Four solid spheres,each of diameter sqrt{5} | vedclass

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(A) Let the side of the square be $a = 4 \ cm = 0.04 \ m$. The radius of each sphere is $R = \frac{\sqrt{5}}{2} \ cm = \frac{\sqrt{5}}{2} 	imes 10^{-

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(A) Let the side of the square be $a = 4 \ cm = 0.04 \ m$. The radius of each sphere is $R = \frac{\sqrt{5}}{2} \ cm = \frac{\sqrt{5}}{2} 	imes 10^{-2} \ m$. The mass of each sphere is $m = 0.5 \ kg$.Let the diagonal of the square be the axis of rotation. Two spheres lie on this diagonal,and two spheres are at a perpendicular distance $d = \frac{a}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \ cm = 2\sqrt{2} 	imes 10^{-2} \ m$ from the diagonal.For the two spheres on the diagonal,the moment of inertia is $I_1 = I_2 = \frac{2}{5}mR^2$.For the two spheres off the diagonal,using the parallel axis theorem,$I_3 = I_4 = \frac{2}{5}mR^2 + md^2$.The total moment of inertia $I = 2(\frac{2}{5}mR^2) + 2(\frac{2}{5}mR^2 + md^2) = \frac{8}{5}mR^2 + 2md^2$.Substituting the values: $I = \frac{8}{5}(0.5)(\frac{5}{4} 	imes 10^{-4}) + 2(0.5)(8 	imes 10^{-4}) = 0.5 	imes 10^{-4} + 8 	imes 10^{-4} = 8.5 	imes 10^{-4} \ kg \cdot m^2$. Wait,re-evaluating the geometry: The distance of the spheres at the corners from the diagonal is $d = \frac{a}{\sqrt{2}}$. The two spheres on the diagonal have $I = \frac{2}{5}mR^2$. The two spheres off the diagonal have $I = \frac{2}{5}mR^2 + md^2$. Total $I = \frac{8}{5}mR^2 + 2md^2 = 1.6(0.5)(1.25 	imes 10^{-4}) + 2(0.5)(8 	imes 10^{-4}) = 1.0 	imes 10^{-4} + 8 	imes 10^{-4} = 9 	imes 10^{-4} \ kg \cdot m^2$. Thus,$N = 9$.

Column-$I$	Column-$II$
$(1)$ Perpendicular Axis Theorem	$(a)$ $I = I_C + Md^2$
$(2)$ Parallel Axis Theorem	$(b)$ $I_z = I_x + I_y$

Four solid spheres,each of diameter $\sqrt{5} \ cm$ and mass $0.5 \ kg$,are placed with their centers at the corners of a square of side $4 \ cm$. The moment of inertia of the system about the diagonal of the square is $N \times 10^{-4} \ kg \cdot m^2$. Find $N$.

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Match Column-$I$ with Column-$II$:
Column-$I$ Column-$II$
$(1)$ Perpendicular Axis Theorem $(a)$ $I = I_C + Md^2$
$(2)$ Parallel Axis Theorem $(b)$ $I_z = I_x + I_y$

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