A rod of length `1.0 m` and mass `0.5 kg` fixed at ond is initially hanging vertical. The other end is now raised until it makes an angle `60^@` with the vertical. How much work is required?

Explanation

To solve the problem of how much work is required to raise a rod of length 1.0 m and mass 0.5 kg from a vertical position to an angle of 60° with the vertical, we can follow these steps:

### Step 1: Understand the Initial and Final Positions
- The rod is initially hanging vertically, which means its center of mass is at a height of \( \frac{L}{2} \) from the pivot point (the fixed end).
- When the rod is raised to an angle of 60° with the vertical, we need to determine the new height of the center of mass.

### Step 2: Calculate the Height Change (ΔH)
- The center of mass of the rod is located at a distance of \( \frac{L}{2} = \frac{1.0}{2} = 0.5 \) m from the pivot.
- When the rod is raised to an angle of 60°, the vertical height of the center of mass can be found using trigonometry:
\[ \text{New height} = \frac{L}{2} \cos(60°) = 0.5 \cdot \frac{1}{2} = 0.25 \text{ m} \]
- The change in height (ΔH) is then:
\[ \Delta H = \text{Initial height} - \text{New height} = 0.5 \text{ m} - 0.25 \text{ m} = 0.25 \text{ m} \]

### Step 3: Calculate the Work Done (W)
- The work done in raising the rod is equal to the change in potential energy, which can be calculated using the formula:
\[ W = m g \Delta H \]
- Here, \( m = 0.5 \) kg, \( g = 10 \) m/s² (approximating the acceleration due to gravity), and \( \Delta H = 0.25 \) m.
- Substituting the values:
\[ W = 0.5 \cdot 10 \cdot 0.25 = 1.25 \text{ joules} \]

### Final Answer
The work required to raise the rod is **1.25 joules**.

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