Calculate the radius of gyration of a cylindrical rod of mass M and length L about an axis of rotation perpendicular to its length and passing through its centre.

Answer

To calculate the radius of gyration of a cylindrical rod of mass \( M \) and length \( L \) about an axis of rotation perpendicular to its length and passing through its center, we can follow these steps:

### Step 1: Define the Moment of Inertia
The moment of inertia \( I \) of a uniform rod of length \( L \) and mass \( M \) about an axis perpendicular to its length and passing through its center is given by the formula:
\[ I = \frac{1}{12} M L^2 \]

### Step 2: Use the Formula for Radius of Gyration
The radius of gyration \( k \) is defined as:
\[ k = \sqrt{\frac{I}{M}} \]
where \( I \) is the moment of inertia and \( M \) is the mass of the rod.

### Step 3: Substitute the Moment of Inertia into the Radius of Gyration Formula
Substituting the expression for \( I \) into the formula for \( k \):
\[ k = \sqrt{\frac{\frac{1}{12} M L^2}{M}} \]

### Step 4: Simplify the Expression
The mass \( M \) cancels out:
\[ k = \sqrt{\frac{1}{12} L^2} \]
This simplifies to:
\[ k = \frac{L}{\sqrt{12}} \]

### Step 5: Further Simplify the Radius of Gyration
We can express \( \sqrt{12} \) as \( 2\sqrt{3} \):
\[ k = \frac{L}{2\sqrt{3}} \]

### Final Answer
Thus, the radius of gyration \( k \) of the cylindrical rod about the specified axis is:
\[ k = \frac{L}{2\sqrt{3}} \]

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