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Chapter 6: Problem 61

The moment of inertia of a circular wire of mass \(m\) and radius \(R\) about its diameter is (A) \(m R^{2} / 2\) (B) \(m R^{2}\) (C) \(2 m R^{2}\) (D) \(m R^{2} / 4\)

Short Answer

Expert verified
The moment of inertia of a circular wire of mass \(m\) and radius \(R\) about its diameter is (A) \(m R^{2} / 2\).

Step by step solution

01

Identify the formula for the moment of inertia of a circular wire

Use the formula for moment of inertia of a mass element, which is given by \(I = \int r^{2} dm\), where \(I\) is the moment of inertia, \(r\) is the distance of the mass element from the axis of rotation, and \(dm\) is the mass of the element.
02

Express mass of an element in terms of density

The mass of an element can be written in terms of the linear density (\(位\)) of the wire. The linear density is given by \(位 = \frac{m}{2蟺R}\). And the mass of an element of the circular wire is given by its linear density times its length, or: \(dm = 位dl\).
03

Set up the integral for the moment of inertia

Now, let's find the integral of the moment of inertia for the circular wire by substituting the expression for \(dm\) in terms of linear density: \(I = \int r^{2} 位 dl\).
04

Integrate over the length of the circular wire

The circular wire subtends an angle of \(2蟺\) at the center. Using polar coordinates, the integral transforms to: \(I = 位\int_{0}^{2蟺} R^2 \cos^2胃 d胃\), with the limits \(0\) and \(2蟺\) representing the entire length of the wire.
05

Calculate the integral

We can now perform the integral: 1. Use the identity \(\cos^2胃 = \frac{1}{2}(1 + \cos 2胃)\). 2. Substitute and integrate: \(I = 位 \frac{1}{2}\int_{0}^{2蟺} (1 + \cos 2胃) d胃\). 3. The integral becomes: \(I = \frac{位}{2}(胃 + \frac{1}{2}\sin 2胃)\Big|_{0}^{2蟺}\). 4. Evaluate the integral: \(I = \frac{位}{2}(2蟺)\). 5. Replace 位 with its expression, \(位 = \frac{m}{2蟺R}\), so the moment of inertia becomes: \(I = \frac{m}{2蟺R}\frac{R^2}{2}(2蟺) = \frac{1}{2}mR^2\).
06

Identify the correct option

Comparing our result to the given options, we can see that the correct option for the moment of inertia of a circular wire of mass `m` and radius `R` about its diameter is: (A) \(m R^{2} / 2\)

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