ABC is a triangular plate of uniform thicknes | vedclass

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(C) The moment of inertia $I$ of a body depends on the distribution of its mass relative to the axis of rotation. The farther the mass is from the axi

Vedclass · Accepted Answer

$I_{BC} > I_{AB}$

Vedclass · Answer

(C) The moment of inertia $I$ of a body depends on the distribution of its mass relative to the axis of rotation. The farther the mass is from the axis,the greater the moment of inertia.For a triangular plate of mass $M$ and height $h$ relative to a base $b$,the moment of inertia about the base is $I = \frac{Mh^2}{6}$.Let the sides be $AB = 4k$,$BC = 3k$,and $AC = 5k$. The area of the triangle is $A = \frac{1}{2} 	imes 4k 	imes 3k = 6k^2$.$1$. For axis $AB$ (base $4k$,height $3k$): $I_{AB} = \frac{M(3k)^2}{6} = 1.5 Mk^2$.$2$. For axis $BC$ (base $3k$,height $4k$): $I_{BC} = \frac{M(4k)^2}{6} = 2.67 Mk^2$.$3$. For axis $AC$ (base $5k$,height $h'$): The height $h'$ is found by $A = \frac{1}{2} 	imes 5k 	imes h' = 6k^2$,so $h' = 2.4k$. Thus,$I_{AC} = \frac{M(2.4k)^2}{6} = 0.96 Mk^2$.Comparing the values: $I_{BC} (2.67 Mk^2) > I_{AB} (1.5 Mk^2) > I_{AC} (0.96 Mk^2)$.Therefore,$I_{BC} > I_{AB}$ is the correct relation.

$ABC$ is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. $I_{AB}, I_{BC}, I_{CA}$ are the moments of inertia of the plate about axes $AB, BC,$ and $CA$ respectively. Which one of the following relations is correct?

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