Work done by conservative force is equal to

Question

Prepp Team · Accepted Answer

Understanding Work Done by a Conservative Force

In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Examples of conservative forces include gravity and the elastic force of a spring. A key characteristic of conservative forces is their association with potential energy.

Relationship Between Work and Potential Energy

For any conservative force, the work done by the force on an object as it moves from an initial position \( i \) to a final position \( f \) is defined in terms of the change in potential energy (\( \Delta PE \)) associated with that force. The relationship is given by:

\[ W_c = - \Delta PE \]

Here, \( W_c \) is the work done by the conservative force. \( \Delta PE \) represents the change in potential energy, calculated as the final potential energy minus the initial potential energy:

\[ \Delta PE = PE_{final} - PE_{initial} \]

Substituting this into the work equation:

\[ W_c = - (PE_{final} - PE_{initial}) \] \[ W_c = PE_{initial} - PE_{final} \]

The term \( PE_{initial} - PE_{final} \) represents the difference between the initial potential energy and the final potential energy. If \( PE_{initial} \) is greater than \( PE_{final} \), the potential energy has decreased (\( \Delta PE \) is negative), and the work done by the conservative force is positive (\( W_c = -(\text{negative}) = \text{positive} \)). If \( PE_{initial} \) is less than \( PE_{final} \), the potential energy has increased (\( \Delta PE \) is positive), and the work done by the conservative force is negative (\( W_c = -(\text{positive}) = \text{negative} \)).

Therefore, the work done by a conservative force is equal to the decrease in potential energy.

Conservative Force Work and Kinetic Energy

The Work-Energy Theorem states that the net work done by all forces acting on an object is equal to the change in its kinetic energy (\( \Delta KE \)).

\[ W_{net} = \Delta KE \]

If the conservative force is the only force doing work, then the net work is simply the work done by the conservative force (\( W_{net} = W_c \)). In this specific case:

\[ W_c = \Delta KE \]

Combining this with \( W_c = -\Delta PE \), we get \( \Delta KE = -\Delta PE \), which means \( \Delta KE + \Delta PE = 0 \). This shows that the total mechanical energy (\( KE + PE \)) is conserved when only conservative forces do work.

However, the question asks what the work done by a conservative force is equal to. The fundamental and always true relationship for the work done by a conservative force, regardless of whether other forces are present or doing work, is its relation to potential energy.

Analyzing the Options

Let's look at the given options in light of our understanding:

Decrease in potential energy: As derived, \( W_c = PE_{initial} - PE_{final} \), which is the decrease in potential energy. This matches our finding.
Increase in kinetic energy: This is only true if the conservative force is the only force doing work (from \( W_c = \Delta KE \)). This is a specific case, not a general definition of the work done by a conservative force.
Increase in potential energy: The work done is equal to the negative of the increase in potential energy (\( W_c = -\Delta PE \)).
Decrease in kinetic energy: The work done is equal to the negative of the change in kinetic energy (\( W_c = \Delta KE \)). This is only true if the conservative force is the only force doing work. A decrease in kinetic energy (\( -\Delta KE \)) would correspond to \( -W_c \), or an increase in potential energy.

Based on the fundamental definition relating the work done by a conservative force to potential energy, the work done is equal to the decrease in potential energy.

Revision Table: Work and Energy Concepts

Concept	Description	Formula
Work Done by Conservative Force (\(W_c\))	Work done is path independent; related to potential energy.	\(W_c = -\Delta PE\) or \(W_c = PE_{initial} - PE_{final}\)
Potential Energy (\(PE\))	Energy stored due to position or configuration relative to a conservative force.	Depends on the force (e.g., \(mgh\) for gravity, \(\frac{1}{2}kx^2\) for spring).
Kinetic Energy (\(KE\))	Energy of motion.	\(KE = \frac{1}{2}mv^2\)
Work-Energy Theorem	Net work by all forces equals change in kinetic energy.	\(W_{net} = \Delta KE\)
Conservation of Mechanical Energy	Total mechanical energy (\(KE+PE\)) is constant if only conservative forces do work.	\(\Delta KE + \Delta PE = 0\) or \(KE_i + PE_i = KE_f + PE_f\)

Additional Information: Exploring Conservative Forces

Understanding conservative forces is crucial in physics, especially when studying energy conservation.

Path Independence: The work done by a conservative force only depends on the starting and ending points, not on the specific route taken. This is not true for non-conservative forces like friction.
Potential Energy: A potential energy function can only be defined for a conservative force. This potential energy represents stored energy that can be converted into kinetic energy or work.
Negative Work and Potential Energy: When a conservative force does positive work (e.g., gravity pulling an object down), the potential energy decreases. Conversely, when work is done against a conservative force (e.g., lifting an object against gravity), the potential energy increases, and the force itself does negative work.
Relationship to Force: A conservative force can be expressed as the negative gradient of its associated potential energy function (\(\mathbf{F}_c = -\nabla PE\)).