Four point masses of 10 kg each are placed at the corners of a square PQRS of side 3 cm. Calculate the moment of inertia of the square about an axis coinciding with the side PQ of the square.

Answer

To calculate the moment of inertia of the square about an axis coinciding with the side PQ, we will follow these steps:

### Step 1: Understand the Configuration
We have a square PQRS with each side measuring 3 cm. At each corner of the square, there is a point mass of 10 kg. The square is oriented such that the axis of rotation is along the side PQ.

### Step 2: Identify the Relevant Masses
The masses at points P and Q will not contribute to the moment of inertia about the axis PQ because their distance from the axis is zero. Therefore, we only need to consider the masses at points R and S.

### Step 3: Calculate the Distance of R and S from the Axis
The distance of point R from the axis PQ is equal to the side length of the square, which is 3 cm (or 0.03 m). The same applies to point S.

### Step 4: Use the Moment of Inertia Formula
The moment of inertia \( I \) for a point mass is given by the formula:
\[ I = m r^2 \]
where \( m \) is the mass and \( r \) is the distance from the axis of rotation.

### Step 5: Calculate the Moment of Inertia for Points R and S
For point R:
\[ I_R = m \cdot r^2 = 10 \, \text{kg} \cdot (0.03 \, \text{m})^2 = 10 \cdot 0.0009 = 0.009 \, \text{kg m}^2 \]

For point S:
\[ I_S = m \cdot r^2 = 10 \, \text{kg} \cdot (0.03 \, \text{m})^2 = 10 \cdot 0.0009 = 0.009 \, \text{kg m}^2 \]

### Step 6: Sum the Contributions
The total moment of inertia \( I \) about the axis coinciding with side PQ is the sum of the moments of inertia of points R and S:
\[ I = I_R + I_S = 0.009 \, \text{kg m}^2 + 0.009 \, \text{kg m}^2 = 0.018 \, \text{kg m}^2 \]

### Final Answer
The moment of inertia of the square about the axis coinciding with side PQ is:
\[ I = 0.018 \, \text{kg m}^2 \]

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