7.4 Conservative Forces and Potential Energy

Conservative Forces and Potential Energy

Conservative forces and potential energy

A conservative force is one where the work done on an object depends only on where it starts and where it ends, not on the path it takes to get there. Gravity is the classic example: whether you drop a ball straight down or roll it along a ramp, the work done by gravity between the same two heights is identical.

Common conservative forces include gravitational force, spring (elastic) force, and electric force. Non-conservative forces like friction do depend on the path, which is why they don't fit this framework.

The connection between conservative forces and potential energy is captured by this relationship:

$W = -\Delta U$

This says the work done by a conservative force equals the negative change in potential energy. If a conservative force does positive work (like gravity pulling a ball downward), potential energy decreases. If you do work against a conservative force (like lifting that ball back up), potential energy increases.

Potential energy is energy stored in a system due to its position or configuration. The main types you'll encounter in this course:

• Gravitational potential energy (object at a height)
• Elastic potential energy (compressed or stretched spring)
• Electric potential energy (charged particle in an electric field)

Conservative forces can also be described using a force field, which maps out the force acting at every point in space. Gravity near Earth's surface is a uniform force field pointing downward, for instance.

Potential energy in spring systems

Springs follow Hooke's law, which says the force a spring exerts is proportional to how far it's displaced from its natural (equilibrium) position:

$F_s = -kx$

• $k$ is the spring constant, measured in N/m. A larger $k$ means a stiffer spring.
• $x$ is the displacement from equilibrium. The negative sign means the force always pushes back toward equilibrium (a restoring force).

The elastic potential energy stored in a spring is:

$U_s = \frac{1}{2}kx^2$

Notice this depends on $x^2$, so it doesn't matter whether the spring is compressed or stretched by the same amount. Either way, the stored energy is the same.

For example, a spring with $k = 200 \text{ N/m}$ compressed by 0.1 m stores:

$U_s = \frac{1}{2}(200)(0.1)^2 = 1 \text{ J}$

The work done by the spring as it moves from one displacement to another is:

$W_s = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2$

Note the sign carefully here: this is $-\Delta U_s$, consistent with $W = -\Delta U$. The spring does positive work when it returns toward equilibrium (potential energy decreasing) and negative work when you stretch or compress it further.

Work-energy theorem and conservation

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

$W_{net} = \Delta K$

When only conservative forces act, you can substitute $W_{net} = -\Delta U$, which gives:

$-\Delta U = \Delta K$

Rearranging: $\Delta K + \Delta U = 0$

This is the conservation of mechanical energy. It says that in a system where only conservative forces do work, the total mechanical energy (kinetic + potential) stays constant:

$K_i + U_i = K_f + U_f$

This is one of the most powerful problem-solving tools in introductory physics. Instead of tracking forces and accelerations along a complicated path, you just compare energies at two points.

Three classic applications:

1. Pendulums: At the highest point, energy is all gravitational potential. At the lowest point, it's all kinetic. The pendulum swings back and forth converting between the two.
2. Springs: A mass on a spring oscillates between maximum displacement (all elastic potential energy) and the equilibrium point (all kinetic energy).
3. Roller coasters: At the top of a hill, a car has maximum gravitational potential energy. As it descends, that converts to kinetic energy. This analysis works well when friction and air resistance are small enough to neglect.

Potential energy analysis

A potential energy curve is a graph of $U$ vs. position. These graphs pack a lot of information into one picture.

The force at any point is related to the slope of the curve. Specifically, the force points in the direction where potential energy decreases. Where the curve is steep, the force is large; where it's flat, the force is zero.

• A minimum on the curve is a stable equilibrium. If you nudge the object away from this point, the force pushes it back. Think of a ball sitting at the bottom of a bowl.
• A maximum on the curve is an unstable equilibrium. A small nudge causes the object to accelerate away. Think of a ball balanced on top of a hill.
• A flat region means no net force acts on the object there.

These curves are especially useful for visualizing where an object with a given total energy can and cannot go: it can only exist in regions where its total energy is greater than or equal to the potential energy (since kinetic energy can't be negative).