PART II KNOWLEDGE STRUCTURE
Conservation Laws

Key Findings

What are the overarching findings of Part II?

Work causes a system’s energy to change. It is a transfer of energy to or from the environment.
Energy can be transformed within a system, but the total energy of an isolated system does not change.
An impulse causes a system’s momentum to change.
Momentum can be exchanged within a system, but the total momentum of an isolated system does not change.

What laws of physics govern energy and momentum?

Energy principle	$Δ E_{sys} = Δ K + Δ U + Δ E_{th} = W_{ext} or K_{i} + U_{i} + W_{ext} = K_{f} + U_{f} + Δ E_{th}$
Conservation of energy	For an isolated system $(W_{ext} = 0)$ , the total energy $E_{sys} = K + U + E_{th}$ is conserved. $Δ E_{sys} = 0$ . For an isolated, nondissipative system, the mechanical energy $E_{mech} = K + U$ is conserved.
Momentum principle	$Δ \vec{p} = \vec{J}$
Conservation of momentum	For an isolated system $(\vec{J} = \vec{0})$ , the total momentum is conserved. $Δ \vec{P} = \vec{0}$ .

What are the most common models for using conservation laws?

Energy is a property of the system.
Energy is transformed within the system without loss.
Energy is transferred to and from the system by forces that do work.
- $W > 0$ for energy added.
- $W < 0$ for energy removed.
- $Δ E_{sys} = Δ K + Δ U + Δ E_{th} = W_{ext}$

Perfectly inelastic collision: Objects stick together and move with a common final velocity.
- Momentum is conserved.
- $(m_{1} + m_{2}) v_{f x} = m_{1} (v_{i x})_{1} + m_{2} (v_{i x})_{2}$
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Perfectly elastic collision: Objects bounce apart with no loss of energy.
- Momentum and energy are both conserved.

An isolated system does not interact with its environment.
- For energy, an isolated system has no work done on it.
- For momentum, an isolated system experiences no impulse.
Thermal energy is the microscopic energy of moving atoms and stretched bonds.
An ideal spring obeys Hooke’s law for all displacements: $(F_{Sp})_{s} = - k Δ s$ .
The impulse approximation ignores forces that are small compared to impulsive forces during the brief time of a collision or explosion.

What are the most important tools for using energy and momentum?

Kinetic energy:

$K = \frac{1}{2} m v^{2}$
- Kinetic energy is an energy of motion.
Gravitational potential energy:

$U_{G} = m g y$
Elastic potential energy:

$U_{Sp} = \frac{1}{2} k {(Δ s)}^{2}$
- Potential energy is an energy of position.
Momentum: $\vec{p} = m \vec{v}$ .
Force acting through a displacement does work.
- Constant force:
  
  $W = \vec{F} \cdot Δ \vec{r} = F (Δ r) cos θ$
- Variable force:
  
  $\begin{array}{l} W & = \int_{s_{i}}^{s_{f}} F_{s} d s \\ = area under the F_{s} - versus - s curve \end{array}$
Force acting over time delivers an impulse:

$\begin{array}{l} J_{x} & = \int_{t_{i}}^{t_{f}} F_{x} d t \\ = area under the F_{x} - versus - t curve \end{array}$
Energy diagrams

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- Visualize speed changes and turning points. $F_{x} =$ negative of the slope of the curve.
Bar charts
- Momentum and energy conservation.
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Before-and-after pictorial representation.
- Important problem-solving tool.