Question

Moment of inertia of a circular wire of mass M and radius R about its diameter is:

• A. $\frac{MR^2}{2}$
• B. $MR^2$
• C. $2MR^2$
• D. $\frac{MR^2}{4}$

Answer 1

Answer: (A)

$\frac{MR^2}{2}$

Answer 2

Using the perpendicular axis theorem:

$$I_{\perp} = I_x + I_y$$

For a circular wire of mass $M$ and radius $R$, the moment of inertia about an axis perpendicular to its plane is:

$$I_{\perp} = MR^2$$

Since the wire is symmetric, $I_x = I_y$. Thus:

$$2I_\text{diameter} = MR^2 \implies I_\text{diameter} = \frac{MR^2}{2}$$