A metre stick of mass 400 g is pivoted at one end and displaced through an angle `60^(@)` . The increase in its potential energy is

Options

A. 2 J
B. 3 J
C. 0 J
D. 1 J

Explanation

To solve the problem, we need to calculate the increase in potential energy of a meter stick that is pivoted at one end and displaced through an angle of 60 degrees. Here’s a step-by-step solution:

### Step 1: Understand the System
The meter stick has a mass of 400 g (which is 0.4 kg) and is pivoted at one end. When it is displaced to an angle of 60 degrees, the center of mass of the stick will rise to a new height.

### Step 2: Calculate the Initial Height of the Center of Mass
The center of mass of a uniform meter stick is located at its midpoint, which is at a distance of \( \frac{L}{2} \) from the pivot point. For a meter stick (L = 1 m):
- Initial height of the center of mass (h_initial) = \( \frac{1}{2} \) m = 0.5 m.

### Step 3: Calculate the Final Height of the Center of Mass
When the stick is tilted at an angle of 60 degrees, the height of the center of mass can be calculated using trigonometry:
- The height of the center of mass after displacement (h_final) = \( \frac{L}{2} \times \cos(60^\circ) \).
- Since \( \cos(60^\circ) = \frac{1}{2} \):
\[ h_{final} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \, \text{m} = 0.25 \, \text{m}. \]

### Step 4: Calculate the Increase in Height
The increase in height (Δh) of the center of mass when the stick is tilted is:
\[ \Delta h = h_{final} - h_{initial} = 0.25 \, \text{m} - 0.5 \, \text{m} = -0.25 \, \text{m}. \]
(Note: The height decreases, indicating a negative change, but we need the absolute value for potential energy calculation.)

### Step 5: Calculate the Change in Potential Energy
Potential energy (PE) is given by the formula:
\[ PE = mgh, \]
where:
- m = mass of the stick = 0.4 kg,
- g = acceleration due to gravity = 10 m/s²,
- h = change in height.

The initial potential energy (PE_initial):
\[ PE_{initial} = mgh_{initial} = 0.4 \times 10 \times 0.5 = 2 \, \text{J}. \]

The final potential energy (PE_final):
\[ PE_{final} = mgh_{final} = 0.4 \times 10 \times 0.25 = 1 \, \text{J}. \]

### Step 6: Calculate the Increase in Potential Energy
The increase in potential energy (ΔPE) is:
\[ \Delta PE = PE_{final} - PE_{initial} = 1 \, \text{J} - 2 \, \text{J} = -1 \, \text{J}. \]
Since we are interested in the increase, we take the absolute value:
\[ \Delta PE = 1 \, \text{J}. \]

### Final Answer
The increase in potential energy is **1 Joule**.

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