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Week 5: Energy

[1] Relevant Equations/Concepts⚓︎

Tip: Memorizing Formulae

The equation sheet they give you has ~50-100 formulae...? In reality, you only need to memorize 1-5 formulae per unit, tops.

I'll write necessary equations/concepts in each of these weekly posts. Previous weeks will be filled in the near future.

For "Work and Energy", the formula sheet has 9 entries. I don't think that's very helpful.

Instead, I recommend you memorize these four formulae, and pay close attention to how "work" works.

[1.1] Mechanical Energy⚓︎

The first three equations are fairly straightforward "plug-and-chug":

\[\begin{equation*} \begin{aligned} E_{\text{Kinetic}} & =\tfrac{1}{2} mv^{2}\\ E_{\text{Potential: Gravity}} & =mg\Delta h\\ E_{\text{Mechanical}} & =E_{K} +E_{P:G} \end{aligned} \end{equation*}\]

You can choose whatever variables make sense to you(1)— I chose the variables above to emphasize they're all types of energy (with the same unit of Joules).

  1. These are all equivalent:
    \(E_{\text{Kinetic}} \leftrightarrow E_{\text{K}} \leftrightarrow KE \leftrightarrow K\)
    \(E_{\text{Potential: Gravity}} \leftrightarrow E_{\text{P:G}} \leftrightarrow PE_g \leftrightarrow U_g\)

These are usually sufficient for problems where you're not considering complicated forces like friction, or a hand/rope pushing/pulling a box across a surface.

 

[1.2] Work⚓︎

(Preface: This one is a bit more tricky, not just because of the symbols, but because of what it represents conceptually. Personally, if you find yourself needing to use it, I would start with a strong conceptual understanding of what should happen before you move onto the math, as opposed to mindlessly plugging in numbers...)

 

If a force pushes/pulls an object through a distance, then it does work on the object, which is just another word for saying that it transfers energy to/from the object:

\[\begin{equation} (W\equiv \Delta E) = F d \cos( \theta _{\angle _{d}^{F}}) \end{equation}\]

This quantity is positive or negative, depending on whether the directions of the force and motion are same or opposite (that's what the cosine term tries to capture).

 

But that's not all!!! You know that some amount of energy is being transferred to/from the object, but you ALSO need to know whether the total mechanical energy of the object will change...

  1. If the force is non-conservative, then that's basically the end of the story — the object's total mechanical energy (kinetic + potential) really does increase or decrease by that amount.

    For example, consider a box sliding across a rough surface until it comes to a halt. It starts with some kinetic energy, but friction does negative mechanical work (i.e. it slows down), so the total mechanical energy decreases.

    • In this course, 99% of the time, the only non-conservative force will be friction.

  2. However, if the force is conservative, then the work shows up as a trade between kinetic and potential energy, so the total mechanical energy stays constant.

    For example, consider a ball rolling down a hill. It starts with zero kinetic energy, then gravity does positive kinetic work as it rolls down (i.e. it speeds up, so \(K\) increases) — but gravitational potential energy decreases by that same amount — so total mechanical energy stays the same.

 

[1.3] Conservation of Energy⚓︎

I like the analogy of "pouring liquid" between two "cups" of kinetic and potential energy, where the total amount of liquid is analogous to the total mechanical energy (which may or may not stay constant, depending on whether forces are conservative like gravity or nonconservative like friction). The following video (5 mins) is a nice, clear, and simple animation of that concept as applied to various examples:

[Professor Dave Explains] Conservation of Energy: Free Fall, Springs, and Pendulums

 

 

[2] Helpful Resources⚓︎

 

 

[3] Recitation Prep⚓︎

In addition to section 1, I recommend reviewing the following concepts in the videos above. They will probably come up in recitation problems:

  • Graphs where force is on one axis, and displacement is on the other.
  • Pushing box across rough surface at an angle.
  • Box sliding down rough incline.