Limiting friction


Limiting Friction

Friction is a force that resists the relative motion between two surfaces in contact. It plays a crucial role in everyday life, providing the grip needed for walking, driving, and holding objects. Limiting friction is a specific type of frictional force that comes into play when an object is on the verge of sliding over the surface of another object. Understanding limiting friction is essential for studying the laws of motion and designing mechanical systems.

Definition

Limiting friction is the maximum static frictional force that must be overcome before an object starts to move over the surface of another object. It acts when an object is subjected to an external force that tends to move it, but the object is still stationary. Once the applied force exceeds the limiting friction, the object begins to slide, and kinetic friction takes over.

Factors Affecting Limiting Friction

Several factors affect the magnitude of limiting friction:

  1. Nature of the Surfaces: The roughness or smoothness of the contacting surfaces influences the amount of limiting friction.
  2. Normal Reaction: The force exerted by a surface perpendicular to the object resting on it, known as the normal reaction, directly affects the limiting friction.
  3. Area of Contact: Contrary to what one might expect, the limiting friction is independent of the apparent area of contact for most practical surfaces.
  4. Additives: Substances like lubricants can reduce the limiting friction by altering the surface characteristics.

Formula

The limiting friction ($F_{\text{limiting}}$) can be calculated using the coefficient of static friction ($\mu_s$) and the normal reaction ($N$):

$$ F_{\text{limiting}} = \mu_s \cdot N $$

Where:

  • $F_{\text{limiting}}$ is the limiting frictional force,
  • $\mu_s$ is the coefficient of static friction, and
  • $N$ is the normal reaction force.

Table: Differences Between Static, Limiting, and Kinetic Friction

Property Static Friction Limiting Friction Kinetic Friction
Definition Frictional force when an object is at rest Maximum static friction before the object starts moving Frictional force when an object is in motion
Magnitude Variable, up to a maximum value Maximum value of static friction Usually less than limiting friction and constant for given conditions
Dependency on Normal Force Directly proportional Directly proportional Directly proportional
Dependency on Surface Area Independent Independent Independent
Coefficient Coefficient of static friction ($\mu_s$) Coefficient of static friction ($\mu_s$) at its maximum Coefficient of kinetic friction ($\mu_k$)

Examples

Example 1: Threshold of Motion

A box weighing 10 kg is resting on a horizontal surface. The coefficient of static friction between the box and the surface is 0.5. What is the limiting friction?

Solution:

First, calculate the normal reaction ($N$), which is equal to the weight of the box (assuming no other vertical forces):

$$ N = m \cdot g = 10 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 98 \, \text{N} $$

Now, use the formula for limiting friction:

$$ F_{\text{limiting}} = \mu_s \cdot N = 0.5 \cdot 98 \, \text{N} = 49 \, \text{N} $$

The limiting friction is 49 N.

Example 2: Overcoming Limiting Friction

A person is pushing a heavy crate across the floor. The crate has a mass of 50 kg, and the coefficient of static friction is 0.4. How much force must the person apply to start moving the crate?

Solution:

Calculate the normal reaction ($N$):

$$ N = m \cdot g = 50 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 490 \, \text{N} $$

Calculate the limiting friction:

$$ F_{\text{limiting}} = \mu_s \cdot N = 0.4 \cdot 490 \, \text{N} = 196 \, \text{N} $$

The person must apply a force greater than 196 N to start moving the crate.

Conclusion

Limiting friction is a fundamental concept in the study of motion and mechanics. It represents the threshold at which an object transitions from rest to motion due to an external force. Understanding and calculating limiting friction is essential for designing mechanical systems and solving problems related to motion on surfaces.