*8–52. beam ab has a negligible mass and thickness, and supports the 200-kg uniform block. it is pinned at a and rests on the top of a post, having a mass of 20 kg and negligible thickness. determine the two coefficients of static friction at b and at c so that when the magnitude of the applied force is increased to p=300 n, the post slips at both b and c simultaneously.

μ₁ = 0.1048

μ₂ = 0.1375

Explanation:

Using  static equation can find in both point the moment and the forces so:

∑ M = F *d  , ∑ F = 0

∑ M A = 0

N₁ * 3 - 200 * 9.81 * 1.5 = 0

N₁ = 981  

∑ M y = 0

N₂ + 300 * ³/₅ - 981 - 20 * 9.81 = 0

N₂ = 997.2 N

∑ M C = 0

F₁ * 1.75  - 300 * ⁴/₅  * 0.75 = 0

F₁ = 102.86

∑ M B = 0

300 * ⁴/₅ * 1 - F₂ * 1.75 = 0

F₂ = 137.14 N

The Force F1 and F2 related the coefficients of static friction

F₁ = μ₁ * N₁   ⇒  102.86 N = μ₁ * 981 ⇒ μ₁ = 0.1048

F₂= μ₂ * N₂  ⇒  137.14 N = μ₂ * 997  ⇒ μ₂ = 0.1375

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