A uniform disc of mass M and radius R can oscillate in its own plane about a fixed horizontal axis passing through a point A on its rim, as shown in the figure. For small angular oscillations under gravity, what is its time period?

Question

G1 Physics · Accepted Answer

For a uniform disc, the moment of inertia about an axis through its centre and perpendicular to the plane is I(CM) = (1 ÷ 2)MR². However, the axis shown in the source figure is in the plane of the disc and tangent at the rim. About a diameter through the centre, I(diameter) = (1 ÷ 4)MR². Using the parallel-axis theorem for the tangent axis A: I(A) = I(diameter)+MR² = (1 ÷ 4)MR²+MR² = (5 ÷ 4)MR². For a small angular displacement θ, the restoring torque is τ ≈ -MgRθ. Hence, I(A)θ̈ = -MgRθ. Theref…