The moment of inertia of a non-uniform semicircular wire about a line perpendicular to the plane of the wire through the center can be calculated by considering small elements of the wire and summing up their individual moments of inertia.

1. Definition of moment of inertia:
The moment of inertia of an object is a measure of its resistance to rotational motion. It depends on the mass distribution of the object and the axis of rotation. For a continuous object, it is calculated by integrating the mass elements over the object.

2. Mass distribution of the semicircular wire:
In this case, the semicircular wire has a non-uniform mass distribution. This means that different parts of the wire have different masses. To calculate the moment of inertia, we need to consider small elements of the wire.

3. Consideration of small elements:
We can divide the semicircular wire into small elements of equal length. Let's denote the length of each small element as dl. Since the wire is semicircular, the angle subtended by each small element at the center is dθ, where dθ = dl/r (r is the radius of the wire).

4. Mass of each small element:
The mass of each small element can be calculated using the linear mass density, λ, which is defined as the mass per unit length of the wire. The mass of each small element, dm, is given by dm = λdl.

5. Moment of inertia of each small element:
The moment of inertia of each small element, dI, about the axis perpendicular to the plane of the wire through the center can be calculated using the formula for the moment of inertia of a point mass, which is given by dI = (dm)(r^2).

6. Summation of moments of inertia:
To calculate the total moment of inertia, we need to sum up the moments of inertia of all the small elements. This can be done by integrating the moment of inertia expression over the length of the wire. The integral can be written as I = ∫dI = ∫(dm)(r^2).

7. Integration of the mass element:
Substituting the expression for dm = λdl and the relationship dθ = dl/r into the integral, we get I = ∫(λdl)(r^2) = λr^2∫dl.

8. Integration limits:
The integration limits depend on the length of the semicircular wire. Let's assume the length of the wire is L. Since we are considering only the semicircle, the integration limits will be from 0 to π.

9. Integration and final result:
Evaluating the integral, we get I = λr^2∫dl = λr^2L. Finally, substituting the value of λ = m/L (mass divided by length), we have I = (m/L)(r^2)L = m(r^2).

10. Final result and conclusion:
The moment of inertia of the non-uniform semicircular wire about a line perpendicular to the plane of the wire through the center is given by I = m(r^2), where m is the mass of the wire and r is the radius.

By following this step-by-step approach, we can calculate the moment of inertia of a non-uniform semicircular wire accurately.

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