Friction Problems in Mechanics

Overview of the Problem

• Block Dimensions: 2A by 2A
• Forces: F1 and F2; F2 varies from 0 to 50, 90, and 100 Newtons
• Coefficients:
  • Static friction: 0.3
  • Dynamic friction: 0.25
• Objective: Determine the reaction position on the block and frictional behavior under varying forces.

Key Concepts

Understanding Frictional Forces

• Direction of Friction: Always opposite to the applied force (F2).
• Notations: Use small f for frictional force instead of mu times normal reaction unless motion is impending.
• Normal Reaction: Must not be assumed at the centroid; determined from moment equilibrium.

Equilibrium Equations

1. For horizontal forces: ( \Sigma F_x = 0 )
2. For vertical forces: ( \Sigma F_y = 0 )
3. Moment about point C to find distances.

Calculating Forces and Distances

• Weight of block = 200 N, External force = 100 N results in normal reaction ( N = 300 N ).
• Moment equilibrium equation:
  • ( -N \cdot x + 100 \cdot \frac{A}{2} - 1.5A imes F2 = 0 )
• Distance from calculations:
  • ( x = \frac{A}{300} (50 - 1.5 F2) )

Case Analysis of F2 Values

1. F2 = 0: No frictional forces develop.
2. F2 = 50 N:
   • Resulting distance ( x = -0.088A )
   • Frictional force = F2; not at max value.
3. F2 = 90 N:
   • Resulting distance ( x = -0.28A )
   • Frictional force reaches max (90 N); impending motion.
4. F2 = 100 N:
   • Distance ( x = 0.33A )
   • Max friction (75 N); block begins to slide as friction cannot balance the force.

Guidelines for Solving Friction Problems

• Always determine the direction of friction before calculations.
• Use equilibrium equations to find unknown frictional forces, especially when motion is not impending.
• For impending motion, replace frictional forces with ( \mu imes N ).
• Be cautious about the centroid placement for normal force; analyze based on moment equilibrium.

Microscopic and Macroscopic Friction Behavior

• Surface Roughness: Microscopic projections and depressions affect friction.
• Elastic Deformation: Occurs below maximum static friction; reversible once the force is removed.
• Plastic Deformation: Leads to sliding friction; energy lost as heat.

Surface Finish Impact

• Ground surfaces have lower friction until a certain limit.
• Highly polished surfaces may lead to higher friction due to adhesion.

Rolling Resistance

• Definition: Resistance when a body rolls on a surface; distinct from dry friction due to deformation.
• Common Models: Rolling friction can be expressed as a coefficient times normal force.
• Applications: Ball and roller bearings minimize resistance.

Block and Tipping Analysis

• Analyze potential motions: sliding vs tipping.
• If the point of reaction lies within the block, sliding occurs; if outside, tipping occurs.

Example Problems

1. Two Block Problem: Determine weight of Block A for equilibrium under applied forces.
   • Investigate three scenarios: block sliding, tipping, or both sliding together.
   • Use free body diagrams and equilibrium equations.
2. Wedge Problem: Determine the force required to lift a block on a wedge with friction.
   • Recognize directions of forces and friction based on applied force.
   • Calculate forces in equilibrium and determine self-locking conditions.

Self-Locking Concept

• Self-locking occurs when the wedge remains in place after the applied force is removed.
• Condition for self-locking depends on friction coefficient and wedge angle.

Conclusion

• Understanding friction is crucial for solving mechanical problems effectively.
• Always analyze potential motions and directions of forces before calculations.