e object and the axis of rotation.

For a continuous object like a semicircular ring, we can calculate the moment of inertia by integrating over the entire object. Here's how we can approach this problem:

1. Divide the semicircular ring into small mass elements: Imagine the semicircular ring divided into infinitesimally small mass elements, each with mass $$dm$$.

2. Calculate the moment of inertia of each element: The moment of inertia of each element about the axis passing through the center and perpendicular to the plane of the ring is given by $$dI = dmR^2$$, where R is the radius of the ring.

3. Integrate to find the total moment of inertia: To find the total moment of inertia, we need to integrate $$dI$$ over the entire ring. This means integrating from $$0$$ to $$\pi$$ (the angle spanned by the semicircle) with respect to the angle $$\theta$$.

4. Relate $$dm$$ to the total mass: Since the ring has a uniform mass distribution, we can express the mass of each element $$dm$$ as a fraction of the total mass $$M$$: $$dm = \frac{M}{πR} Rd\theta = \frac{M}{\pi} d\theta$$.

Now, let's perform the integration:

$$I = \int_{0}^{\pi} dI = \int_{0}^{\pi} dmR^2 = \int_{0}^{\pi} \frac{M}{\pi} d\theta R^2$$

$$I = \frac{MR^2}{\pi} \int_{0}^{\pi} d\theta = \frac{MR^2}{\pi} [\theta]_{0}^{\pi}$$

$$I = \frac{MR^2}{\pi} [\pi - 0] = MR^2$$

Therefore, the moment of inertia of the semicircular ring about the given axis is $$MR^2$$. Comparing this to the given formula, we find that $$x = \boxed{1}$$.

Rotational Motion

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