Consider a large disk of radius R and two smaller disks, each of radius r = R/50, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation Δθ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities ω and 2ω while the large disk is held stationary. The time τ at which the smaller disks are again in contact is: [Use sin(Δθ) = Δθ and ignore gravity.]

Question

G1 Physics · Accepted Answer

Using the rolling constraint, the relative angular speed of the two small disks around the large disk is 3ω/51. From the initial contact geometry and the small-angle approximation specified in the question, the initial angular separation is 4/51 rad. For the next contact, the relative angular displacement is 2π − 4/51. Therefore τ = [2π − 4/51]/(3ω/51) = 51(2π − 4/51)/(3ω).