Potential Energy and Energy Conservation (Chapter 7) Study Notes

Potential Energy and Energy Conservation

Goals for Chapter 7

• To use gravitational potential energy in vertical motion

• To use elastic potential energy for a body attached to a spring

• To solve problems involving conservative and nonconservative forces

• To determine the properties of a conservative force from the corresponding potential-energy function

• To use energy diagrams for conservative forces

Introduction to Energy Concepts

Energy can be stored and transformed from one form to another. For example, as a duck descends, its gravitational potential energy is converted into kinetic energy.

Types of Forces

• Conservative forces: Gravity, spring force

• Nonconservative forces: Friction, tension

Conservative Forces

A force is conservative if the work it does on an object moving between two points is independent of the path taken.

• Work done depends only on initial and final positions ($r_i$ and $r_f$)

• If an object moves in a closed path ($r_i = r_f$), total work done by the force is zero

Nonconservative Forces

Work done by a nonconservative force depends on the path taken and dissipates energy (e.g., friction).

Properties of Conservative and Nonconservative Forces

• Conservative forces allow conversion between kinetic and potential energy

• Work done by a conservative force:

  • Can be expressed in terms of a potential energy function

  • Is reversible

  • Is independent of the path

  • Is zero for closed paths

• Nonconservative (dissipative) forces (e.g., friction) do not store energy as potential energy

Mathematical Definition of Conservative Force

• $\vec{F}$ is conservative if the work it does around a closed curve is zero

• Equivalent: Work is independent of the path connecting initial and final points

Path Independence

For conservative forces, the work done along any path between two points is the same:

• $W = \int_C \vec{F} \cdot d\vec{r}$

• For a closed path: $\int_C \vec{F} \cdot d\vec{r} = 0$

Potential Energy and Force

Potential energy is defined such that the work done by a conservative force is related to the change in potential energy:

• $\vec{F} = -\nabla U$

• In one dimension: $F_x = -\frac{dU}{dx}$

• In two dimensions: $F_x = -\frac{\partial U}{\partial x}$, $F_y = -\frac{\partial U}{\partial y}$

Zero of Potential Energy

• The value of potential energy $U$ is arbitrary up to an additive constant

• Only changes in potential energy ($\Delta U$) are physically meaningful

• The "zero" of potential energy can be chosen for convenience

Gravitational Potential Energy

Constant Gravitational Force

• For an object of mass $m$ at height $y$:

• $U_g = mgy$

• Work done by gravity: $W_g = -\Delta U = U_i - U_f$

Example Calculation

• Moving a 4-kg object from ground to 1 m shelf: $\Delta U = mgy = 4 \times 9.8 \times 1 = 39.2$ J

• Moving from 1 m to 2 m shelf: $\Delta U = 4 \times 9.8 \times (2-1) = 39.2$ J

Work and Energy Along a Curved Path

• The change in gravitational potential energy depends only on the vertical displacement $\Delta y$, not the path taken

Elastic (Spring) Potential Energy

Spring Potential Energy

• For a spring with force constant $k$ and displacement $x$ from equilibrium:

• $U_s = \frac{1}{2}kx^2$

Elastic Potential Energy

• A body is elastic if it returns to its original shape after deformation

• Elastic potential energy is stored in elastic bodies (e.g., springs)

• Graph of $U_{el}$ vs. $x$ is a parabola, minimum at $x=0$

Mechanical Energy and Conservation

Mechanical Energy

• Mechanical energy $E$ is the sum of kinetic and potential energies:

• $E = U + K$

• In an isolated system with only conservative forces, $E$ is constant

• $E_f = E_i \rightarrow U_f + K_f = U_i + K_i$

Conservation of Energy

• For any net force: $W = K_f - K_i$

• For conservative forces: $W = U_i - U_f$

• Combination: $K_i + U_i = K_f + U_f$

• Total mechanical energy remains constant throughout motion

The Conservation of Mechanical Energy

• Total mechanical energy is conserved when only conservative forces act

• Example: A falling object converts potential energy to kinetic energy, but $E = K + U$ remains constant

Work Done by Nonconservative Forces

• Nonconservative forces (e.g., friction) change the amount of mechanical energy in a system

• $W_{nc} = \Delta E$

• Summary of work formulae:

  • $W_{total} = \Delta K$

  • $W_c = -\Delta U$

  • $W_{nc} = \Delta E$

General Law of Conservation of Energy

• Energy is never created or destroyed, only transformed

• Nonconservative forces change the internal energy of a system

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

Elastic Potential Energy and Harmonic Oscillator

• Total energy $E$ of a harmonic oscillator:

• Kinetic energy: $K = \frac{1}{2}mv^2$

• Potential energy: $U = \frac{1}{2}kx^2$

• As $x$ decreases, $K$ increases (energy conversion)

Energy Conservation in Oscillators

• Combining work expressions:

• $W = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2$ and $W = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

• Total energy $E$ does not change when block is moving:

• $E_i = E_f$ with $E = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$

Gravitational Potential Energy: Conservation Law

• Work done by gravity when falling from $h_i$ to $h_f$:

• $W = mgh_i - mgh_f$

• From work-energy theorem:

• $mgh_i - mgh_f = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

• Conservation law:

• $mgh_i + \frac{1}{2}mv_i^2 = mgh_f + \frac{1}{2}mv_f^2$

Situations with Both Gravitational and Elastic Forces

• Total potential energy is the sum:

• $U = U_{grav} + U_{el}$

• Example: A person and dog jumping on a trampoline experience both gravitational and elastic potential energy

Simultaneous Presence of Conservative and Non-Conservative Forces

• Work done by resultant force from $x_i$ to $x_f$:

• $W = \int_{x_i}^{x_f} (F_c + F_{NC}) dx$

• Change in kinetic energy: $K_f - K_i = U_i - U_f + \int_{x_i}^{x_f} F_{NC} dx$

• Final energy: $K_f + U_f = K_i + U_i + \int_{x_i}^{x_f} F_{NC} dx$

Example: Block on Incline with Friction

• Block starts at height $H$, moves down incline, then up another incline with friction

• Initial energy: $E_i = m g H$

• Final energy: $E_f = \frac{1}{2} m v_f^2 + m g y_f$

• Frictional work reduces final energy

• Equation for final velocity:

• $v_f = \sqrt{2g[H - y_f - \mu_k y_f \cot \theta]}$

Damped Oscillator

• Harmonic oscillator with friction

• Frictional force always opposes motion, reducing velocity and energy

• Work for motion from $x_i$ to new stopping point $x_f$:

• $W = \int_{0}^{x_f} (-kx - \mu_k m g) dx = -\frac{1}{2} k x_f^2 - \mu_k m g x_f$

• Equation for stopping position:

• $x_f^2 = \frac{m}{k} V^2 - 2 \frac{\mu_k m g}{k} x_f$

• With friction, stopping distance is less than in frictionless case

Energy Diagrams and Equilibrium

Energy Diagrams

• Plot of potential energy $U(x)$ vs. position $x$

• Total energy $E$ is a horizontal line; vertical distance between $E$ and $U(x)$ gives kinetic energy $K$

• Turning points: Where $U(x) = E$ (kinetic energy is zero)

• Equilibrium points: Where slope of $U(x)$ is zero ($F = 0$)

• Stable equilibrium: Minimum of $U(x)$

• Unstable equilibrium: Maximum of $U(x)$

Comparison: Momentum vs. Energy

Momentum	Energy
Conserved if no external force present	Conserved if only conservative forces present
Change due to impulse (force × time)	Change due to work (force × distance)
Vector quantity	Scalar quantity

Important Points:

• Potential energy depends only on position; work done by conservative force on a closed path is zero

• Absolute value of potential energy depends on reference point; only changes are physically meaningful

• For small changes: $\Delta U = -F_x \Delta x$; in the limit $F_x = -\frac{dU}{dx}$

• Energy diagrams help visualize motion and equilibrium

Additional info: These notes expand on the original slides by providing full definitions, equations, and context for each concept, ensuring a self-contained study guide for exam preparation.