The mass of a unifprm rod is 0.4 kg and its length is 2 m. .Calculate the radius of gyration of the rod about an axis passing through its centre of mass and perpendicular to its length ?

Explanation

To calculate the radius of gyration of a uniform rod about an axis passing through its center of mass and perpendicular to its length, we can follow these steps:

### Step 1: Identify the given values.
- Mass of the rod (m) = 0.4 kg
- Length of the rod (l) = 2 m

### Step 2: Use the formula for the moment of inertia (I) of a uniform rod about an axis through its center of mass and perpendicular to its length.
The moment of inertia (I) for a uniform rod about this axis is given by the formula:
\[ I = \frac{m l^2}{12} \]

### Step 3: Substitute the values into the moment of inertia formula.
Substituting the given values into the formula:
\[ I = \frac{0.4 \times (2)^2}{12} \]
Calculating \( (2)^2 \):
\[ (2)^2 = 4 \]
Now substituting this back into the equation:
\[ I = \frac{0.4 \times 4}{12} \]

### Step 4: Simplify the equation.
Calculating the numerator:
\[ 0.4 \times 4 = 1.6 \]
Now substituting this into the equation:
\[ I = \frac{1.6}{12} \]
Calculating the division:
\[ I = 0.1333 \text{ kg m}^2 \]

### Step 5: Use the relationship between moment of inertia and radius of gyration.
The relationship between the moment of inertia (I) and the radius of gyration (k) is given by:
\[ I = m k^2 \]
Rearranging this gives:
\[ k^2 = \frac{I}{m} \]

### Step 6: Substitute the values to find k.
Substituting the values of I and m into the equation:
\[ k^2 = \frac{0.1333}{0.4} \]
Calculating the division:
\[ k^2 = 0.33325 \]

### Step 7: Take the square root to find k.
Taking the square root of both sides:
\[ k = \sqrt{0.33325} \approx 0.577 \text{ m} \]

### Final Answer:
The radius of gyration (k) of the rod about the specified axis is approximately **0.577 m**.

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