## Backlinks

So let's talk about energy!

## 1 Types of Energy

- Potential Energy \(PE_{grav}=mgh\) (which is work (force times distance) for moving stuff up \(\vec{F} \cdot \vec{h}\))
- Kinetic Energy \(KE_{translational} = \frac{1}{2}mv^2\) + \(KE_{rotational} = \frac{1}{2}I \omega^2\)

Where…

- \(I\): moment of inertia
- \(\omega\): rotational velocity

## 2 Work

\(W = \vec{F} \cdot \vec{d}\), where \(\vec{F}\) force and \(\vec{d}\) change of distance that the force manifest.

=> \(W = |\vec{F}|\cos{\theta} \times |\vec{d}|\)

which, => \(W = |\vec{d}|\cos{\theta} \times |\vec{F}|\)

so, essentially, work is either displacement times parallel as part of force, or visa versa.

Why?

### 2.1 The Dot Product, a review

#### 2.1.1 What is it

The Dot product is a measure of the "pararllelity" of \(\vec{F}\) with \(\vec{D}\).

=> Dot product: the component of the first vector parallel to the second vector multiplied to the magnitude of d.

\(\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos{\theta}\)

#### 2.1.2 Calculating it

Given two vectors

- \(\vec{V_1} =
\) - \(\vec{V_2} =
\)

The dot product is…

\(\vec{V1} \times \vec{V2} = a_x b_x + a_y b_y + a_z b_z\)

## 3 Potential Energy

Potential energy exists because of a force field. There is an object "propping" it up pending release of energy.

### 3.1 Where did \(\Delta PE = W = mg \Delta h\) come from?

So, define \(PE = -W_{AB}\). Which is "potential energy of A to B." Gravity will do a certain amount of work from one point to anther, it will do the opposite the other way.

\(\Delta PE_g = -W_{AB} = -\vec{F} \cdot \vec{d}\)

\(\Delta PE_g = -((-mg) \cdot \Delta h)\) The negative again! $ is