A metallic disc is melted and casted in to a solid sphere. Its moment of inertia about the axis thro

A metallic disc is melted and casted in to a solid sphere. Its moment of inertia about the axis through its Centre will :
(a) decrease (b) increase
(c) remain unchanged (d) data is incomplete
📌 Understanding the Change in Moment of Inertia When a Disc is Melted into a Sphere 🔥⚙️
When a metallic disc is melted and recast into a solid sphere, an important question arises: How does the *moment of inertia* change? Let's explore this concept in detail.
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🌟 *Moment of Inertia: A Quick Recap*
The *moment of inertia (I)* of an object depends on its *mass distribution* about a given axis. It is mathematically given by:
\[
I = \sum m r^2
\]
where *m* is the mass element and *r* is its perpendicular distance from the axis of rotation. The greater the mass is distributed away from the axis, the higher the moment of inertia.
---
🛞 *Moment of Inertia of a Disc*
For a *thin uniform disc* of mass *M* and radius **R**, the moment of inertia about its central axis (perpendicular to the plane of the disc) is given by:
\[
I_\textdisc = \frac12M R^2
\]
This formula shows that half of the mass is concentrated closer to the center, but a significant portion still remains at the outer edge, contributing to a higher moment of inertia.
---
🏀 *Moment of Inertia of a Solid Sphere*
Now, when the disc is melted and reshaped into a *solid sphere**, the mass distribution changes. The **moment of inertia of a solid sphere* of mass *M* and radius *R'* about its central axis is given by:
\
Since the sphere is more *compact**, the mass is closer to the axis of rotation compared to the disc. This results in a **lower* moment of inertia.
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🔄 *Comparing the Two Moments of Inertia*
We know that in the transformation process:
*Mass remains constant* → \
Thus, the *moment of inertia decreases* when the disc is melted and reshaped into a solid sphere.
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✅ *Final Answer:*
The moment of inertia of the metallic object *decreases* when recast into a sphere.
🔹 *Correct option: (a) decrease*
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🚀 *Concept Application & Exam Relevance*
🔸 *JEE & NEET Physics:* Understanding moment of inertia is crucial in *mechanics and rotational motion* questions.
🔸 *Real-World Application:* This concept applies to *engineering designs* where rotational stability is a factor.
🔸 *Competitive Exams:* A frequently tested problem in **IIT JEE, NEET, and physics Olympiads**.
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🔥 *Boost Your Preparation with More Resources!*
For more detailed physics explanations, tricks, and shortcut methods, follow me on *Instagram* 📲: [Physics Behind Everything](www.instagram....)
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