Radius of Gyration of Disc Shortcut

  

❓ Question

A solid sphere of radius $R=10$cm is shown. An axis is at a distance $d=15$ cm from the center of the sphere.

If the radius of gyration about the shown axis is $\sqrt{n}$ cm, find $n$.

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✍️ Short Explanation

For a solid sphere,

$$I_{CM}=\frac{2}{5}M{R}^2$$

Using the Parallel Axis Theorem:

$$I=I_{CM}+M{d}^2$$

where

$$R=10\text{ cm},d=15\text{ cm}$$

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🔷 Step 1 — Calculate Moment of Inertia

$$I=\frac{2}{5}M(10{)}^2+M(15{)}^2$$
$$I=\frac{2}{5}(100)M+225M$$
$$I=40M+225M$$
$$I=265M$$

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🔷 Step 2 — Use Radius of Gyration

By definition,

$$I=M{k}^2$$

Therefore,

$$M{k}^2=265M$$
$$k^2=265$$
$$k=\sqrt{265}\text{ cm}$$

Given

$$k=\sqrt{n}\text{ cm}$$

hence

$$\boxed{{\displaystyle n=265}}$$

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✅ Final Answer

$$\boxed{{\displaystyle 265}}$$

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