A uniform rod o length one meter is bent at its midpoint to make `90^@`. The distance of centre of mass from the centre of rod is (in `cm`)

Options

A. `20.2`
B. `13.4`
C. `15`
D. `17.6`

Explanation

To find the distance of the center of mass from the center of a bent uniform rod, we can follow these steps:

### Step 1: Understand the Configuration
The rod is 1 meter long and is bent at its midpoint to form a right angle (90 degrees). This means that each segment of the rod is 0.5 meters long.

### Step 2: Identify the Center of Mass of Each Segment
Since the rod is uniform, the center of mass of each straight segment (0.5 meters long) will be at its midpoint. Therefore, the center of mass of each segment is located at 0.25 meters from the bend (the midpoint).

### Step 3: Set Up the Coordinate System
Let’s place the bend at the origin (0,0) of a coordinate system:
- The first segment (horizontal) extends from (-0.25, 0) to (0.25, 0).
- The second segment (vertical) extends from (0, -0.25) to (0, 0.25).

### Step 4: Calculate the Center of Mass
The coordinates of the center of mass for each segment can be calculated as follows:
- For the horizontal segment:
\[ (x_1, y_1) = (0.25, 0) \]
- For the vertical segment:
\[ (x_2, y_2) = (0, -0.25) \]

### Step 5: Use the Center of Mass Formula
The center of mass (CM) of the entire system can be calculated using the formula:
\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
\[ y_{CM} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \]
Since the rod is uniform, we can assume \(m_1 = m_2 = m\) (mass of each segment is equal):
\[ x_{CM} = \frac{m(0.25) + m(0)}{m + m} = \frac{0.25m}{2m} = 0.125 \text{ m} \]
\[ y_{CM} = \frac{m(0) + m(-0.25)}{m + m} = \frac{-0.25m}{2m} = -0.125 \text{ m} \]

### Step 6: Calculate the Distance from the Center of the Rod
The center of the rod is at (0,0). The distance from the center of mass to the center of the rod can be calculated using the distance formula:
\[ d = \sqrt{(x_{CM} - 0)^2 + (y_{CM} - 0)^2} = \sqrt{(0.125)^2 + (-0.125)^2} \]
\[ d = \sqrt{0.015625 + 0.015625} = \sqrt{0.03125} = 0.176776695 \text{ m} \]

### Step 7: Convert to Centimeters
To convert meters to centimeters, we multiply by 100:
\[ d = 0.176776695 \times 100 = 17.6776695 \text{ cm} \approx 17.68 \text{ cm} \]

### Final Answer
The distance of the center of mass from the center of the rod is approximately **17.68 cm**.

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