Day 15 - Potential Energy and Stability

Day 15 - Potential Energy and Stability#

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Mexican Hat/Sombrero Potential \(\longrightarrow\)

Mexican Hat Potential#

\(V(\phi) = -5|\phi|^2 + |\phi|^4\)

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Announcements#

  • HW 4 is due today

  • Midterm 1 is available today (Due Feb 24th)

Seminars this week#

MONDAY, February 17, 2025#

  • QuIC Seminar, 12:30 pm, 1400 BPS, Dr. Michael Hilke, The history of quantum computing

  • High Energy Physics Seminar, 1:30pm, BPS 1400 BPS, Joshua Isaacson, Event Generation for Next-Gen HEP Experiments

  • Condensed Matter Seminar 4:10 pm, 1400 BPS, Lisa Lapidus, The Physics of Biomolecular Condensation

Seminars this week#

TUESDAY, February 18, 2025#

  • Theory Seminar, 11:00am., FRIB 1200 lab, Ibrahim Abdurahman, Investigating Fission Dynamics within Time-Dependent Density Functional Theory Extended to Superfluid Systems

Seminars this week#

WEDNESDAY, February 19, 2025#

  • Astronomy Seminar, 1:30 pm, 1400 BPS, Aaron Bello-Arufa, The atmospheres of small exoplanets with JWST

  • PER Seminar, 3:00 pm., BPS 1400, Anthony Escuardo, OPTYCS: A Community of Practice Supporting Teaching and Scholarship at Two-Year College

  • FRIB Nuclear Science Seminar, 3:30pm., FRIB 1300 Auditorium, Elise Novitski, A new approach to measuring neutrino mass

Seminars this week#

THURSDAY, February 20, 2025#

  • High Energy Physics Seminar, 1:30pm, BPS 1400 BPS, Ben Assi, Precision QCD and EFT for Next-Generation Collider Studies

  • Physics and Astronomy sColloquium, 3:30 pm, 1415 BPS, Eric Hudson, Laser spectroscopy of a nucleus

This Week’s Goals#

  • Understand the concept of potential energy

  • Determine the equilibrium points of a system using potential energy

  • Analyze the stability of equilibrium points

  • Define and begin to apply conservation of linear and angular momentum

Reminders: Conservative Forces#

  • Conservative forces are those with zero curl

\[\nabla \times \vec{F} = 0\]

  • The work done by a conservative force is path-independent; on a closed path, the work done is zero

\[\oint \vec{F} \cdot d\vec{r} = 0\]

  • The force can be written as the gradient of a scalar potential energy function

\[\vec{F} = - \nabla U\]

Clicker Question 15-1#

Here’s the graph of the potential energy function \(U(x)\) for a pendulum.

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What can you say about the equilibrium points? There is/are:

  1. One stable point

  2. Two stable points

  3. One stable and one unstable point

  4. Two unstable and one stable point

Clicker Question 15-2#

Here’s a potential energy function \(U(x)\) for a pendulum:

\[U(\phi) = -mgL\cos(\phi) + U_0\]

  1. Find the equilibrium points (\(\phi^*\)) of the pendulum by setting:

\[\frac{dU(\phi^*)}{d\phi} = 0.\]

  1. Characterize the stability of the equilibrium points (\(\phi^*\)) by examining the second derivative:

\[\frac{d^2U(\phi^*)}{d\phi^2}>0? \qquad \frac{d^2U(\phi^*)}{d\phi^2}<0?\]

Click when done.

Clicker Question 15-3#

A double-well potential energy function \(U(x)\) is given by

\[U(x) = -\frac{1}{2}kx^2 + \frac{1}{4}kx^4.\]

We assume we have scaled the potential energy so that all the units are consistent.

How many equilibrium points does this system have?

  1. 1

  2. 2

  3. 3

  4. 4

Clicker Question 15-4#

A double-well potential energy function \(U(x)\) is given by

\[U(x) = -\frac{1}{2}kx^2 + \frac{1}{4}kx^4.\]

  1. Find the equilibrium points (\(x^*\)) of the pendulum by setting:

\[\frac{dU(x^*)}{dx} = 0.\]

  1. Characterize the stability of the equilibrium points (\(x^*\)) by examining the second derivative:

\[\frac{d^2U(x^*)}{dx^2}>0? \qquad \frac{d^2U(x^*)}{dx^2}<0?\]

Click when done.

Clicker Question 15-5#

Here’s a graph of the potential energy function \(U(x)\) for a double-well potential.

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Describe the motion of a particle with the total energy, \(E=\)

  1. \(0.4\,\mathrm{J}\), \(<\) barrier height

  2. \(1.2\,\mathrm{J}\), \(>\) barrier height

  3. \(1.0\,\mathrm{J}\), \(=\) barrier height

Click when done.