§7.4 — Potential Energy

Not all forces behave the same. Some forces (gravity, springs, electric forces) are conservative: they return all the energy you invest if you reverse the process. Others (friction, air drag) dissipate energy irreversibly. This section defines potential energy for conservative forces and explains how energy "hides" in the configuration of a system.

Conservative forces and path independence

A force is conservative if the work it does is path-independent: the work depends only on the starting and ending positions, not on the route taken.

Example: Gravity (conservative force)

Climbing a 100 m cliff:

• Path 1: Vertical climb (steep). Displacement up: 100 m
• Path 2: Spiral trail (gentle). Displacement up: still 100 m (vertically); horizontal distance: longer
• Work against gravity: Both paths require the same work: \( W = mgh = m \times 9.8 \times 100 \)

Gravity only "cares" about vertical displacement. The horizontal component of motion does no work against gravity.

Example: Friction (non-conservative force)

Walking on a rough floor:

• Path 1: Straight line from A to B (distance 10 m). Friction work: \( W = -f \times 10 \)
• Path 2: Zigzag from A to B (distance 30 m). Friction work: \( W = -f \times 30 \)

Friction is non-conservative because the longer path dissipates more energy.

Potential energy definition

For a conservative force, we define a potential energy function \( U(x) \) such that:

\[ W_{\text{conservative}} = -\Delta U = -(U_f - U_i) = U_i - U_f \]

Equivalently, if we rearrange:

\[ U_f = U_i - W_{\text{conservative}} \]

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

Recovering force from potential energy

Given the potential energy \( U(x) \), we can recover the force:

\[ F(x) = -\frac{dU}{dx} \]

This is read: "force is the negative gradient (derivative) of potential energy."

Why the negative sign? A force naturally points toward regions of lower potential energy. An object "wants" to slide downhill in potential energy. The negative sign ensures this.

Common potential energies

Gravitational potential energy (near Earth's surface):

Close to Earth, gravity is approximately constant: \( g \approx 9.8 \) m/s². At height \( h \) above a reference point:

\[ U_{\text{grav}}(h) = mgh \]

where m is mass [kg], g is gravitational acceleration [m/s²], and h is height [m].

Verification: The force is: \[ F = -\frac{dU}{dh} = -\frac{d(mgh)}{dh} = -mg \]

Correct — the gravitational force is \( -mg \) (downward, if we take up as positive).

Elastic potential energy (spring):

From §7.2, when a spring is compressed or stretched by distance \( x \) from equilibrium:

\[ U_{\text{spring}}(x) = \frac{1}{2} kx^2 \]

Verification: The force is: \[ F = -\frac{dU}{dx} = -\frac{d(\frac{1}{2}kx^2)}{dx} = -kx \]

Correct — Hooke's law.

Zero-point of potential energy

Potential energy is defined only up to a constant. We choose the zero-point arbitrarily:

For gravity: We might set \( U = 0 \) at ground level, or at sea level, or at the center of Earth. The difference in potential energy is what matters, not the absolute value.

Example:

• If \( U_{\text{grav}} = 0 \) at ground level: \( U(10 \text{ m}) = mg(10) \)
• If \( U_{\text{grav}} = 0 \) at the rooftop: \( U(10 \text{ m}) = -mg(10) \) (10 m below the rooftop)
• But \( \Delta U \) between any two points is the same in both choices

The work done by gravity is always \( W = -\Delta U \), independent of where we place the zero-point.

Example: Roller coaster

A 1000 kg roller coaster car starts at the top of a hill (height 40 m) with zero velocity. It slides down a frictionless track. What is its speed at height 10 m?

Initial state (top of hill):

• Height: 40 m
• Velocity: 0 m/s
• \( U_i = mgh_i = 1000 \times 9.8 \times 40 = 392{,}000 \) J
• \( KE_i = 0 \) J

Final state (at height 10 m):

• Height: 10 m
• Velocity: \( v_f \) (unknown)
• \( U_f = mgh_f = 1000 \times 9.8 \times 10 = 98{,}000 \) J
• \( KE_f = \frac{1}{2} m v_f^2 \) (unknown)

Using §7.3 (work–energy theorem): The only force doing work is gravity (a conservative force). The work done by gravity is:

\[ W_{\text{grav}} = -(U_f - U_i) = -(98{,}000 - 392{,}000) = 294{,}000 \text{ J} \]

By the work–energy theorem:

\[ W_{\text{grav}} = KE_f - KE_i \]

\[ 294{,}000 = \frac{1}{2} \times 1000 \times v_f^2 - 0 \]

\[ v_f^2 = \frac{2 \times 294{,}000}{1000} = 588 \]

\[ v_f = \sqrt{588} \approx 24.2 \text{ m/s} \]

The car reaches speed 24.2 m/s at height 10 m.

Classification: which forces are conservative?

Conservative forces:

• Gravity (uniform or inverse-square)
• Electric force (point charge or field)
• Elastic force (springs, etc.)
• Nuclear force

All of these can be written as \( \mathbf{F} = -\nabla U \) for some potential energy \( U \).

Non-conservative (dissipative) forces:

• Kinetic friction
• Air drag
• Viscous damping
• Any force depending on velocity

These forces cannot be expressed as the gradient of a potential energy. They dissipate mechanical energy into heat.

Connection to §7.3 (Work–Energy Theorem)

Recall the work–energy theorem:

\[ W_{\text{net}} = \Delta KE \]

If only conservative forces act, then:

\[ W_{\text{net}} = W_{\text{conservative}} = -\Delta U \]

Therefore:

\[ -\Delta U = \Delta KE \]

\[ 0 = \Delta KE + \Delta U \]

This leads to the principle of mechanical energy conservation (§7.5).

Connection to Chapter 5 (Newton's Laws)

In Chapter 5, we wrote Newton's second law as \( \mathbf{F} = m\mathbf{a} \). For conservative forces, we can also write:

\[ \mathbf{F} = -\nabla U = -\frac{dU}{dx} \hat{x} \]

This shows that force is the gradient of the potential energy landscape. The object accelerates in the direction of decreasing potential energy.

What you should be able to do

After this section, you should be able to:

• Define conservative vs. non-conservative forces and give examples of each
• Given a potential energy \( U(x) \), find the force \( F(x) = -dU/dx \)
• Given a force (like gravity or spring), derive the potential energy
• Explain why potential energy has a free choice of zero-point, and how this affects calculations
• Apply potential energy to solve dynamics problems using the work–energy theorem

Preview of §7.5: Mechanical Energy Conservation

If only conservative forces act on a system, the total mechanical energy is conserved:

\[ E_{\text{mech}} = KE + U = \text{constant} \]

This is one of the most powerful results in physics. §7.5 develops this principle and explores what happens when non-conservative forces (friction) are present.

📚 See also: Halliday Vol 1, Ch 7, §7.5 — Potential Energy and the Work Done by Conservative Forces. 📖 Open reference: OpenStax University Physics Vol 1 — Chapter 7.3: Gravitational Potential Energy. 🎓 Video: MIT 8.01 Lecture 10 (Walter Lewin) — "Conservative Forces and Potential Energy".

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