The potential energy of a system increased if work is done

Options

A. by the system against a conservative force
B. by the system against a nonconservative force
C. upon the system by a conservative force
D. upon the system by a nonconservative force

Explanation

To solve the question, "The potential energy of a system increased if work is done," we need to understand the relationship between work, forces, and potential energy.

### Step-by-Step Solution:

1. **Understanding Potential Energy (PE)**:
- Potential energy is the energy stored in an object due to its position relative to other objects. For example, gravitational potential energy depends on the height of an object above the ground.

2. **Work Done Against Forces**:
- Work is defined as the force applied to an object times the distance over which that force is applied, in the direction of the force. Mathematically, \( W = F \cdot d \cdot \cos(\theta) \), where \( \theta \) is the angle between the force and the direction of motion.

3. **Conservative Forces**:
- A conservative force is one where the work done is independent of the path taken. The most common example is gravitational force. The work done against a conservative force results in an increase in potential energy.

4. **Example of Lifting an Object**:
- Consider lifting a ball of mass \( M \) from the ground (height = 0) to a height \( H \). The gravitational potential energy at height \( H \) is given by the formula:
\[ U = Mgh \]
- Here, \( g \) is the acceleration due to gravity. When the ball is lifted, work is done against the gravitational force.

5. **Calculating Work Done**:
- The work done \( W \) in lifting the ball to height \( H \) is equal to the increase in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} = Mgh - 0 = Mgh \]
- Thus, the work done on the system (lifting the ball) results in an increase in potential energy.

6. **Conclusion**:
- Therefore, the potential energy of a system increases if work is done by the system against a conservative force. In this case, the correct option is:
- **By the system against a conservative force**.

### Final Answer:
The potential energy of a system increases if work is done **by the system against a conservative force**.

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