any further increase in the applied force will break the interlocking of the surface asperities, and motion will commence.
Mathematical Representation:
$F_{max} = \mu_s \cdot R$
Where $F_{max}$ is the limiting friction, $\mu_s$ is the coefficient of static friction, and $R$ (or $N$ ) is the Normal Reaction.

5. Laws of Friction (Coulomb's Laws)

These empirical laws govern the behavior of dry friction.

Laws of Static Friction:

The force of friction always acts in a direction opposite to that in which the body tends to move.
The magnitude of the force of friction is exactly equal to the applied force until the limiting friction is reached.
The magnitude of the limiting friction bears a constant ratio to the normal reaction between the two surfaces ( $\frac{F}{R} = \text{constant}$ ).
The force of friction is independent of the apparent area of contact between the two surfaces.
The force of friction depends upon the roughness/smoothness (nature) of the surfaces and the material they are made of.

Laws of Dynamic (Kinetic) Friction:

The force of friction always acts in a direction opposite to that of the body's motion.
The magnitude of dynamic friction bears a constant ratio to the normal reaction, but this ratio ( $\mu_k$ ) is slightly less than that of static friction ( $\mu_s$ ).
For moderate speeds, the force of dynamic friction remains practically constant and is independent of the velocity of motion.

6. Angle of Friction, Angle of Repose, and Cone of Friction

A. Angle of Friction ( $\phi$ )

When a body is on the verge of sliding, it is subjected to two contact forces: the Normal Reaction ( $R$ ) and the Limiting Frictional Force ( $F_{max}$ ).

Definition: The angle made by the resultant of the normal reaction and the limiting frictional force with the normal reaction is called the Angle of Friction.
Formula:
$\tan \phi = \frac{F_{max}}{R} = \frac{\mu_s R}{R} = \mu_s$
Therefore, the coefficient of friction is equal to the tangent of the angle of friction.

B. Angle of Repose ( $\alpha$ )

Definition: The maximum angle of inclination of a rough plane with the horizontal, at which a body placed on it is just on the verge of sliding down by its own weight.
Relationship: By resolving the weight of the body along and perpendicular to the inclined plane, it can be mathematically proven that the Angle of Repose is exactly equal to the Angle of Friction.
$\alpha = \phi$

C. Cone of Friction

If the resultant reaction vector is rotated $360^\circ$ around the normal reaction vector while keeping the angle $\phi$ constant, it generates a right circular cone.

Significance: As long as the resultant force acting on the body falls within this cone, the body will remain at rest (equilibrium). If it falls outside, motion will occur.

7. Motion of Bodies

Analyzing the motion (or impending motion) of bodies involves drawing Free Body Diagrams (FBDs) and applying the conditions of equilibrium ( $\sum F_x = 0$ , $\sum F_y = 0$ ) or Newton's Second Law ( $\sum F = ma$ ).

A. Body on a Horizontal Plane

Consider a body of weight $W$ resting on a rough horizontal plane, subjected to a pulling force $P$ at an angle $\theta$ to the horizontal.

Normal Reaction ( $R$ ): The vertical components must balance.
$R + P \sin \theta = W \implies R = W - P \sin \theta$
Frictional Force ( $F$ ): Opposes impending motion.
$F = \mu R = \mu (W - P \sin \theta)$
Equation of Motion/Equilibrium: For impending motion (limiting case):
$P \cos \theta = F \implies P \cos \theta = \mu (W - P \sin \theta)$

B. Body on an Inclined Plane

Consider a body of weight $W$ resting on a plane inclined at an angle $\alpha$ to the horizontal.

Case 1: Body moving UP the inclined plane
An external force $P$ is applied parallel to the plane to pull the body up.

Friction ( $F$ ) acts down the plane.
Weight component down the plane = $W \sin \alpha$ .
Normal Reaction: $R = W \cos \alpha$ .
Equation for impending motion:
$P = W \sin \alpha + F \implies P = W \sin \alpha + \mu (W \cos \alpha)$

Case 2: Body moving DOWN the inclined plane
An external force $P$ may be applied to prevent it from sliding, or to pull it down. Assuming it is on the verge of sliding down naturally (or being pushed):

Friction ( $F$ ) acts up the plane.
Normal Reaction: $R = W \cos \alpha$ .
If a force $P$ is applied parallel to the plane to prevent sliding:
$P + F = W \sin \alpha \implies P = W \sin \alpha - \mu (W \cos \alpha)$
(Note: If $\alpha < \phi$ , the body will not slide down under its own weight).

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