Question

The wheels of a wagon can be approximated as the combination of a thin outer hoop, of radius rh = 0.527 m and mass 4.70 kg, and two thin crossed rods of mass 9.52 kg each. You would like to replace the wheels with uniform disks that are 0.0462 m thick, made out of a material with a density of 5530 kilograms per cubic meter. If the new wheel is to have the same moment of inertia about its center as the old wheel about its center, what should the radius of the disk be?

Answer 1

here,

radius of wheel, rw = 0.527 m

mass of wheel, mw = 4.70 kg

mass of thin rods, m = 9.52 kg each

thickness if disks, t = 0.0462 m

density of material, rho = 5530 kg/m^3

Moment of inertia of wheel
Iw = mw*rw^2 + 2/3 * m * (rw)^2
Iw = 4.70 * 0.527^2 + 2/3 * 9.52 * (0.527)^2
Iw = 3.068 kg.m^2

moment of inertia of disk
Id = 0.5 * mass of disk * r^2
Id = 0.5 * (rho * pi * r^2 * t) * r^2

Since inertia is same for both of them :

3.068 = 0.5 * (rho * pi * r^2 * t) * r^2

3.068 = 0.5 * (5530 * pi * r^2 * 0.0462) * r^2

radius of disk, r = 0.296 m