Day 13 - Conservative Forces

Day 13 - Conservative Forces#

Conservative Forces bg right 100%

\[\vec{F} = - \nabla U\]
\[U = - \int \vec{F} \cdot d\vec{r}\]
\[\nabla \times \vec{F} = 0\]

Announcements#

  • HW 4 is due Friday

    • Friday’s Class: We will work HW 4 Exercise 4 together + Q&A

    • Friday Office Hours (14:00-16:00; 1248 BPS)

  • Midterm 1 is available now

    • We will talk about it Monday (come with questions)

    • There will be no office hours on Oct 3rd (DC traveling)

Seminars this week#

WEDNESDAY, September 24, 2025#

Astronomy Seminar, 1:30 pm, 1400 BPS, In Person and Zoom, Host~ Speaker: Kosuke Namekata, NAOJ Title: Zoom Link: https://msu.zoom.us/j/887295421?pwd=N1NFb0tVU29JL2FFSkk0cStpanR3UT09 Meeting ID: 887-295-421 Passcode: 002454

Seminars this week#

THURSDAY, September 25, 2025 (Promotion Talk!)#

Colloquium, 3:30 pm, 1415 BPS, in person and zoom. Host ~
Refreshments and social half-hour in BPS 1400 starting at 3 pm Speaker: Wolfgang Kerzendorf, MSU - (PTRC) Title: Calibrating Stellar Explosions as Probes of the Evolving Universe Background: For more information and to schedule time with the speaker, see the colloquium calendar at https://pa.msu.edu/news-events-seminars/colloquium-schedule.aspx Zoom Link: https://msu.zoom.us/j/94951062663 Password: 2002 Or complete link: https://msu.zoom.us/j/94951062663?pwd=c48uM25P9UsRVuR74rkOioOWgpoxgC.1

Seminars this week#

FRIDAY, September 26, 2025#

QuIC Seminar, 12:30pm, -1:30pm, 1300 BPS, In Person
Speaker: Alexei Bazavov, MSU Title: Efficient State Preparation for the Schwinger Model Full Scheule is at: https://sites.google.com/msu.edu/quic-seminar/ For more information, reach out to Ryan LaRose

Seminars this week#

FRIDAY, September 26, 2025#

IReNA Online Seminar, 2:00 pm, In Person and Zoom, FRIB 2025 Nuclear Conference Room, Light refreshments will be served at 1:50pm. Hosted by: Steffen Turkat (TU Dresden, Germany) Speaker: Dominik Koll, HZDR, Germany Title: The search for freshly synthesized radionuclides from stellar explosions on Earth Zoom Link: https://msu.zoom.us/j/827950260 Password: JINA

This Week’s Goals#

  • Remind ourselves of the concept of energy and energy conservation

  • Apply the conservation of energy to a variety of systems

  • Develop the mathematical tools to analyze energy conservation in more complex systems

  • Connect our new understanding of energy conservation to our previous work on forces and motion

Reminders#

  • Energy is conserved in every process; our choice of system determines how we account for energy.

  • Closed, isolated systems are often the simplest to analyze.

  • A point particle is a model that allows us to ignore the internal structure of an object.

  • The Work-Energy Theorem is just a statement of the conservation of energy for a point particle.

Conservation of Energy#

General Principle: Energy is conserved in every process.

\[\Delta E_{sys} = W + Q\]

Isolated System: No work or heat is exchanged with the surroundings.

\[\Delta E_{sys} = 0\]

Point Particle: A model that allows us to ignore the internal structure of an object.

\[\Delta K = W_{\text{ext}}\]

The Potential Energy Function#

Simple Harmonic Oscillator (\(F_{s} = -kx\))#

\[\Delta K = W_{s}\]
\[\dfrac{1}{2} m v_f^2 - \dfrac{1}{2} m v_i^2 = \int_{x_i}^{x_f} F_s dx = - \int_{x_i}^{x_f} kx dx\]
\[\dfrac{1}{2} m v_f^2 - \dfrac{1}{2} m v_i^2 = - \dfrac{1}{2} k x_f^2 + \dfrac{1}{2} k x_i^2\]
\[\dfrac{1}{2} m v_f^2 + \dfrac{1}{2} k x_f^2 = \dfrac{1}{2} m v_i^2 + \dfrac{1}{2} k x_i^2\]
\[K_f + U_{s,f} = K_i + U_{s,i}\]
\[U_s = \dfrac{1}{2} k x^2\]

Clicker Question 13-1#

The gravitation force near the Earth’s surface is given by \(\vec{F} = -mg\hat{z}\). What is the potential energy function for this force? Choose \(+\hat{z}\) to be up.

  1. \(U = -mgz\)

  2. \(U = mgz\)

  3. \(U = -mgz + U_0\)

  4. \(U = mgz + U_0\)

  5. None of the above

Clicker Question 13-2#

A model for a lattice chain acting on a electron is given by \(F(x) = - F_0 \sin\left(\dfrac{2\pi x}{b}\right)\). What is the potential energy function for this force?

  1. \(U = -F_0 \cos\left(\dfrac{2\pi x}{b}\right)\)

  2. \(U = F_0 \cos\left(\dfrac{2\pi x}{b}\right)\)

  3. \(U = -\dfrac{F_0 b}{2\pi} \cos\left(\dfrac{2\pi x}{b}\right)\)

  4. \(U = \dfrac{F_0 b}{2\pi} \cos\left(\dfrac{2\pi x}{b}\right)\)

  5. None of the above

Clicker Question 13-3#

I say “Stokes’ Theorem” and you say…

Gravedigger bg left

  1. HELL YEAH BROTHER 🤘

  2. I’m not sure what that is 🤷

  3. DEAR GOD WHY?!?! 😭

Clicker Question 13-4#

The curl of a vector field is given by \(\nabla \times \vec{F}\). If the curl of a vector field is zero, what can we say about the vector field?

  1. It is a conservative force

  2. It is a non-conservative force

  3. It is a constant force

  4. It is a force that does no work

Clicker Question 13-5#

Which of the following fields have no divergence?

A. A B. B

  1. A

  2. B

  3. Both A and B

  4. Neither A nor B

Clicker Question 13-5#

Which of the following fields have no curl?

A. A B. B

  1. A

  2. B

  3. Both A and B

  4. Neither A nor B

Clicker Question 13-6#

Consider a vector field with zero curl: \(\nabla \times \vec{F} = 0\). Which of the following statements is true?

  1. The field is conservative

  2. \(\int \nabla \times \vec{F} \cdot d\vec{A} = 0\)

  3. \(\oint \vec{F} \cdot d\vec{r} \neq 0\)

  4. \(\vec{F}\) is the gradient of some scalar function, e.g., \(\vec{F} = - \nabla U\)

  5. Some combination of the above

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