A uniform metal rod of length 1m is bent at `90^(@)`, so as to form two arms of equal length. The centre of mass of this bent rod is

Options

A. on the bisector of the angle, `(1/sqrt(2))` m from vertex
B. on the bisector of angle, `(1/(2sqrt(2)))` m from vertex
C. on the bisector of the angle, `(1/2)` m from vertex
D. on the bisector of the angle, `(1/(4sqrt(2)))`m from vertex

Explanation

To find the center of mass of a uniform metal rod bent at a 90-degree angle, we can follow these steps:

### Step 1: Understand the Geometry of the Bent Rod
The rod has a total length of 1 meter and is bent at a right angle, forming two arms of equal length. Since the total length is 1 meter, each arm will have a length of:
\[ L = \frac{1 \text{ m}}{2} = 0.5 \text{ m} \]

### Step 2: Define the Coordinate System
We can place the bent rod in a coordinate system. Let's assume:
- Point B (the vertex of the angle) is at the origin (0, 0).
- Point A (end of the first arm) is at (0, 0.5) on the y-axis.
- Point C (end of the second arm) is at (0.5, 0) on the x-axis.

### Step 3: Determine the Center of Mass of Each Arm
For each arm, the center of mass is located at its midpoint:
- The center of mass of arm AB (vertical arm) is at:
\[ \text{CM}_{AB} = \left(0, \frac{0 + 0.5}{2}\right) = \left(0, 0.25\right) \]

- The center of mass of arm BC (horizontal arm) is at:
\[ \text{CM}_{BC} = \left(\frac{0 + 0.5}{2}, 0\right) = \left(0.25, 0\right) \]

### Step 4: Calculate the Overall Center of Mass
Since the rod is uniform, we can treat each arm as having equal mass. Let’s denote the mass of each arm as \( m \). The coordinates of the center of mass (CM) of the entire bent rod can be calculated using the formula:
\[ X_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
\[ Y_{CM} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \]

Substituting the values:
- For the x-coordinates:
\[ X_{CM} = \frac{m \cdot 0 + m \cdot 0.25}{m + m} = \frac{0 + 0.25m}{2m} = \frac{0.25}{2} = 0.125 \]

- For the y-coordinates:
\[ Y_{CM} = \frac{m \cdot 0.25 + m \cdot 0}{m + m} = \frac{0.25m + 0}{2m} = \frac{0.25}{2} = 0.125 \]

### Step 5: Final Coordinates of the Center of Mass
Thus, the coordinates of the center of mass of the bent rod are:
\[ \text{CM} = (0.125, 0.125) \]

### Step 6: Distance from the Vertex
To find the distance from the vertex (point B) to the center of mass:
\[ \text{Distance} = \sqrt{(0.125)^2 + (0.125)^2} = \sqrt{0.015625 + 0.015625} = \sqrt{0.03125} = \frac{1}{4\sqrt{2}} \text{ m} \]

### Conclusion
The center of mass of the bent rod is located at a distance of \( \frac{1}{4\sqrt{2}} \) meters from the vertex along the bisector of the angle formed by the two arms.

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