The Extended Work-Energy Theorem and Non-Conservative Forces

When we talk about the Total Mechanical Energy ($E_{mech}$), we are referring to the sum of these two parts: $E_{mech} = K + U$

Work Done by Non-conservative Forces

In a perfect world, forces like gravity are "conservative." This means if you lift a ball and then let it fall, the work done by gravity is "recovered" as kinetic energy; the total mechanical energy remains constant.

Non-conservative forces, however, do not play by these rules. These include:

• Friction: Converts mechanical energy into thermal energy (heat).
• Air Resistance: A drag force that opposes motion.
• Applied Forces: An external person or motor pushing an object.

Unlike conservative forces, the work done by non-conservative forces ($W_{nc}$) results in a permanent change to the system's total mechanical energy. If friction acts on a sliding block, it does negative work, effectively "stealing" mechanical energy and turning it into heat.

The Extended Work-Energy Theorem

The Extended Work-Energy Theorem provides the mathematical framework to account for these energy losses or gains. It states that the work done by non-conservative forces is exactly equal to the change in the system's total mechanical energy:

$W_{nc} = \Delta E_{mech}$

By expanding the definition of mechanical energy, we can see how non-conservative forces influence both motion and position simultaneously:

$W_{nc} = \Delta K + \Delta U$ $W_{nc} = (K_f - K_i) + (U_f - U_i)$

Where:

• $K_f$ and $K_i$ are the final and initial kinetic energies.
• $U_f$ and $U_i$ are the final and initial potential energies.

Key Insight: If $W_{nc}$ is zero (meaning only conservative forces like gravity are acting), then $\Delta E_{mech} = 0$, and we say mechanical energy is conserved. If $W_{nc}$ is negative, the system loses mechanical energy. If $W_{nc}$ is positive, an external agent is adding energy to the system.

Problem-Solving Strategy

Applying this theorem to complex problems—like a block sliding across a floor with friction and then up a ramp—requires a disciplined approach.

1. Identify Important Events: Pinpoint the specific moments in time where the state changes (e.g., the moment the block leaves a spring, or the moment it hits a rough patch of floor).
2. Define the System: Clearly decide what objects you are analyzing.
3. Choose a Coordinate System: Select a reference point for your potential energy (e.g., setting the floor as $h = 0$).
4. Write Energy Expressions: For every "event" identified in step 1, write down the expressions for $K$ and $U$.
5. Apply the Theorem: Use $W_{nc} = \Delta K + \Delta U$ to link the initial state to the final state.

Worked Example: The Sled on an Incline

Consider a sled of mass $m = 70\text{ kg}$ that starts from rest and slides down a $10^\circ$ incline for $80\text{ m}$. It then travels $20\text{ m}$ horizontally before climbing an $8^\circ$ incline for $60\text{ m}$ until it comes to rest. What is the magnitude of the net work done by friction?

Solution Approach:

1. Initial State ($i$): The sled is at the top of the first incline. $K_i = 0$. $U_i = mgh_{initial}$.
2. Final State ($f$): The sled is at rest at the top of the second incline. $K_f = 0$. $U_f = mgh_{final}$.
3. Identify $W_{nc}$: The only non-conservative force doing work is friction.
4. Apply Theorem: $W_{friction} = (K_f + U_f) - (K_i + U_i)$.

By calculating the change in height between the starting point and the stopping point, you can determine exactly how much energy was "lost" to the work of friction.

Exam Tip: Always check if the problem mentions "frictionless." If it does, you can simplify your work by setting $W_{nc} = 0$ and simply using $\Delta E_{mech} = 0$.