Three solid spheres each of mass m and diamet | vedclass

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Question

(A) The radius of each sphere is $r = d/2$. The distance from the centroid of the equilateral triangle to the center of each sphere is $R = \frac{d}{\

Vedclass · Accepted Answer

$\frac{13}{23}$

Vedclass · Answer

(A) The radius of each sphere is $r = d/2$. The distance from the centroid of the equilateral triangle to the center of each sphere is $R = \frac{d}{\sqrt{3}}$.Using the parallel axis theorem for each sphere about the axis passing through the centroid $(I_0)$:$I_0 = 3 	imes [I_{cm} + mR^2] = 3 	imes [\frac{2}{5}m(d/2)^2 + m(d/\sqrt{3})^2]$$I_0 = 3 	imes [\frac{1}{10}md^2 + \frac{1}{3}md^2] = 3 	imes [\frac{3+10}{30}]md^2 = 3 	imes \frac{13}{30}md^2 = \frac{13}{10}md^2$.Now,for the axis passing through the center of one sphere $(A)$ and perpendicular to the plane,we use the parallel axis theorem between the centroid axis and the axis at $A$:$I_A = I_0 + 3mR^2 = \frac{13}{10}md^2 + 3m(d/\sqrt{3})^2 = \frac{13}{10}md^2 + md^2 = \frac{23}{10}md^2$.The ratio is $\frac{I_0}{I_A} = \frac{13/10}{23/10} = \frac{13}{23}$.

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