Four hollow spheres, each with a mass of 1 kg and a radius R = 10 cm , are connected with massless rods to form a square with side L = 50 cm . In case-1, the masses rotate about an axis that bisects t

Question

Four hollow spheres, each with a mass of 1 kg and a radius $R = 10 cm$ , are connected with massless rods to form a square with side $L = 50 cm$ . In case-1, the masses rotate about an axis that bisects two sides of the square. In case-2, the masses rotate about an axis that passes through the diagonal of the square, as shown in the figure. Compute the ratio of the moments of inertia $I_{1} / I_{2}$ for the two cases.

TestHub · Accepted Answer

Hollow sphere about its diameter: 2MR²/3

Case 1: every sphere is 25 cm = 2.5R from the axis

$I_{1} = 4 [\frac{2}{3} M R^{2} + M (2.5 R)^{2}] = \frac{83}{3} M R^{2}$

Case 2: two spheres on the axis, two at distance L/√2, d² = 12.5R²

$I_{2} = 4 \times \frac{2}{3} M R^{2} + 2 M (12.5 R^{2}) = \frac{83}{3} M R^{2}$

$\frac{I _{1}}{I _{2}} = 1$

Answer: D

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