A uniform solid ball is placed at rest on an incline of slope angle . What is the minimum value of the coefficient of static friction between ball and incline so that the ball will roll down the incline without slipping?
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There are three forces:
Gravity parallel to ramp = M g s i n θ Normal reaction = M g c o s θ Critical Friction force = μ M g c o s θ
The sphere moment of inertia is I = 5 2 M R 2 . Examine the critical condition in which the maximum possible friction force is required. For rolling without slipping (suppose α is the rotation angle of the sphere about its own central axis):
a C M = R α ¨ = R I τ M M g s i n θ − μ M g c o s θ = R 5 2 M R 2 R μ M g c o s θ s i n θ − μ c o s θ = 2 5 μ c o s θ s i n θ = 2 7 μ c o s θ 7 2 t a n θ = μ