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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