A semicircular ring of mass m and radius r is released from rest in the position shown with its lower edge resting on a horizontal surface. The minimum coefficient of static friction μₛ which is necessary to prevent any initial slipping of the ring is x/10. Value of x is
Physics · Mixed Chapter
JEE Mains

Question

A semicircular ring of mass m and radius r is released from rest in the position shown with its lower edge resting on a horizontal surface. The minimum coefficient of static friction μₛ which is necessary to prevent any initial slipping of the ring is x/10. Value of x is

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The centre of mass of the uniform semicircular ring is 2r/π to the left of the circle centre and r above its lowest contact point. Step 2: At release ω=0. About the ground contact, the initial moment of inertia is I about contact=2mr² and the gravitational torque is mg(2r/π), giving angular acceleration α=g/(πr). Step 3: Initial horizontal centre-of-mass acceleration is −αr=−g/π; its vertical acceleration is −(2r/π)α=−2g/π². Step 4: Hence |f|=mg/π and N=mg(1−2/π²). The required coeffici…