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If the particle moves opposite to the co...

If the particle moves opposite to the conservative field, work done by the field will be ________ and so change in potential energy will be _________, i.e. potential energy wil ________ .

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To solve the question, we need to analyze the scenario where a particle moves opposite to a conservative field. Let's break it down step by step. ### Step-by-Step Solution 1. **Understanding the Direction of Motion**: - When a particle moves opposite to a conservative field (for example, against the force of gravity), it means that the particle is moving in the direction opposite to the force acting on it. 2. **Work Done by the Field**: - The work done (W) by a conservative force is given by the formula: \[ W = F \cdot s \cdot \cos(\theta) \] where \( F \) is the magnitude of the force, \( s \) is the displacement, and \( \theta \) is the angle between the force and the displacement vector. - In this case, since the particle moves opposite to the conservative field, \( \theta = 180^\circ \). Thus, \( \cos(180^\circ) = -1 \). - Therefore, the work done by the field becomes: \[ W = -F \cdot s \] - Since the force \( F \) can be represented as \( mg \) (weight of the particle), and if the particle moves a distance \( h \) upwards, the work done by the field is: \[ W = -mg \cdot h \] - This indicates that the work done by the conservative field is **negative**. 3. **Change in Potential Energy**: - The change in potential energy (\( \Delta U \)) is related to the work done by the conservative field: \[ W = -\Delta U \] - Since we found that \( W = -mgh \), we can substitute this into the equation: \[ -mgh = -\Delta U \] - This simplifies to: \[ \Delta U = mgh \] - This indicates that the change in potential energy is **positive**. 4. **Conclusion about Potential Energy**: - Since the change in potential energy is positive, it means that the potential energy of the particle has increased as it moves against the conservative field. ### Final Answers - Work done by the field will be **negative**. - Change in potential energy will be **positive**. - Potential energy will **increase**.
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