PART III KNOWLEDGE STRUCTURE
Applications of Newtonian Mechanics

Key Findings

What are the overarching findings of Part III?

Newton’s laws of motion together with the conservation laws for energy and momentum form what is called Newtonian mechanics. Part III has introduced relatively little new physics and instead has focused on applying Newtonian mechanics to new situations.
Rotational motion is analogous to linear motion.
With Newton’s law of gravity, we can use Newtonian mechanics to understand the motions of satellites and planets.
Liquids and gases cannot be modeled as particles. Even so, fluids still obey Newtonian laws of statics and motion.

What laws of physics govern these applications?

Newton’s second law for rotation	A net torque causes an extended object to have angular acceleration: $τ_{net} = I α$ .
Conservation of angular momentum	For an isolated system $(τ_{net} = 0)$ , the total angular momentum is conserved: $Δ \vec{L} = \vec{0}$ .
Newton’s law of gravity	The attractive force between two objects separated by $r$ is $F_{m on M} = F_{M on m} = G M m / r^{2}$ .
Bernoulli’s equation	For two points along a streamline of a flowing fluid, $p_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g y_{1} = p_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g y_{2}$ .

What are the most important models of Part III?

A rigid body is an extended object whose size and shape do not change.
- Particle-like atoms held together by bonds that are massless, rigid rods.
A rigid body that experiences a constant torque undergoes constant angular acceleration.
- Obeys Newton’s second law for rotation.
  
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An extended object is in equilibrium if and only if there are no net force and no net torque.

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A gas fills its container.
- Molecules move freely.
- Molecules are far apart.
- A gas is compressible.
A liquid has a well-defined surface.
- Molecules are loosely bound.
- Molecules are close together.
- A liquid is incompressible.
An ideal fluid obeys Bernoulli’s equation.
- Incompressible
- Nonviscous
- Smooth, laminar flow

What are the most important tools introduced in Part III?

Rotational/linear analogs:

Moment of inertia I Mass m

Torque $τ$ Force F

Angular acceleration $α$ Acceleration a

Angular momentum $\vec{L}$ Momentum $\vec{p}$
Rotational kinetic energy: $K = \frac{1}{2} I ω^{2}$
Torque: $τ = r F sin ϕ = r F_{∥} = d F$
- d is the moment arm or lever arm.
- Vector torque is $\vec{τ} = \vec{r} \times \vec{F}$ .
Angular momentum: $\vec{L} = I \vec{ω}$
Rolling motion:
- Rolling without slipping: $v_{cm} = R ω$
- $K = K_{cm} + K_{rot}$

Newton’s theory of gravity predicts Kepler’s three observational laws. We can summarize them as:
- Planets and satellites move in elliptical orbits.
- Angular momentum is conserved.
- For circular orbits, the square of the period is proportional to the cube of the orbit’s radius.
Gravitational potential energy:

$U_{G} = - \frac{G M m}{r}$
- The potential energy is negative with a zero at infinity
- Escape speed is the minimum speed needed to reach infinity.

Pressure $p = F / A$ exists at all points in a fluid.
- Gas pressure is primarily thermal.
- Liquid pressure is primarily gravitational.
- Hydrostatic pressure at depth d:
  
  $p = p_{0} + ρ g d$
Archimedes’ principle says that the upward buoyant force equals the weight of the displaced liquid.
- An object floats if its average density is less than the fluid density.
Equation of continuity: $v_{1} A_{1} = v_{2} A_{2}$
- Relates two points on a streamline.
- v is the flow speed.
- A is the cross-section area.