Answer to Question #269587 in Mechanics | Relativity for Lhen

Question

Four identical spheres of masses 2.0 kg each and radius 0.25 m are situated at the four corners of a square. One side of the square measures 3.00 m. Find the moment of inertia about an axis (a) passing through the center of the square and perpendicular to its plane, (b) passing through one of the masses and perpendicular to its plane, (c) passing through two adjacent masses parallel to the plane, and (d) in the plane running diagonally through two meters.

Answer 1

(a) The moment of inertia of a single hollow sphere:

$$I_1=\frac23mr^2.$$

Apply the parallel axis theorem to determine the moment of inertia of a sphere rotating around the center of the square at length equal to half diagonal of the square:

$$I_s=I_1+I_a=\frac23 mr^2+m\bigg(a\frac{\sqrt2}{2}\bigg)^2=\frac23mr^2+\frac{ma^2}2.\\\space\\ I=4I_s=2m\bigg(\frac43r^2+a^2\bigg)=36.3\text{ kg}·\text{m}^2.$$

For (b), (c), (d), see the figure below:

(b) Here, we have 3 different moments of inertia that make up the total:

$$I_1=\frac23mr^2.\\\space\\ I_2=\frac23mr^2+ma^2,\\\space\\ I_3=\frac23mr^2+\frac{ma^2}2.\\\space\\ I=I_1+2I_2+I_3,\\\space\\ I=\frac83mr^2+2.5ma^2=45.3\text{ kg}·\text{m}^2.$$

(c) Determine what is required:

$$I_1=\frac23mr^2.\\\space\\ I_2=\frac23mr^2+ma^2,\\\space\\ I=2I_1+2I_2,\\\space\\ I=\frac83mr^2+2ma^2=36.3\text{ kg}·\text{m}^2.$$

(d) Determine what is required:

$$I_1=\frac23mr^2.\\\space\\ I_2=\frac23mr^2+m\bigg(a\frac{\sqrt2}2\bigg)^2=\frac23mr^2+0.5ma^2,\\\space\\ I=2I_1+2I_2,\\\space\\ I=\frac83mr^2+ma^2=18.3\text{ kg}·\text{m}^2.$$