Work Energy Theorem

The Work-Energy Theorem for conservative forces explains the relationship between the change in an object's energy and the work done by a force. When a conservative force acts on a body, a decrease in its potential energy (like when a body falls) is completely turned into kinetic energy. Conversely, if an object moves against a conservative force, its kinetic energy decreases. This decrease is equal to the work done against the force. This fundamental principle is known as the work-energy theorem for conservative forces.

"If there is a decrease in the potential energy (like a body falling down) due to a conservative force, it is entirely converted into kinetic energy. Work done by the force then appears as kinetic energy. Vice versa if an object is moving against a conservative force its kinetic energy decreases by an amount equal to the work done against the force."

The source mentions two main cases (types) illustrating the theorem:

• Case I: Focuses on the work-energy theorem purely for a conservative force (like gravity) in a simplified scenario.
• Case II: Discusses the application when an accelerating conservative force (like gravity) acts alongside a retarding non-conservative force (like air resistance).

Since the features are few and primarily relate to the two cases, they are listed simply:

• It applies when work is done by a conservative force (like gravity).
• A decrease in potential energy is entirely converted into kinetic energy.
• Work done by the force appears as kinetic energy.
• It can be extended to include non-conservative forces like air resistance.

The source provides a step-by-step mathematical demonstration for Case I (constant opposing force).

Process Explanation (Case I)

1. Start with the equation of motion: Consider an object of mass m moving with initial velocity \[\vec u\], slowed down to \[\vec v\] over displacement \[\vec s\] by a constant opposing force \[\vec F\].

2. Apply the kinematic equation:

   v² - u² = 2(-a)s

   (The acceleration, a, is negative because the force is opposing/retarding.)

3. Substitute acceleration: The magnitude of acceleration is a = \[\frac{F}{m}\].

4. Rearrange the equation: Multiply the entire equation by \[\frac {m}{2}\]​:

   \[\frac{1}{2}mu^2-\frac{1}{2}mv^2\] = (ma) . s = F . s

Conclusion of Process

• The Left-hand side (\[\frac {1}{2}\]mu² - \[\frac {1}{2}\]mv²) represents the decrease in the kinetic energy.

• The Right-hand side (F.s) represents the work done by the force.

• This shows that the change in kinetic energy is equal to the work done by the conservative force, which is in accordance with the work-energy theorem.

The source only provides two scenarios as examples:

1. Body Falling Down: A body falling down is an example where a decrease in potential energy (due to a conservative force like gravity) is entirely converted into kinetic energy (if air resistance is negligible).

2. Object Dropped Through Air (Non-Ideal): An object dropped from a height h falls through the air.

• A decrease in potential energy happens.
• Work is done against non-conservative air resistance.
• Some energy appears as heat, sound, etc. (Work is not entirely converted to KE).