Figure shows two smooth-surfaced solid cylinder of radius a and mass m each are placed touching along their length on a horizontal surface. A third cylinder of the same material but twice the radius i

Question

Figure shows two smooth-surfaced solid cylinder of radius a and mass m each are placed touching along their length on a horizontal surface. A third cylinder of the same material but twice the radius is placed on them so that the system remains at rest. If all three cylinders have the same length, calculate the minimum value of the coefficient of static friction between the smaller cylinders and the horizontal surface.

TestHub · Accepted Answer

Geometry: cos θ = 3 R R ​ = 3 1 ​ , so sin θ = 3 2 2 ​ ​ . The big cylinder has twice the radius, hence 4 times the mass: 2 N sin θ = 4 m g ⇒ N = 2 ​ 3 m g ​ . For a small cylinder: friction f = N cos θ = 2 ​ m g ​ and normal from ground N 0 ​ = m g + N sin θ = 3 m g . μ min ​ = N 0 ​ f ​ = 3 2 ​ 1 ​ - (D)

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