Mass on an Incline

Question

A block with mass m₁ = 8.6 kg is on an incline with an angle θ = 33° with respect to the horizontal. For the first question there is no friction, but for the rest of this problem the coefficients of friction are: μ_k = 0.29 and μ_s = 0.319.

Now a new block with mass m₂ = 15.7 kg is attached to the first block. The new block is made of a different material and has a greater coefficient of static friction. What minimum value for the coefficient of static friction is needed between the new block and the plane to keep the system from accelerating?

Answer 1

Don't see a first question, but will proceed to answer the second: sketch a FBD for the combined mass M to show weight Mg downwards, normal force F_N perpendicular to ramp, and friction force f up along the ramp.

Sum the perpendicular and parallel forces: ∑F_⊥ = F_N - Mg cos 33° = 0, ∑F_|| = f - Mg sin 33° = 0?

From these, F_N = (8.6 + 15.7 kg)(9.81) /(cos 33 = 0.8387) = 284 N; f ≥ (24.3 kg)(9.81)(sin 33 = 0.5446) = 130 N. Let μ_s be the minimum static coefficient for which there is zero acceleration down the ramp; then μ_s F_N = 130 N and μ_s = 130/284 = 0.457. This is a weighted average of the two coefficients, μ_s = (8.6 • 0.319 + 15.7 • μ₂) / (8.6 + 15.7), 15.7 μ₂ = (24.3)(0.457) - (8.6 • 0.319) = 8.3₆ and the new block's static coefficient is μ₂ ≥ 0.53.