Question

What is the moment of inertia uniform rectangular wire frame as diagonal?

Answer 1

Let mass of rectangle is M , length is L and breadth is B.

we can wirte moment of inertia of rectangle about an axis passing through its diagonal = 2 × moment of inertia of right angled triangle about an axis passing through hypotenuse
[ to understand, see figure ]

as we know, moment of inertia of right angled triangle about an axis passing through hypotenuse and parallel to its plane , I = 1/6 Mr² , where r = 1/6 mr²
where m = M/2 and r = LB/√(L² + B²)

inertia of rectangle = 2 × {1/6 × (M/2) × L²B²/(L² + B²) }

= 1/6 M ( L²B²)/(L² + B²)

Answer 2

Hello dear,

● Answer-
Id = m(l^2+b^2)/24

● Explanation-
Let's consider,
Id = M.I. about diagonal
Il = M.I. about length = ml^2/12
Ib = M.I. about breadth = mb^2/12

By perpendicular axis theorem-
Il + Ib = 2Id
Id = (Il + Ib) / 2
Id = (ml^2/12 + mb^2/12) / 2
Id = m/24 (l^2+b^2)


M.I. of rectangle about its diagonal is given by m/24 (l^2+b^2).



Hope that is useful...