# Lecture 8: Potential Energy and Conservation of Energy

## Physics I — Mechanics & Thermodynamics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Define and calculate gravitational potential energy and elastic potential energy
2. Distinguish between conservative and non-conservative forces
3. State and apply the law of conservation of mechanical energy
4. Use energy conservation to solve problems more efficiently than Newton's laws alone
5. Incorporate non-conservative forces (friction) into energy accounting via the work-energy theorem
6. Interpret potential energy diagrams and identify equilibrium points

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## Lecture Content

### I. Conservative and Non-Conservative Forces

A **conservative force** is one for which the work done depends only on the initial and final positions, not on the path taken. Equivalently, the work done by a conservative force around any closed path is zero. Examples include gravity, the spring force, and the electrostatic force. A **non-conservative force** is one for which the work depends on the path. Kinetic friction, air resistance, and forces applied by muscles are all non-conservative. These forces convert mechanical energy into other forms such as thermal energy and sound. Only conservative forces have an associated **potential energy**.

### II. Gravitational Potential Energy

For an object of mass m at height y near Earth's surface, the gravitational potential energy is U_g = mgy. The choice of reference point (y = 0) is arbitrary because only **changes** in potential energy are physically meaningful: Delta U_g = mg Delta y = mg(y_f - y_i). The relationship between work done by gravity and the change in gravitational potential energy is W_gravity = -Delta U_g = -(U_f - U_i) = mgy_i - mgy_f. When an object rises, its gravitational potential energy increases and gravity does negative work. When it falls, its potential energy decreases and gravity does positive work.

### III. Elastic Potential Energy

For a spring obeying Hooke's law (F = -kx), the elastic potential energy is U_s = (1/2) k x^2, where x is the displacement from the spring's natural (equilibrium) length. The reference is automatically x = 0, where U_s = 0. Elastic potential energy is always non-negative because the spring stores energy whether it is compressed or stretched. The work done by the spring is W_spring = -Delta U_s = (1/2)k x_i^2 - (1/2)k x_f^2.

### IV. Conservation of Mechanical Energy

**Mechanical energy** is defined as E = K + U, the sum of kinetic and potential energy. When **only conservative forces** do work, the total mechanical energy is conserved: E_i = E_f, or K_i + U_i = K_f + U_f. For gravity, this becomes (1/2)mv_i^2 + mgy_i = (1/2)mv_f^2 + mgy_f. Energy is **transformed** between kinetic and potential forms, but the total mechanical energy remains constant.

This principle is an enormously powerful problem-solving tool. It requires no knowledge of the path or the time, works for curved paths and variable forces (as long as they are conservative), and produces a scalar equation with no vector components to manage.

<image>A roller coaster track showing a car at three positions: top of the first hill (high U, low K), bottom of a valley (low U, high K), and partway up a second hill (intermediate U and K). Bar charts beside each position show the relative amounts of K and U, with the total E = K + U remaining constant (same total bar height). Dashed lines indicate the height reference level at the bottom.</image>

### V. Including Non-Conservative Forces

When non-conservative forces such as friction are present, the energy equation must be modified: K_i + U_i + W_nc = K_f + U_f, where W_nc = Delta K + Delta U = Delta E_mech. For friction, W_nc = W_friction = -f_k d, which is always negative. This means friction reduces the total mechanical energy, with the lost energy going into thermal energy: Delta E_thermal = f_k d.

The generalized energy conservation equation is E_i + W_nc = E_f, which expands to (1/2)mv_i^2 + mgy_i - f_k d = (1/2)mv_f^2 + mgy_f. If an external agent does work W_ext (for example, a push), the equation becomes E_i + W_ext - f_k d = E_f.

### VI. Potential Energy Diagrams

A plot of U(x) versus x provides a powerful visual tool for understanding motion. On such a diagram, the **total mechanical energy** E appears as a horizontal line, and the **kinetic energy** at any point is K = E - U(x). Since K must be non-negative, the object can only exist in regions where E >= U(x).

**Turning points** are locations where E = U(x), so K = 0 and the object reverses direction. **Equilibrium positions** are points where dU/dx = 0. A **stable equilibrium** occurs at a local minimum of U (d^2U/dx^2 > 0), where a slight displacement produces a restoring force that pushes the object back. An **unstable equilibrium** occurs at a local maximum (d^2U/dx^2 < 0), where any displacement causes the object to move farther away. A **neutral equilibrium** exists where U is flat (d^2U/dx^2 = 0). The force at any point can be recovered from the potential energy curve as F(x) = -dU/dx, which always pushes the object toward lower potential energy.

<image>A potential energy curve U(x) with several features: a local minimum (labeled "stable equilibrium"), a local maximum (labeled "unstable equilibrium"), and smoothly rising walls on either side. A horizontal line at energy level E intersects the curve at two turning points. The region between the turning points is shaded and labeled "classically allowed region." Arrows show the direction of force at various points along the curve (always pointing toward decreasing U). The kinetic energy K = E - U is indicated as the vertical gap between the E line and the U curve.</image>

### VII. Problem-Solving Strategy with Energy Conservation

The energy conservation approach follows a clear sequence. First, define the system (object + Earth for gravity, or object + spring for elastic potential energy). Then identify the initial and final states, and classify all forces as conservative or non-conservative. If only conservative forces act, set K_i + U_i = K_f + U_f. If non-conservative forces are present, include them: K_i + U_i + W_nc = K_f + U_f. Choose a convenient reference level for potential energy, substitute known values, and solve for the unknown.

Common applications include the pendulum, where h = L - L cos(theta) and energy conservation gives the speed at the bottom; the spring-launched projectile, where (1/2)kx^2 = (1/2)mv^2 + mgh; and sliding down a curved ramp, where energy methods avoid the need to know the exact shape of the path.

<image>A spring-mass system on a horizontal surface. Panel A (initial): The spring is compressed by distance x from equilibrium, mass is at rest. Energy bar chart shows all energy as elastic PE. Panel B (final): The spring is at natural length, mass moves with speed v. Energy bar chart shows all energy as kinetic energy. If friction is present, Panel C shows the mass with a slightly smaller speed, and the bar chart includes a thermal energy component, with the total energy bar still equal to the initial elastic PE.</image>