Three identical billiard balls each of mass 1 kg are placed as shown in the figure. The upper ball is smooth. Find minimum coefficient of static friction between lower balls and ground, so that balls remain stationary

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$\frac{1}{\sqrt{3}}$

$\frac{1}{3 \sqrt{3}}$

$\frac{2}{\sqrt{3}}$

$0 \cdot 5$

answer is B.

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Detailed Solution

Problem Statement

Three identical billiard balls, each of mass m = 1 kg, are placed in a triangular arrangement with two balls on the ground and one smooth ball resting on top. Find the minimum coefficient of static friction (μ) between the lower balls and the ground so that the balls remain stationary.

Solution Approach

Step 1: Geometry Analysis

For three identical spheres in contact with radius r:

Distance between centers of lower balls = 2r
The upper ball forms an equilateral triangle with the lower balls
The angle between vertical and the line joining centers = 30°

Step 2: Forces on Upper Ball A

The upper smooth ball experiences:

Weight: mg (downward)
Normal force N₁ from ball B (perpendicular to contact, at 30° from vertical)
Normal force N₂ from ball C (perpendicular to contact, at 30° from vertical)

Equilibrium conditions:

Horizontal: N₁ sin(30°) = N₂ sin(30°) → N₁ = N₂ = N

Vertical: 2N cos(30°) = mg

N = mg / (2 cos(30°)) = mg / √3

Step 3: Forces on Lower Ball B (or C)

Each lower ball experiences:

Weight: mg (downward)
Normal force from ball A: N (at 30° from vertical, pushing outward)
Normal force from ground: N_ground (upward)
Friction force from ground: f (inward, preventing slip)

Equilibrium conditions:

Horizontal: f = N sin(30°) = (mg/√3) × (1/2) = mg/(2√3)

Vertical: N_ground = mg + N cos(30°) = mg + (mg/√3) × (√3/2) = 3mg/2

Step 4: Minimum Coefficient of Friction

For the system to remain stationary:

f ≤ μ N_ground

mg/(2√3) ≤ μ × (3mg/2)

μ ≥ mg/(2√3) × 2/(3mg) = 1/(3√3) = √3/9

Minimum Coefficient of Static Friction: μ_min = √3/9 ≈ 0.192

Physical Interpretation: The friction must be sufficient to prevent the lower balls from sliding outward due to the outward horizontal component of the normal force from the upper ball. A coefficient of approximately 0.19 or higher is needed to keep the system stable.

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