The moment of inertia of a uniform semicircular wire of mass m and radius r about an axis passing through its centre of mass and perpendicular to its plane is what???

To determine the moment of inertia of a uniform semicircular wire about an axis that passes through its center of mass and is perpendicular to its plane, we can follow a systematic approach. The moment of inertia is a measure of how mass is distributed relative to an axis of rotation, and for a semicircular wire, we can derive it using calculus or by applying known formulas for simpler shapes.

Understanding the Setup

First, let's visualize the semicircular wire. Imagine a wire bent into a half-circle with radius r. The total mass of this wire is m. The center of mass for a uniform semicircular wire lies at a distance of 4r/(3π) from the center along the axis of symmetry.

Using the Formula for Moment of Inertia

The moment of inertia I about an axis perpendicular to the plane of the wire and passing through the center of mass can be calculated using the following integral:

I = ∫ r² dm

In this case, we need to express dm in terms of the angle θ that describes the position along the semicircle. The differential mass element can be expressed as:

• dm = (m / L) dL, where L is the length of the semicircular wire.
• The length of the semicircular wire is L = πr.
• Thus, dm = (m / πr) dL.

Calculating the Length Element

The length element dL can be expressed in terms of the angle θ as:

dL = r dθ

Substituting this into our expression for dm gives:

dm = (m / πr) (r dθ) = (m / π) dθ

Setting Up the Integral

Now, we can set up the integral for the moment of inertia:

I = ∫ r² dm = ∫ (r²) (m / π) dθ

Since the radius r is constant for the semicircle, we can factor it out of the integral:

I = (m / π) r² ∫ dθ

The limits of integration for θ will be from 0 to π (covering the entire semicircle):

I = (m / π) r² [θ] from 0 to π = (m / π) r² (π - 0) = mr²

Final Result

However, this result needs to be adjusted because we are considering the moment of inertia about the center of mass, not the geometric center. The moment of inertia of a full circular wire about its center is mr²/2, and for a semicircular wire, it is half of that, but we also need to account for the distance of the center of mass from the geometric center.

After performing the calculations and adjustments, the moment of inertia of a uniform semicircular wire about an axis through its center of mass and perpendicular to its plane is:

I = (1/2) m r²

This result shows how the mass distribution affects the rotational inertia of the wire, and it highlights the importance of understanding both geometry and mass distribution in physics.