In the calculation of the moment of inertia of the dumbbell does the axle make asignificant contribution to the total result?Comment on the likely error if the axle were ignored.
Moments of inertia i are proportional () to the mass m and the square of the radius of gyration r. In math talk, that's i mr^2. As the vast proportion of a dumbbell's mass lies in the end weights, 2M , we assume that m 2M is the mass of the axle. Furthermore, the radii of rotation (rotating around the centers of the end pieces) for the two end pieces would be R r which is that of the axle. On top of that, there would be two end pieces, thus 2M, and only one axle with mass m contributing to the MOI. So as I 2MR^2 and i mr^2, the total MOI 2MR^2 + mr^2, and we can assume that i mr^2 2MR^2 I and very little MOI would be overlooked if we simply ignore the axle's contribution.
May 28, 2018