7.5 Nonconservative Forces

Nonconservative Forces and Mechanical Energy

Effects of nonconservative forces

Conservative forces like gravity have a useful property: the work they do depends only on where you start and where you end up, not the path you take. Nonconservative forces are the opposite. Forces like friction and air resistance are path-dependent, meaning the work they do changes depending on the route an object travels.

Why does this matter? Because nonconservative forces alter the total mechanical energy of a system. They dissipate energy out of the "useful" mechanical forms (kinetic and potential) and convert it into other forms like heat.

• Friction transforms kinetic energy into thermal energy. A book sliding across a table slows down because friction heats up the surfaces in contact.
• Air resistance reduces kinetic energy by doing negative work on moving objects. A parachute, for example, dramatically increases air resistance to slow a skydiver's descent.

When nonconservative forces act on a system, total mechanical energy is not conserved. A block sliding down a rough incline arrives at the bottom with less kinetic energy than it would on a frictionless surface, because friction has converted some of that energy into heat.

The key relationship:

$\Delta E = W_{nonconservative}$

The change in mechanical energy equals the work done by nonconservative forces.

Work-energy theorem applications

The work-energy theorem states that the net work done on an object equals the change in its kinetic energy:

$W_{net} = \Delta KE$

When both conservative and nonconservative forces act on an object, you split the net work into two parts:

$W_{net} = W_{conservative} + W_{nonconservative}$

• $W_{conservative}$ is work done by forces like gravity or a spring force
• $W_{nonconservative}$ is work done by forces like friction or air resistance

Since the work done by conservative forces equals the negative change in potential energy ($W_{conservative} = -\Delta PE$), you can rearrange to get:

$\Delta KE + \Delta PE = W_{nonconservative}$

This is the same as $\Delta E = W_{nonconservative}$, confirming that nonconservative forces account for any change in total mechanical energy.

Problem-solving steps:

1. Identify all forces acting on the object and classify each as conservative or nonconservative.
2. Calculate the work done by each nonconservative force (e.g., $W_{friction} = -f_k \cdot d$, where $f_k$ is the kinetic friction force and $d$ is the distance traveled).
3. Determine the initial and final kinetic and potential energies.
4. Apply $E_{initial} + W_{nonconservative} = E_{final}$ to solve for the unknown quantity.

Energy changes with nonconservative forces

The total mechanical energy of a system is the sum of kinetic and potential energy:

$E = KE + PE$

When nonconservative forces do work, you can track the energy change using:

$E_{initial} + W_{nonconservative} = E_{final}$

The sign of $W_{nonconservative}$ tells you what happens to the system's energy:

• Positive $W_{nonconservative}$: Mechanical energy increases. This happens when an external force adds energy to the system, like when you push a box up a rough incline (your push does more work than friction removes).
• Negative $W_{nonconservative}$: Mechanical energy decreases. Friction on a block sliding down a rough incline converts some mechanical energy into heat, so the block ends up with less total mechanical energy than it started with.
• Zero $W_{nonconservative}$: Mechanical energy is conserved. This is the special case where no nonconservative forces act, like a frictionless pendulum swinging back and forth.

Energy Conservation and Thermodynamics

Even when friction or air resistance removes mechanical energy from a system, that energy doesn't vanish. The law of conservation of energy still holds: energy is only converted from one form to another.

Dissipative forces like friction convert mechanical energy into thermal energy (internal energy). The surfaces in contact heat up, and the system's internal energy increases by exactly the amount of mechanical energy lost. If you've ever rubbed your hands together to warm them up, you've felt this conversion directly.

Thermodynamics provides the broader framework for tracking these energy transfers between a system and its surroundings. For this course, the main takeaway is straightforward: nonconservative forces don't destroy energy. They just move it out of the mechanical category and into thermal or other forms that are much harder to recover as useful work.