Potential Energy and Energy Conservation (Chapter 7 Study Notes)

Potential Energy and Energy Conservation

Learning Outcomes

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

• Apply elastic potential energy concepts to objects attached to springs.

• Distinguish between conservative and nonconservative forces.

• Interpret energy diagrams to analyze motion under conservative forces.

Potential Energy

Definition and Relationship to Work and Kinetic Energy

• Potential energy (U) is the energy stored in the configuration of a system of objects that exert forces on each other.

• Potential energy is related to work and kinetic energy through the principle of energy conservation.

Gravitational Potential Energy

Definition and Calculation

• Gravitational potential energy: $U_g = mgy$

• Change in gravitational potential energy: $\Delta U_g = U_g(y) - U_g(y_0) = mg(y - y_0) = mgh$

• Work done by gravity when lifting a mass to height $h$: $W_g = -mgh$

• Relationship: $\Delta U_g = -W_g$

Physical Interpretation

• When a particle is in Earth's gravitational field, it has gravitational potential energy.

• As an object descends, gravitational potential energy is converted to kinetic energy.

• When the object moves up, work done by gravity is negative, and potential energy increases.

Example: Power Produced by Niagara Falls

Problem Setup

• 5520 m3 of water falls 49.0 m every second.

• Mass of 1 m3 of water = 1000 kg.

• Work done by falling water equals the change in gravitational potential energy.

• Average power is work per unit time.

Calculation

• Average power: $\bar{P} = \frac{W}{t} = \frac{mgh}{t} = \left(\frac{m}{t}\right)gh$

• Substitute values: $\frac{m}{t} = 5520 \times 1000 = 5.52 \times 10^6$ kg/s

• $\bar{P} = (5.52 \times 10^6 \text{ kg/s})(9.81 \text{ m/s}^2)(49.0 \text{ m}) = 2.65 \times 10^9$ W = 2.65 GW

Discussion

• Result is comparable to large power plants.

• Not all water goes over the falls; much is diverted for power generation.

Energy Storage

• Work can be stored as potential energy by lifting a mass.

• Hydroelectric plants use this principle for energy management.

• Not all forces allow reversible storage of potential energy.

Work and Energy Along a Curved Path

• The expression for gravitational potential energy applies to both straight and curved paths.

• $W_{grav} = mg y_1 - mg y_2$

Elastic Potential Energy

Definition and Formula

• An object is elastic if it returns to its original shape after deformation.

• Elastic potential energy (for a spring): $U_{el} = \frac{1}{2}kx^2$

• x is the extension or compression from equilibrium.

Physical Example

• The Achilles tendon acts like a spring, storing and releasing elastic potential energy during running.

Work Done by a Spring

• Springs do positive work when returning to equilibrium and negative work when stretched or compressed further.

• The graph of $U_{el}$ vs. $x$ is a parabola; $U_{el}$ is never negative.

Situations with Both Gravitational and Elastic Forces

• Total potential energy is the sum: $U = U_{grav} + U_{el}$

Conservative and Nonconservative Forces

Definitions

• Conservative force: Work done depends only on endpoints, not path. Examples: gravity, spring force.

• Nonconservative force: Work done depends on path; energy is dissipated (e.g., friction).

Properties of Conservative Forces

• Work can be expressed as a potential energy function.

• Work is reversible and path-independent.

• Total work over a closed path is zero.

Friction as a Nonconservative Force

• Friction always opposes motion and dissipates energy as heat.

• Work done by friction over a closed path is negative.

Conservation of Mechanical Energy

• Total mechanical energy (kinetic + potential) is conserved if only conservative forces act.

• For an isolated system: $K + U = \text{constant}$

• For conservative forces: $\Delta U = -W$ and $\Delta K = W$ so $\Delta K + \Delta U = 0$

Force and Potential Energy in One Dimension

• For a conservative force in one dimension: $F(x) = -\frac{dU}{dx}$

• Where $U(x)$ changes rapidly, force is large.

• Force pushes the system toward lower potential energy.

Force and Potential Energy in Three Dimensions

• Components of a conservative force: $F_x = -\frac{\partial U}{\partial x}$, $F_y = -\frac{\partial U}{\partial y}$, $F_z = -\frac{\partial U}{\partial z}$

• The vector sum of these components is called the gradient of $U$.

Energy Diagrams and Equilibrium

• An energy diagram plots $U(x)$ and total mechanical energy $E$.

• Points where $\frac{dU}{dx} = 0$ are equilibrium points.

• Stable equilibrium: $U(x)$ is at a minimum; small displacements result in oscillations.

• Unstable equilibrium: $U(x)$ is at a maximum; small displacements lead to acceleration away from equilibrium.

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 (usually negative)
Examples	Gravity, Spring Force	Friction, Air Resistance

Key Equations

• Gravitational Potential Energy: $U_g = mgy$

• Elastic Potential Energy: $U_{el} = \frac{1}{2}kx^2$

• Work by Gravity: $W_g = -mgh$

• Conservation of Mechanical Energy: $K + U = \text{constant}$

• Force from Potential Energy: $F(x) = -\frac{dU}{dx}$