#import "@preview/gentle-clues:0.3.0": important, note

= Chapter 10: Energy and Work

== The Basic Energy Model

Every system has a total energy $E$

=== Energy Transfers and Work

Energy can be transferred between a system and its environment through work and heat

/ Work: Mechanical transfer of energy to or from a system by pushing or pulling on it
/ Heat: Nonmechanical transfer of energy between system and environment due to temperature difference between the two

Work is change in total energy:
$ W = Delta E $

Work is positive when energy transferred into system and negative when energy transferred out of system

/ Isolated system: No energy transferred into or out of system

=== Law of Conservation of Energy

The total energy of an isolated system remains constant

== Work

Work is done on a system by *external forces*

$ W = F d $

Work is a *scalar* even though force and displacement are vectors

Units of work are $J$ (joules), same as energy

A force does no work on an object if the object undergoes no displacement (or the force is perpendicular to the displacement)

=== Kinetic Energy

/ Translational kinetic energy: Energy of motion in a line

$ K = 1/2 m v^2 $

/ Rotational kinetic energy: Way of expressing sum of kinetic energy of all parts of a rotating object.
  Moment of inertia takes place of mass and angular velocity takes place of linear velocity.

$ K_"rot" = 1/2 I omega^2 $

== Potential Energy

/ Conservative forces: Forces that can store useful energy, e.g. gravity, elastic forces
/ Nonconservative forces: Forces that can't store useful energy, e.g. friction

=== Gravitational Potential Energy

$ U_upright(g) = m g h $

Only the height ($h$) matters, not the path the object took to get there

=== Elastic Potential Energy

$ U_upright(s) = 1/2 k x^2 $

== Conservation of Energy

In an isolated system, $W = 0$

Carefully choose an isolated system to solve problems in

== Energy Diagrams

*Energy diagrams* graph potential energy as a function of position

- Free fall energy diagrams are linear graphs of gravitational potential energy
- A spring's energy diagram is a parabola showing spring potential energy

#align(center)[
  #image("/Images/10-Energy-Diagrams.png", width: 45%)
]

- The distance from the axis to the PE line is the potential energy
- The distance from the PE line to the energy line E is the kinetic energy
- The object cannot be at a position where the PE curve is above the E line (because E is total energy)
- A position where the E line crosses the PE curve is a turning point where the object reverses direction
- If the E line crosses the PE curve at two positions, the object will oscillate between those two positions
- Speed will be at a maximum where the PE curve is a minimum (because that's where KE is at a maximum)

== Power

/ Power: Rate at which energy is transformed or transferred (scalar), measured in watts (W)

$ P = (Delta E) / (Delta t) = W / (Delta t) $

/ Output power: A force doing work transfers energy, and the rate at which the force transfers energy is called output power

$ P = F v $

where $P$ is $F$'s output power, and $v$ is the velocity of the object that $F$ is acting on
