Coefficient of Static Friction from Angle of Repose: Solved Example

 How to Calculate Static Friction Using Inclined Plane Angle

-Dr.Sanjaykumar Pawar 

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Laws of Friction Explained

Static vs Kinetic Friction

Free Body Diagrams in Physics

Inclined Plane Problems and Solutions

Newton's Laws of Motion

Force Equilibrium Concepts

Applications of Friction in Daily Life

JEE Mechanics Important Questions

NEET Physics Friction Chapter Notes

Work, Energy and Power Fundamentals

Free-body diagram of a block on an inclined plane showing the relationship between angle of repose and coefficient of static friction.

Example 4.8

A mass of 4 kg rests on a rough horizontal plane. The plane is gradually inclined. When the angle becomes 15°, the block just begins to slide. Find the coefficient of static friction (μₛ).

Given Data

• Mass of block, m = 4 kg
• Angle of inclination, θ = 15°
• Find μₛ

Step 1: Understand the Situation

As the plane is tilted, the component of weight acting down the plane increases.

Static friction opposes this motion and keeps the block at rest.

At θ = 15°, the block is just about to move. At this point, static friction reaches its maximum value.

Maximum Static Friction = μₛN

Step 2: Forces Acting on the Block

The following forces act on the block:

• Weight (mg) acting vertically downward
• Normal reaction (N) acting perpendicular to the plane
• Static friction (fₛ) acting upward along the plane

Step 3: Resolve Weight into Components

Weight mg is resolved into two components:

Along the plane = mg sin θ

This component tends to pull the block downward.

Perpendicular to the plane = mg cos θ

This component presses the block against the surface.

Step 4: Apply Equilibrium Conditions

Since the block is still at rest:

fₛ = mg sin θ

N = mg cos θ

Step 5: Use Maximum Static Friction Formula

fₛ = μₛN

Substitute the values:

mg sin θ = μₛ (mg cos θ)

Cancel mg from both sides:

sin θ = μₛ cos θ

Divide by cos θ:

μₛ = tan θ

Step 6: Substitute θ = 15°

μₛ = tan 15°

μₛ = 0.268

μₛ ≈ 0.27

Final Answer: μₛ = 0.27

Important Formula

μₛ = tan θ

This formula is used when a block is just about to slide on an inclined plane.

Key Points for Beginners

• Static friction prevents motion.
• Static friction adjusts itself according to the applied force.
• At the point of sliding, static friction becomes maximum.
• Maximum static friction = μₛN.
• The angle at which sliding begins is called the angle of repose.
• For angle of repose, μₛ = tan θ.
• The value of μₛ does not depend on the mass of the block.

One-Line Summary

When a block just begins to slide on an inclined plane, the coefficient of static friction is equal to the tangent of the angle of inclination.

μₛ = tan 15° = 0.27