6.3 Conservative and Non-conservative Forces

Forces shape our physical world, determining how objects move and interact. This section explores conservative and non-conservative forces, highlighting their key differences and impacts on energy in systems.

Potential energy is a crucial concept linked to conservative forces. Understanding these principles is vital for grasping energy conservation and work in various scenarios, from simple gravity to complex mechanical systems.

Types of Forces and Energy

Conservative vs. non-conservative forces

The distinction between these two types of forces comes down to one question: does the work depend on the path the object takes?

• Conservative forces do work that is independent of the path taken. Only the starting and ending positions matter. This property is what allows us to define potential energy for these forces, and the work they do is reversible. Examples: gravitational force, elastic (spring) force, electrostatic force. A useful test: if an object travels along a closed loop (returns to where it started), a conservative force does zero net work.
• Non-conservative forces do work that depends on the path taken. A longer or more winding path means more (or less) work. Because of this path dependence, you can't define a potential energy for them, and the work they do is irreversible. Examples: friction, air resistance, tension in a moving rope. Friction, for instance, always opposes motion, so a longer path means more energy lost to heat.

Potential energy and conservative forces

Potential energy ($U$ or $PE$) is energy stored in a system due to position or configuration. It can only be defined for conservative forces.

The work done by a conservative force equals the negative change in potential energy:

$W = -\Delta U$

The negative sign means that when a conservative force does positive work (like gravity pulling an object downward), the potential energy decreases. You can also go the other direction and find the force from the potential energy function:

$F = -\frac{dU}{dx}$

Two common types you'll use repeatedly:

• Gravitational potential energy: $U_g = mgh$, where $h$ is the height above a chosen reference point. The reference point is arbitrary, but you must keep it consistent within a problem.
• Elastic potential energy: $U_e = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from the spring's natural (unstretched) length.

Energy Conservation and Work

Conservation of mechanical energy

When only conservative forces act on a system, total mechanical energy stays constant:

$KE_i + PE_i = KE_f + PE_f$

This is one of the most powerful tools in introductory physics because it lets you relate speeds and heights (or spring compressions) without ever needing to know the details of the motion in between.

Problem-solving steps:

1. Identify your initial and final states clearly (draw a sketch if possible).
2. Choose a reference point for potential energy (e.g., ground level for gravity).
3. Write out $KE$ and $PE$ at both the initial and final states.
4. Set $KE_i + PE_i = KE_f + PE_f$ and solve for the unknown.

For example, if a 2 kg ball is dropped from 5 m, you can find its speed just before hitting the ground: $mgh = \frac{1}{2}mv^2$, giving $v = \sqrt{2gh} = \sqrt{2(9.8)(5)} \approx 9.9 \text{ m/s}$.

Non-conservative forces and energy

When non-conservative forces are present, mechanical energy is not conserved. These forces typically convert mechanical energy into other forms like heat or sound.

The generalized work-energy theorem accounts for this:

$\Delta KE = W_{total} = W_{conservative} + W_{non-conservative}$

You can also write this as:

$KE_f + PE_f = KE_i + PE_i + W_{nc}$

Since non-conservative forces like friction generally remove mechanical energy from the system, $W_{nc}$ is usually negative. For instance, friction converts kinetic energy into thermal energy, so the system ends up with less mechanical energy than it started with.

Work of forces in scenarios

Conservative forces have path-independent work, so you calculate it using the change in potential energy:

$W_c = -\Delta U$

• Gravity: $W_g = -mg(h_f - h_i) = mg(h_i - h_f)$. Gravity does positive work when an object moves downward.
• Spring: $W_s = -\frac{1}{2}k(x_f^2 - x_i^2)$. The spring does positive work when it returns toward its natural length.

Non-conservative forces have path-dependent work, so you calculate it directly from force and displacement:

$W_{nc} = \int \vec{F} \cdot d\vec{s}$

• For a constant kinetic friction force over a distance $d$: $W_f = -f_k d$. The negative sign reflects that friction always opposes the direction of motion.
• Air resistance is trickier because the force depends on velocity, so you generally need to integrate or use energy methods to find the work done.