Question

Consider a block of mass m at rest on an inclined plane of angle theta. An acrobat of mass m_A is standing on the top corner of the block so that their weight presses vertically (perpendicular to the flat ground under the incline) downward. If the coefficient of static friction between the block and the incline is mu_s find an expression relating m_A, m, theta, and mu_s just before the block begins slipping. If the condition for slipping does not involve some of these parameters, leave them out.
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Answer 1

Answer:

μ = tan θ

Explanation:

For this exercise let's use the translational equilibrium condition.

Let's set a datum with the x axis parallel to the plane and the y axis perpendicular to the plane.

Let's break down the weight of the block

         sin θ = Wₓ / W

         cos θ = W_y / W

         Wₓ = W sin θ

         W_y = W cos θ

The acrobat is vertically so his weight decomposition is

         sin θ = = wₐₓ / wₐ

         cos θ = wₐ_y / wₐ

         wₐₓ = wₐ  sin θ

         wₐ_y = wₐ cos θ

let's write the equilibrium equations

Y axis  

     N- W_y - wₐ_y = 0

     N = W cos θ + wₐ cos θ

X axis

        Wₓ + wₐ_x - fr = 0

         fr = W sin θ + wₐ sin θ

the friction force has the formula

         fr = μ N

         fr = μ (W cos θ + wₐ cos θ)

we substitute

         μ (Mg cos θ + mg cos θ) = Mgsin θ + mg sin θ

         μ = $\frac{(M +m) \ sin \ \theta }{(M +m) \ cos \ \theta }$

 

         μ = tan θ

this is the minimum value of the coefficient of static friction for which the system is in equilibrium.