Day 12 - Conservation of Energy

Day 12 - Conservation of Energy#

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We observe that energy is conserved $\(\Delta E_{system} = W + Q\)$

Announcements#

  • Homework 4 is posted (due Friday; late after Sunday)

  • First midterm is coming up (assigned Monday, 16 Feb)

    • One exercise will ask you to get started on your final project planning.

    • Who are you gonna work with? What are you interested in studying? Start thinking about this!

  • There will be no class/office hours on Friday

  • Need help this week? Make an appointment with Danny: https://cal.com/dannycaballero/phy-321

Reminder of our Midterm Procedures#

  • The take-home midterms will be open for almost two weeks; you can often start some exercises early as they cover older material.

  • They are meant to be challenging, but we will provide you with the resources and support you need to complete them.

  • There is no homework due during the period in which the midterm is assigned.

  • In contrast to homework assignments, DC will not work any exercises directly

You may work closely together with me, Mihir, and your classmates, but you and your partner must write up your own solutions.

Seminars this week#

WEDNESDAY, February 11, 2026#

Astronomy Seminar, 1:30 p.m., BPS 1400 & Zoom Speaker: Evan Kirby, Notre Dame Title: TBA Zoom Link: https://msu.zoom.us/j/93334479606?pwd=OtIXPWhRPBfzYu53sl3trSJlaBYI7C.1 Passcode: 825824

Seminars this week#

WEDNESDAY, February 11, 2026#

FRIB Nuclear Science Seminar, 3:30 p.m., FRIB 1300 & Zoom Speaker: Kyle Godbey, FRIB Title: What’s Driving the New Era of Discovery in Nuclear Science? Zoom Link: https://msu.zoom.us/j/99975564296?pwd=e3puzoZ4Yu7m6CiCf7SWiaKjvxgCwu.1 Passcode: 569117

Seminars this week#

FRIDAY, February 13, 2026#

IReNA Online Seminar, 2:00 pm, Zoom Light refreshments at 1:50pm in 2025 Nuclear Conference Room - FRIB Hosted by: Artemis Tsantiri (University of Regina, Canada) Speaker: Thanassis Psaltis, Saint Mary’s University - Canada Title: Nuclear physics constraints on the γ-ray signatures of core-collapse supernovae Zoom Link: https://msu.zoom.us/j/827950260 Password: CENAM

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

Clicker Question 12-1#

Which of the following are true about a point particle? (Use 1/A - True, 2/B - False)

12-1a. A point particle has no size. 12-1b. A point particle can have no mass. 12-1c. A point particle can have no charge. 12-1d. A point particle can have no internal energy.

Clicker Question 12-2#

Einstein’s proposed total energy for a particle of mass \(m\) moving at speed \(v\) is given by \(E = \gamma m c^2\), where \(\gamma = 1/\sqrt{1 - v^2/c^2}\). We take the limit as \(v/c \to 0\) to find the total energy of a particle at rest. Which terms below appear in the Taylor expansion of \(\gamma\) in powers of \(v/c\)?

a

b

c

d

\(1\)

\(v/c\)

\((v/c)^2\)

\((v/c)^3\)

1.) a only 2.) a and b 3.) a and c

4.) b and d 5.) all terms

Clicker Question 12-3#

Which of the following are statements of the conservation of energy?

  1. The total energy of a system is constant.

  2. \(\Delta E_{system} = 0\)

  3. \(\Delta E_{system} = W + Q\)

  4. \(\frac{dE_{system}}{dt} = 0\)

  5. All of the above

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 (SHO, \(F_{s} = -kx\))#

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

Clicker Question 12-5#

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?!?! 😭