Question

Moment of inertia of a circular wire of mass M and radius R about its diameter is [AIEEE 2002]
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Answer 1

Answer:

→ Moment of inertia of a circular wire about it's axis is MR². Consider two diameters XX' and YY'

Moment of inertia about of these two diameters is same, say l

From perpendicular axes theorem

→ l+l = MR²

→ l = MR²/2

Explanation:

Hope this helps you

Answer 2

Answer :

Mass of wire = M

Radius of circle = R

We have to find moment of inertia of circular wire about its diameter $.$

★ We can consider circular wire as a ring.

Moment of inertia of circular ring of mass M and radius R about an axis passing through its centre and perpendicular to the plane of ring is given by

• I = MR²

Theorem of perpendicular axis :

• It states that the MI of a planar lamina about an axis perpendicular to the plane of lamina is equal to the sum of the moments of inertia of the surface about the two axes at right angles to each other, in its own plane.

Mathematically, I(z) = I(x) + I(y)

★ Diagram :

Refer to the attachment for better understanding.

★ Calculation :

Let diameter of circular wire be x or y axis.

→ I(z) = I(x) + I(y)

• I(x) = I(y) = I

→ MR² = I + I

→ 2I = MR²

→ I = MR²/2

Cheers!