Potential Energy and Energy Conservation (Chapter 7 Study Notes)

Potential Energy and Energy Conservation

Learning Goals

• Understand how to use gravitational potential energy in problems involving vertical motion.

• Apply elastic potential energy concepts to systems with moving bodies attached to stretched or compressed springs.

• Distinguish between conservative and nonconservative forces. (Conservative forces always have a corresponding potential-energy function.)

• Use energy diagrams to analyze the motion of objects under conservative forces.

Introduction to Energy Concepts

Energy is a fundamental concept in physics, describing the ability of a system to perform work or produce change. In many physical situations, energy can be stored in different forms and transformed from one type to another. For example, as a sandhill crane descends, its gravitational potential energy is converted into kinetic energy.

Gravitational Potential Energy

Definition and Formula

• Gravitational potential energy (Ug) is the energy associated with an object's position in a gravitational field.

• For an object of mass m at height y above a reference level: $U_g = mgy$

• The change in gravitational potential energy when moving from height y_1 to y_2 is: $\Delta U_g = mg(y_2 - y_1)$

Key Points:

• When an object moves downward, gravity does positive work and Ug decreases.

• When an object moves upward, gravity does negative work and Ug increases.

• The reference level for potential energy is arbitrary; only changes in Ug are physically meaningful.

Example: A basketball descending from a height converts gravitational potential energy into kinetic energy, increasing its speed as it falls.

Conservation of Mechanical Energy

Principle and Application

The total mechanical energy of a system is the sum of its kinetic and potential energies. If only conservative forces (like gravity) do work, the total mechanical energy is conserved:

• $E = K + U$ (where K is kinetic energy, U is potential energy)

• For gravitational systems: $K_1 + U_{g,1} = K_2 + U_{g,2}$

Key Points:

• Mechanical energy is conserved when only conservative forces act.

• When moving upward, kinetic energy decreases and potential energy increases; when moving downward, the reverse occurs.

Example: Throwing a baseball straight up: as the ball rises, its kinetic energy is converted into gravitational potential energy until it momentarily stops at its maximum height.

Elastic Potential Energy

Springs and Hooke's Law

• Elastic potential energy is stored in objects that can be stretched or compressed, such as springs.

• For a spring obeying Hooke's Law: $F = -kx$ where F is the restoring force, k is the spring constant, and x is the displacement from equilibrium.

• The elastic potential energy stored in a spring is: $U_s = \frac{1}{2}kx^2$

• x > 0 for stretching, x < 0 for compression. $U_s$ is always non-negative.

Example: The Achilles tendon acts like a natural spring, storing and releasing elastic potential energy as you run.

Conservative and Nonconservative Forces

Definitions and Properties

• Conservative forces (e.g., gravity, spring force):

  • Work done depends only on initial and final positions, not the path taken.

  • Work done over a closed path is zero.

  • Associated with a potential energy function.

• Nonconservative forces (e.g., friction):

  • Work done depends on the path taken.

  • Convert mechanical energy into other forms (e.g., internal energy, heat).

Example: As a car tire rolls, internal friction (a nonconservative force) increases the tire's temperature and pressure.

Work and Energy Along a Curved Path

The expression for gravitational potential energy applies regardless of whether the path is straight or curved. The work done by gravity depends only on the vertical displacement, not the path taken.

Force and Potential Energy Relationships

One Dimension

• The force associated with a potential energy function U(x) is: $F(x) = -\frac{dU}{dx}$

• At points where $\frac{dU}{dx} = 0$, the force is zero (equilibrium points).

• If U(x) is at a minimum, the equilibrium is stable; if at a maximum, it is unstable.

Three Dimensions

• The components of a conservative force in three dimensions are given by the negative partial derivatives of the potential energy function: $F_x = -\frac{\partial U}{\partial x}$ $F_y = -\frac{\partial U}{\partial y}$ $F_z = -\frac{\partial U}{\partial z}$

• The vector form (gradient): $\vec{F} = -\nabla U$

Energy Diagrams

An energy diagram plots the potential energy function U(x) and the total mechanical energy E. These diagrams help visualize the motion of objects under conservative forces, showing regions of allowed and forbidden motion, and points of equilibrium.

• Where the slope of U(x) is zero, the force is zero (equilibrium).

• Stable equilibrium: U(x) is at a minimum.

• Unstable equilibrium: U(x) is at a maximum.

Example: A glider attached to a spring oscillates around the minimum of its potential energy curve.

Summary Table: Conservative vs. Nonconservative Forces

Property	Conservative Force	Nonconservative Force
Path Dependence	No (depends only on endpoints)	Yes (depends on path)
Potential Energy Function	Exists	Does not exist
Work over Closed Path	Zero	Nonzero
Examples	Gravity, Spring Force	Friction, Air Resistance

Law of Conservation of Energy

The total energy of an isolated system is conserved; it can change forms but cannot be created or destroyed. When nonconservative forces are present, the change in kinetic and potential energy equals the change in internal energy:

• $\Delta K + \Delta U + \Delta U_{int} = 0$

Example: Pushing a block up a ramp with friction increases the internal energy of the block and ramp due to heat.

Additional info: Some examples and context were inferred and expanded for clarity and completeness.