A uniform wire is bent into the form of a rectangle of length `L` and width `W`. The coordinates of its centre of mass from a corner are

Options

A. `(0,0)`
B. `(L/2,W)`
C. `(L,W/2)`
D. `(L/2,W/2)`

Explanation

To find the coordinates of the center of mass of a uniform wire bent into the shape of a rectangle with length \( L \) and width \( W \), we can follow these steps:

### Step 1: Understand the Geometry
The rectangle has a length \( L \) and a width \( W \). We need to determine the coordinates of its center of mass from one of its corners, which we can assume to be the origin (0, 0).

### Step 2: Identify the Center of Mass Formula
For a uniform wire, the center of mass (CM) can be found using the formula:
\[ X_{cm} = \frac{\int x \, dm}{\int dm} \]
\[ Y_{cm} = \frac{\int y \, dm}{\int dm} \]
Where \( dm \) is the mass element of the wire.

### Step 3: Calculate the Mass Element
Since the wire is uniform, we can express the mass per unit length (linear mass density) as:
\[ \lambda = \frac{M}{L + 2W} \]
where \( M \) is the total mass of the wire.

### Step 4: Calculate the X Coordinate of the Center of Mass
1. The wire consists of four segments: two lengths \( L \) and two widths \( W \).
2. The x-coordinates of the segments are:
- For the bottom segment (length \( L \)): \( x = \frac{L}{2} \)
- For the top segment (length \( L \)): \( x = \frac{L}{2} \)
- For the left segment (width \( W \)): \( x = 0 \)
- For the right segment (width \( W \)): \( x = L \)

3. The contribution to the x-coordinate from each segment:
- Bottom segment: \( \frac{L}{2} \cdot L \)
- Top segment: \( \frac{L}{2} \cdot L \)
- Left segment: \( 0 \cdot W \)
- Right segment: \( L \cdot W \)

4. The total x-coordinate of the center of mass:
\[ X_{cm} = \frac{(L \cdot \frac{L}{2}) + (L \cdot \frac{L}{2}) + (W \cdot 0) + (W \cdot L)}{L + 2W} \]
\[ X_{cm} = \frac{L^2 + WL}{L + 2W} = \frac{L}{2} \]

### Step 5: Calculate the Y Coordinate of the Center of Mass
1. The y-coordinates of the segments are:
- For the bottom segment (length \( L \)): \( y = 0 \)
- For the top segment (length \( L \)): \( y = W \)
- For the left segment (width \( W \)): \( y = \frac{W}{2} \)
- For the right segment (width \( W \)): \( y = \frac{W}{2} \)

2. The contribution to the y-coordinate from each segment:
- Bottom segment: \( 0 \cdot L \)
- Top segment: \( W \cdot L \)
- Left segment: \( \frac{W}{2} \cdot W \)
- Right segment: \( \frac{W}{2} \cdot W \)

3. The total y-coordinate of the center of mass:
\[ Y_{cm} = \frac{(0 \cdot L) + (W \cdot L) + (\frac{W}{2} \cdot W) + (\frac{W}{2} \cdot W)}{L + 2W} \]
\[ Y_{cm} = \frac{WL + W^2}{L + 2W} = \frac{W}{2} \]

### Final Result
The coordinates of the center of mass from the corner of the rectangle are:
\[ \left( \frac{L}{2}, \frac{W}{2} \right) \]