Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_108

⚡ Mecademy AIENG정역학 · ch6  Problem Statement The homogeneous disk of mass is resting on the right-angled supporting surfaces shown. The tension in the cord is very gradually increased from zero. If the friction at both and is characterized by , what happens first—does the homogeneous disk slip in place or does it begin to roll up the incline? Determine the value of at which this first movement occurs. Problem 6/108 (a) Analysis of Movement and Equilibrium ● Calculation Process 1. Present Final Formula: To determine what happens first, we consider the equilibrium of the disk. Let the coordinate system be defined with the -axis parallel to surface (up the incline) and the -axis perpendicular to it. The disk is in contact with surfaces and at points and . The equilibrium equations are: where: are normal forces at and . are friction forces at and . A clockwise rotation (induced by ) causes the disk to move down-right relative to surface and up-right relative to surface , thus and both oppose the rotation. 2. Substitute Values: For impending motion (slipping at both points), we set and , with . m PA Bμ = s 0.25 P x Ay ABAB F = ∑ x 0:P+F + A N − B mgsin30= ∘ 0 F = ∑ y 0:N + A F − B mgcos30= ∘ 0 M = ∑ C 0:P⋅r−F ⋅ A r−F ⋅ B r=0⟹P=F + A F B N ,N AB AB F ,F AB ABP A BF A F B F = A μN sA F = B μ N sB μ = s 0.25 ⎩ ⎨ ⎧ P+0.25N +N =0.5mg AB N +0.25N =0.8660mg AB P=0.25N +0.25N AB 3. Partial Operations: Substitute into the first equation: From the second equation: Substitute this into the