A rod of mass M and length L is made to stand vertically. Potential energy of the rod in this position is

Options

A. Mgl
B. `Mgl/2`
C. `Mgl/3`
D. `Mgl/4`

Answer

To find the potential energy of a vertically standing rod of mass \( M \) and length \( L \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Center of Mass**:
The center of mass of a uniform rod is located at its midpoint. For a rod of length \( L \), the center of mass is at a distance of \( \frac{L}{2} \) from either end.

2. **Determine the Height of the Center of Mass**:
When the rod is standing vertically, the center of mass is at a height of \( \frac{L}{2} \) above the ground.

3. **Calculate the Potential Energy**:
The potential energy (PE) of an object at height \( h \) is given by the formula:
\[ PE = mgh \]
where:
- \( m \) is the mass of the object,
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the height above the reference point (ground level in this case).

4. **Substituting Values**:
In our case, the mass of the rod is \( M \) and the height \( h \) is \( \frac{L}{2} \). Therefore, the potential energy of the rod when it is standing vertically is:
\[ PE = M \cdot g \cdot \frac{L}{2} \]

5. **Final Expression**:
Thus, the potential energy of the rod in the vertical position is:
\[ PE = \frac{MgL}{2} \]

### Final Answer:
The potential energy of the rod in the vertical position is \( \frac{MgL}{2} \).

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