PART I KNOWLEDGE STRUCTURE
Newton’s Laws

Key Findings

What are the overarching findings of Part I?

Kinematics is the description of motion. Motion can be described
- Visually
- Graphically
- Mathematically
Forces cause objects to change their motion—that is, to accelerate.
Objects interact by exerting equal but opposite forces on each other.

What laws of physics govern motion?

Newton’s first law

An object will remain at rest or will continue to move with constant velocity if and only if $\vec{F}_{net} = \vec{0}$ .

The object is in mechanical equilibrium.

Newton’s second law

\vec{F}_{net} = m \vec{a}

A net force on an object causes the object to accelerate.

Newton’s third law

\vec{F}_{A on B} = - \vec{F}_{B on A}

For every action, there is an equal but opposite reaction.

What are the most common models for applying the laws of physics to moving objects?

Model the object as a particle.
- Acceleration is in the direction of the net force and is constant.
Mathematically:
- Newton’s second law is
  
  $\vec{F}_{net} = \sum_{i}^{} \vec{F}_{i} = m \vec{a}$
- Use xy-coordinates.
- Constant-acceleration kinematics:
  
  $\begin{array}{l} v_{f s} = v_{i s} + a_{s} Δ t \\ s_{f} = s_{i} + v_{i s} Δ t + \frac{1}{2} a_{s} {(Δ t)}^{2} \\ v_{f s}^{2} = v_{i s}^{2} + 2 a_{s} Δ s \end{array}$
  
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Special cases:
- Uniform motion: $a_{s} = 0$ . The displacement graph is a straight line with slope $v_{s}$ .
- Projectile motion: The only force is gravity. Horizontal motion is uniform; vertical motion has constant $a_{y} = - g$ .

Model the object as a particle.
- The force causes a constant centripetal acceleration. The particle moves around a circle at constant speed and with constant angular velocity.
Mathematically: Newton’s second law is
- $\vec{F}_{net} = (m v^{2} / r$ or $m ω^{2} r$ , toward the center)
- Use rtz-coordinates.
- Uniform-circular-motion kinematics:
  - The tangential velocity is $v_{t} = ω r$ .
  - The centripetal acceleration is $v^{2} / r or ω^{2} r$ .
  - $ω$ and $v_{t}$ are positive for a ccw rotation, negative for a cw rotation.
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General case: Accelerated circular/rotational motion. Angular acceleration is contant.

$\begin{array}{l} ω_{f} = ω_{i} + α Δ t \\ θ_{f} = θ_{i} + ω_{i} Δ t + \frac{1}{2} α {(Δ t)}^{2} \\ ω_{f}^{2} = ω_{i}^{2} + 2 α Δ θ \end{array}$

These equations are analogous to constant-acceleration kinematics.

What are the most important tools for analyzing the physics of motion?

The particle model and motion diagrams

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Vectors

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Free-body diagrams

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Interaction diagrams

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Calculus and graphical analysis

$\begin{array}{l} v_{s} & = & d s / d t = slope of the position graph \\ a_{s} & = & d v_{s} / d t = slope of the velocity graph \\ v_{f s} & = & v_{i s} + \int_{t_{i}}^{t_{f}} a_{s} d t = v_{i s} + area under acceleration curve \\ s_{f} & = & s_{i} + \int_{t_{i}}^{t_{f}} v_{s} d t = s_{i} + area under the velocity curve \end{array}$