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A block is in a limiting equilibrium on a rough horizontal surface .if the net contact force is √3times the normal force the coefficient of static friction



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Contact forces are "Normal force" and "friction"
In this case, the block is in limiting equilibrium means the the static friction is equal to the horizontal force acting on the block and it is not moving.

Let normal force = N (verically upward)
coefficient of static friction = μ
static friction (f)= μN (horizontal)

Net contact force = √((μN)² + N²) = N√(1+μ²)
Given that : N√(1+μ²) = √3 N
⇒√(1+μ²) = √3
⇒1 + μ² = 3
⇒ μ² = 3-1 = 2
⇒ μ = √2

Thus coefficient of static friction is √2

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