# Six Ideas That Shaped Physics Notes

## Unit C: Conservation Laws Constrain Interactions

### Chapter 1: Introduction to Interactions

When an isolated set of objects interacts with one another, we find
that (no matter how the objects interact) certain quantities (which
we call the set's total **energy**, **momentum**, and **angular
momentum**) are *conserved* (i.e., have values that do not change
with time). The laws of physics are independent of *time*, *place*,
and *orientation*.

**Mechanics** is the area of physics that deals with how the physical
interactions between object affect their motion.

**Newton's first law**: Any object that does not interact with
something else moves with a *constant speed* in a *fixed
direction*. An **interaction** is a physical relationship *between two
objects* that, in the absence of other interactions, *changes* the
motion of each. Newton's model sees friction as being an interaction
between a moving object and something else that actively *slows down
the object*.

All matter consists of elementary particles known as **quarks** and
**leptons**. Electrons are an example of the more general class
leptons, while protons and neutrons are constructed of quarks.

There are four fundamental interactions:

**string nuclear****electromagnetic****weak nuclear****gravitational**

Between **macroscopic** objects, we see **long-range** interactions:

**electrostatic****magnetic****gravitational**

and **contact interactions**:

**friction****compression****tension**

Note that at the fundamental level, electrostatic, magnetic, and all contact interactions are different manifestations of the electromagnetic interaction.

#### SI Units

Type | Unit | Abbreviation |
---|---|---|

distance | meter | m |

time | second | s |

mass | kilogram | kg |

temperature | kelvin | K |

molecules | mole | mol |

current | ampere | A |

luminosity | candela | cd |

#### Standard Prefixes

Power | Prefix | Symbol |
---|---|---|

10^12 | tera | T |

10^9 | giga | G |

10^6 | mega | M |

10^3 | kilo | k |

10^-2 | centi | c |

10^-3 | milli | m |

10^-6 | micro | ยต |

10^-9 | nano | n |

10^-12 | pico | p |

### Chapter 2: Vectors

A **scalar** quantity is any quantity (such as speed) that we can
completely describe using a single numerical value. A **vector** is a
quantity (such as velocity) that has **direction** in addition to a
numerical **magnitude**.

**Displacement** is a shift in position from one point in space to
another. See pages 27 and 28 for examples of **vector addition**
(\(\vec{u_1}+\vec{u_2}\)), **vector inverse** (\(\vec{u_1}+\vec{u_2}=0\)),
and **vector difference** (\(\vec{u_1}-\vec{u_2}\)). Note that
\(\vec{u_1}-\vec{u_2}=\vec{u_1}+(-\vec{u_2})\).

A vector is a sum of three component vectors:

\begin{equation} \vec{u}=u_x\hat{x}+u_y\hat{y}+u_z\hat{z} \end{equation}
The \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\) are **directionals**. We call
\(u_x\), \(u_y\), \(u_z\) the **x component**, **y component**, and **z
component**. A **component vector** is the combination of a component
and a directional.

The **magnitude** of an arbitrary vector is as follows:

Pages 32 and 33 relate basic trigonometry to vectors. Note: the three angles of a triangle add to 180 degrees.

As expected, vector addition and difference can be calculated using components in vector notation.

A **reference frame** is an "imaginary cubical grid". We need to
choose the **origin**, and the **+x**, **+y**, and **+z directions**. Note
that we can attach a reference frame to any object that we
please. We can also orient the frame in any way that we like.

We can calculate the displacement between two points with vector difference.

We can use our right hand to determine axis directions in a
**right-handed reference frame**. See figure C2.11a on page 38 for an
example. In **standard orientation**, +x=east, +y=north, +z=up.

### Chapter 3: Interactions Transfer Momentum

A **particle** is a hypothetical object having zero volume, and thus
its position is a mathematical point in space. The position
\(\vec{r}\) of a point in space is its displacement from the reference
frame origin to the point in question. The displacement between two
arbitrary points is the difference of their positions:
\(\Delta\vec{r}\equiv\vec{r_2}-\vec{r_1}\).

A particles **velocity** \(\vec{v}\) at a given instant of time t is a
vector whose magnitude is the particle's speed \(v\) and whose
direction is the particle's direction of motion at that instance of
time. Let \(dt\) be the duration of a time interval containing the
instant \(t\) at which we want to know \(\vec{v}\), and let \(d\vec{r}\)
be the particle's tiny displacement during that tiny time
interval. If \(dt\) is *sufficiently short* that neither the particle's
speed nor its direction of motion changes appreciable during that
time interval, then the particle's velocity is:

**Speed** is the magnitude of a particle's velocity:

Remember from chapter 1 that an **interaction** between two objects is
a physical relationship between them that allows each to affect the
other's motion. **Momentum**, or the "quantity of motion", is defined
as:

The **momentum-transfer principle**: any interaction between two
objects affects their motion by transferring momentum from one to
the other.

Physicists call the amount of momentum that a *particular*
interaction A between two particles transfers to either particle
during a short interval of time the **impulse** \([d\vec{p}]_A\) that
the interaction delievers to that particle during that interval. A
financial analogy may help us understand some subtle distinctions in
meaning between the terms *a particle's momentum*, *impulse*, and
*momentum* in general. Think of the *particle's momentum* as being
like a person's net financial worth. An *interaction* is like a
financial transaction that increases on person's net worth at the
expense of the other's. An *impulse* is like a check that a person
writes or receives in the transaction. The term *momentum* is, like
*money*, a general term for both what each particle (person) has and
what is being transferred.

The change \(d\vec{p}\) in the momentum of a particle that
participates in multiple interactions during a given interval is the
*vector sum* of the impulses it receives during that interval

We define the **force** \(\vec{F}\) that an interaction exerts on a
given object to be the *rate* at which momentum flows into the
object because of that interaction:

The SI unit of force is the **newton**, where \(1 N = 1 kg*m/s^2\).

We can visualize an interaction between two particles as being like
a hose that carries a flow of momentum from one to the other. Since
any momentum that flows *out* of one must flow *into* the other, the
*magnitude* of the rate of momentum flow must be the same for both
particles. This statement and the definition of force directly imply
that: A given interaction between particles A and B must exert a
force on B that is equal in magnitude and opposite in direction the
the force that it exerts on A: \(\vec{F}_{on B} = -\vec{F}_{on
A}\). Physicists call this statement **Newton's third law**.

A particle's **mass** simply expresses the relationship between its
momentum and its velocity, each of which we can define
separately. *Mass* and *weight* are *completely distinct
concepts*. An object's mass expresses its resistance to changes in
its velocity. An object's **weight** \(\vec{F}_g\), however expresses
the force that a gravitational interaction exerts on teh object as a
certain point in space. *Mass* is a *scalar* measured in kilograms;
*weight* is a *vector* measured in newtons.

Where \(\vec{g}\) is the **gravitational field vector** at a certain
point in space. Near the earth's surface, \(\vec{g}\) points toward
the earth's center, and the **gravitational field strength** \(g \equiv
mag(\vec{g}) = 9.8 N/kg\).

There are two momentum flow and motion models discussed in the book:
the **multitap model** and the **three-reservoir model**. See
page 56. Also see page 57-58 for some common mistakes made when
working with vectors.

### Chapter 4: Particles and Systems

A **system** in physics is a set of interacting particles having a
well-defined "boundary" that allows us to determine whether a given
particle is inside or outside the system. An **extended object** is a
material object with well-defined *surface* in space that defines
its boundary and encloses a nonzero volume.

There are two categories of system interaction. If *both* particles
involved in the interaction are inside the system, we call it an
**internal interaction**; if one of the two particles involved in the
interaction is inside the system and the other is outside, then we
call it an **external interaction**.

We define a system's **total momentum** \(\vec{p}_{tot}\) to be the vector
sum of the momenta of its constituent particles:

The **momentum-transfer principal** (from last chapter) means that a
system's internal interactions only transfer momentum back and forth
*within* the system and therefore cannot affect a system's total
momentum. *Only external interactions can transfer momentum into or
out of a system* and thus change its total momentum. If a system
participates in *no* external interactions, we call it an **isolated
system**. Since no momentum can flow either into or out of such a
system, the following holds: **The law of conservation of momentum**:
the total momentum \(\vec{p}_{tot}\) of an *isolated* system is
*conserved*. That is, it does not change with time.

The **center of mass** of *any* system of particles is a mathematical
point whose position we define as follows:

Where \(\vec{r}_1, \vec{r}_2, ..., \vec{r}_N\) are the positions of the system's N particles. \(m_1, m_2, ..., m_N\) are their masses; and \(M \equiv m_1 + ... + m_N\) is the system's mass.

In many systems, there's an outrageously huge number of elementary particles involved. A practical (but approximate) method is to break the object up into a moderate number of macroscopic chunks that are still small enough that we can consider the particles in each to have essentially the same position.

Note that we can calculate the position of the center of mass of any
*set* of extended objects by treating each object as if it were a
point particle with all its mass concentrated at its *individual*
center of mass.

To find a systems total momentum, all that we need to do is to
multiply its total mass M by the velocity of its center of mass
\(\vec{v}_{CM}\), *exactly as if the system were a single particle
located at its center of mass!*

In conjunction with the law of conservation of momentum, the
equation above implies that an *isolated* system's \(\vec{v}_{CM}\) is
a constant. This is simply *Newton's first law*, except we now see
that it applies to arbitrary systems, not just to particles!

**The particle model**: A system's center of mass responds to its
*external* interactions *exactly* as a point particle would respond
to those interactions. This means that every idea of equation that
we have formulated up to now for interacting *particles* also
applies to any *system* of particles, as long as we substitute the
system quantities \(\vec{r}_{CM}\), \(\vec{v}_{CM}\), and
\(\vec{p}_{tot}\) for the particle quantities \(\vec{r}\), \(\vec{v}\),
and \(\vec{p}\), respectively, and put the qualifier *external* in
front of any references to *interactions*.

**Inertial reference frames** are reference frames in which we find
Newton's first law to be obeyed; **noninertial reference frames** are
frames in which we find it to be violated. The models of motion we
have been developing work in inertial reference frames but fail in
noninertial reference frames.

### Chapter 5: Applying Momentum Conservation

A system does not have to be strictly isolated for its total
momentum to be conserved. For example, the gravitation and
compression interations a laptop experiences when it's sitting on a
desk are negligible because they cancel each other out. In general,
a system's momentum is conserved in any process where its external
interactions do not change its total momentum *significantly* during
that process. There are three distinct types of situations in which
this is true:

**Floats in space**: only interacts gravitationally with distant objects**Functionally isolated**: external interactions cancel**Momentarily isolated**: strong external interactions that do*not*cancel, but we only look at the system's momentum*just before*and*just after*a very strong and brief interaction. If the collision process is sufficiently brief, external interactions simply do not have*time*to transfer significant momentum to the system.

#### Problem-Solving Framework

- Translation -> draw a picture
- Conceptual model -> applicable theories/principles
- Algebraic solution -> solve equations symbolically
- Evaluation -> check results make sense

Draw an **interaction diagram** as follows: draw a large circle to
represent the system, and draw one rectangular box inside that
circle for each object inside the system, labeling the box with the
object's name. If the object inside the system interact with object
outside the system, draw a box outside the circle for each relevant
object outside the system, and label these boxes as well. Then draw
lines connecting the boxes to represent the internal and external
interactions involved.

### Chapter 6: Introduction to Energy

@TODO

**Total energy** E is defined as:

Conservation of energy implies that:

\begin{equation} \Delta E = \Delta K_1 + \Delta K_2 + \Delta V(r_{12}) = 0 \end{equation}For a system involving two objects.

In the context of the newtonian model, **kinetic energy** is defined
as:

Experimentally, we find that for the gravitational interaction,

\begin{equation} V(z) = mgz \end{equation}Note that this equation only applies when the object is close to the earth's surface.

Ambiguity in the value of \(V\) allows us to *choose* \(V\) to be zero
at a convenient **reference separation**. If we set up the reference
separation to not be \(z = 0\), but instead \(z_0\), then \(V(z) = mgz -
mgz_0\).

### Chapter 7: Some Potential Energy Functions

@TODO

The potential energy function for the interaction between two
**charged** particles is:

Reference separation: \(V(R) \equiv 0\) when \(r = \infty\). This
equation describes the **electrostatic** potential energy function
\(V(r)\) for an electromagnetic interaction between two charged
particles. In the equation, \(q_1\) and $q_2$d are the two particles'
charges, \(r\) is their separation, and \(k\) is the **Coulomb constant**
= \(8.99\:x\:10^9 J \cdot m/C^2\). The SI unit of charge is the
**coulomb**.

The potential energy function for a gravitational interaction between two particles is:

\begin{equation} V(r) = -G\frac{m_1m_2}{r} \end{equation}
Reference separation: \(V(R) \equiv 0\) when \(r = \infty\). This
equation describes the potential energy function \(V(r)\) for a
gravitational interaction between two particles separated by a
distance \(r\). The symbols \(m_1\) and \(m_2\) are the particles'
masses, and \(G\) is the **universal gravitational constant**
(empirically, G = \(6.67\:x\:10^11 J \cdot m/kg^2\).

We can treat two object connected by a spring as if they were participating in a long-range interaction whose potential energy is very nearly:

\begin{equation} V(r) = \frac{1}{2}k_s(r - r_0)^2 \end{equation}
Reference separation: \(V(r) \equiv 0\) when \(r = r_0\). In the
equation, \(r_0\) is the separation of the objects' centers when the
spring is relaxed, and \(k_x\) is a **spring constant** characterizing
the spring's stiffness.

### Chapter 8: Force and Energy

@TODO

We define the **dot product** \(\vec{u} \cdot \vec{w}\) of two
arbitrary vectors \(\vec{u}\) and \(\vec{w}\) to be the following
*scalar* quantity:

We can therefore express the change in a particle's kinetic energy as being \(dK = \vec{v} \cdot d\vec{p}\). This then implies that:

\begin{equation} [dK] \equiv \vec{F} \cdot d\vec{r} \end{equation} \begin{equation} dK = [dK]_A + [dK]_B + ... \end{equation}
This equation describes how to compute the tiny change \(dK\) in a
particle's kinetic energy from the tiny amount of **k-work** \([dK]\)
that each interaction contributes during a tiny time interval \(dt\).

### Chapter 9: Rotational Energy

@TODO

The **radians** can be defined as:

Where \(s\) is the arclength **subtended** by the angle along the
circumference of a circle of radius \(r\).

The radian is a "unitless unit". We usually display it when we want
to make it clear that the quantity involves measuring an angle or
if we want to convert to or from other angular units, such as the
**degree** or **revolution**.

We define an object's **angular velocity** \(\vec{\omega}\) to be:

This equation only makes sense when applied to a rigid object rotating around a well-defined axis or to a particle moving in a plane around some origin.

An object's **moment of inertia** \(I\) and its **rotational kinetic
energy** \(K^{rot}\) for rotations around a given axis are given by:

Where \(m_i\) is the mass of the $i$th particle in the object, and \(r_i\) is the particle's distance from the rotation axis.

\begin{equation} K^{rot} = \frac{1}{2}I\omega^2 \end{equation}(compare with \(K = \frac{1}{2}mv^2\))

How can we calculate the value of \(I\) for a given object? The
actual number of elementary particles in any extended object is so
large that a *literal* application of the equation above is
impractical. But we can group the particles in an object into a
reasonable number of pieces (each small enough that all its
particles are *approximately* the same distance from the axis of
rotation) and then sum over the pieces instead of the particles.

The total kinetic energy of an object that is simultaneously moving and rotating turns out to be simply:

\begin{equation} K = K^{cm} + K^{rot} = \frac{1}{2}Mv^2_{CM} + \frac{1}{2}I\omega^2 \end{equation}The angular speed \(\omega\) of an object of radius R that rolls without slipping is linked to its center-of-mass speed \(v_CM\) by:

\begin{equation} \omega = \frac{v_{CM}}{R} \end{equation}fRolling without splipping is causes by the friction part of a contact interaction between the object and a surface. Therefore, we can handle such an interaction in a conservation of energy problem by including a rotational kinetic energy term in our conservation of energy equation.

### Chapter 10: Thermal Energy

**Thermal physics** focuses on physical processes involving heat and
temperature change. In 1843, James Prescott Joule showed a
connection between heat and energy. Object are made up of molecules
that are in ceaseless random motion. An object's **temperature** \(T\)
is a measure of the intensity of this motion. The SI unit for
temperature is **kelvins**.

For gases and simple solids under everyday conditions, temperature
is directly proportional to the average *kinetic* energy per
molecule of its molecules' random motions in the substance
(treating the molecule as a point particle):

Where \(k_B = 1.38 x 10^{-23} J/K\) and is called **Boltzmann's
constant**.

An object whose molecules are all at rest has a temperature of
*zero*. Since molecules cannot have negative kinetic energy, this
is the lowest possible temperature an object can have: we therefore
call this temperature **absolute zero**.

When two surfaces are in contact and one moves past the other their
surface molecules often become entangled. The entangled molecules
are stretched away from their normal positions. Their interaction
transforms a tiny bit of kinetic energy to potential energy in the
bonds between molecules. At a certain point, the molecules suddenly
become unstuck and snap back toward their initial positions, where
they oscillate wildly. The kinetic energy apparently lost in
friction and collisions interacts actually is transformed to
*thermal energy*.

As is the case for potential energy, it is the *change* in thermal
energy that is usually important. **Internal energy** \(U\) is the
*total* energy that an object contains within the boundary defining
its surface. When a hot object is placed in contact with a cold
object, energy spontaneously flows across the boundary between them
from the hot object to the cold object until both objects have the
same temperature. In physics, **heat** \(Q\) is any energy that crosses
the boundary between the two objects *because* of the temperature
difference between them. We define **work** \(W\) to be any *other*
kind of energy flowing across an object's boundary. For example, if
I stir a cup of water vigorously and it gets warm as a result, I
have not "heated" the water, I have done *work* on it; the
mechanical energy that flows across the water's boundary flows not
because of a temperature difference but because of my stirring
effort.

Conservation of energy implies that in a given process:

\begin{equation} \Delta U = Q + W \end{equation}
Formally, an object's **thermal energy** \(U^{th}\) is the part of its
internal energy that increases with temperature. Note *heat* is an
energy flow across an object boundary, whereas *thermal energy* is
energy *enclosed* by that boundary. An object can *absorb* heat,
but it never *contains* heat!

An object's thermal energy is generally a complicated function of
temperature, but for relatively small temperature changes not too
close to the temperatures where an object changes **phase** from
solid to liquid or liquid to gas, we have:

Where \(M\) is the object's mass, and c is its **specific heat**. Note
that for this equation to work, the change in temperature \(dT\) must
be small enough that c is mostly constant over the temperature
range.

Keeping track of thermal energies enables us to solve conservation of energy problems involving:

- Impact or friction interactions that do
*work*on objects in the system or **Thermal contact interactions**that transfer heat between objects.

To represent each such interaction, we add a \(\Delta U^{th}\) term to
our conservation of energy equation for *each* object involved.

### Chapter 11: Energy in Bonds

*Note: only sections one and two are included below.*

A **potential energy diagram** for a two-objevt system is simply a
graph of its total interaction potential energy \(V(r)\) versus the
objects' separation \(r\). We represent the system's fixed total
energy \(E\) as a horizontal line on a potential energy diagram. The
light object's kinetic energy \(K\) at any given \(x\) is \(K = E -
V(x)\). Since \(K\) cannot be negative, regions of the x axis where
\(V(x) > E\) are **forbidden regions**, while regions where \(E > V(x)\)
are **allowed regions**. The points where $E = V(x) are **turning
points**: the velocity reverses direction at such points to keep
from entering a forbidden region.

We can read from these diagrams the x component \(F_x\) that the
interaction exerts on the light object at any position. This value
is given \(F_x = \frac{-dV}{dx}\). Points along the x axis where the
slope of \(V(x)\) is zero (implying \(F_x = 0\)) are **equilibrium
positions** where the light object could in principle remain at
rest. If such a point corresponds to the bottom of a valley in the
potential energy function, it is a **stable equilibrium position**;
otherwise it is an **unstable equilibrium position**.

TODO: **bonds**. See page 197

### Chapter 12: Power, Collisions, and Impacts

**Power** expresses *the magnitude of the rate at which energy flows
into or out of a given form* in the course of some interaction or
process.

The SI unit of power is the **watt**, where \(1W \equiv 1J/s\).

If a specific interaction exerts a force on a moving object, the rate at which energy is transformed by that particular interaction is:

\begin{equation} P = \left|\frac{[dK]}{dt}\right| = |\vec{F}\cdot\vec{v}| \end{equation}## Unit N: The Laws of Physics are Universal

### Chapter 1: Newton's Laws

Newton's theories of motion were the first that were able to
explain both terrestial *and* celestial phenomena using the same
theoretical framework. This unification is called the **newtonian
synthesis**.

Conservation of momentum implies that *the total momentum of an
isolated system is conserved*. **Newton's first law** directly
follows from this: "In the absence of external interactions, an
object's (or system's) center of mass moves at a constant
velocity."

A tiny impulse \([d\vec{p}]\) flowing *out* of one object A is the
same as \(-[d\vec{p}]\) flowing *into* A. Force \(\vec{F}\) exerted by
an interaction on an object is defined to be the *rate* at which
the interaction transfers momentum into the object. The rate at
which a given interaction transfers momentum into object B is the
negative of the rate at which it transfers momentum into object
A. **Newton's third law** says that "When object A and B interact,
the force the interaction exerts on A is equal in magnitude and
opposite in direction to the force that it exerts on B." Or
mathematically:

Consider a 10,000-kg truck traveling at 65 mi/h. Imagine that it
hits a parked 500-kg Volkswagen Beetle. Which vehicle exerts the
stronger force on the other during the collision? According to the
third law, the magnitudes of the forces that the contact interation
exerts on either vehicle *must be the same*: any momentum that the
interaction transfers out of the truck must go into the Beetle at
the same rate. This seems counter intuitive. Note that the fact
that the *forces* are the same on each doesn't mean that each has
to *respond* in the same way. Because the Beetle is 20 times less
massive than the truck, a given amount of momentum flowing into the
Beetle will lead to a change in its velocity whose magnitude is 20
times larger than the change in the truck's velocity from the same
momentum flow.

For point particles, **Newton's second law** looks like:

Where a particle's **acceleration** \(\vec{a} \equiv d\vec{v}/dt\). Now
let's consider a *system* of particles that interacts with its
surroundings. Internal interactions only transfer momentum from
particle to particle *inside* the system, so they will not change
the system's *total* momentum.

Note that the object must have a fixed mass. This equation is
**Newton's second law** for *systems* of particles. To understand how
an object's center of mass moves, we only need to be able to
quantify the *net exernal force* acting on that object.

Forces do not cause *motion*, they cause *acceleration*. A force
does not *overcome* another force to cause an object to move in a
certain way: rather all forces acting on the object act in concert
to direct the object's acceleration.

#### Classification of Forces

See pages 11-12 of the the Unit N book for more specific information on these forces.

- Long-Range Interactions
Symbol Name \(\vec{F}_g\) Gravitational force \(\vec{F}_e\) Electrostatic force \(\vec{F}_m\) Magnetic force - Contact Interactions
Symbol Name \(\vec{F}_{sp}\) Spring force \(\vec{F}_T\) Tension force \(\vec{F}_N\) Normal force \(\vec{F}_{SF}\) Static friction \(\vec{F}_{KF}\) Kinetic friction \(\vec{F}_D\) Drag force \(\vec{F}_L\) Lift force \(\vec{F}_{Th}\) Thrust force \(\vec{F}_p\) Pressure force \(\vec{F}_B\) Buoyant force

#### TODO Free-Body Diagrams

See page 13 of the Unit N book!

### Chapter 2: Vector Calculus

An object's **average velocity**:

With a nonzero \(\Delta t\), this equation most closely approximates the object's instantaneous velocity at an instant \(t\) halfway throught the interval \(\Delta t\).

An object's **average acceleration**:

Again, this equation provides a good *approximation* for
instantaneous acceleration midway through the interval.

#### TODO Motion Diagrams

See page 26 of the Unit N book! Note that motion diagrams display both velocity and acceleration.

#### Uniform Circular Motion

**Uniform circular motion** is when an object is moving at a
*constant speed* around a circle of radius \(R\). The acceleration
of an object moving at a constant speed around a circle points
directly toward the center of the circle at every instant. Note
that this *only* applies if the object is moving at a constant
speed. Page 32 gives a superb diagram of the relation between
velocity and acceleration. A mathematical representation of this
acceleration can be seen below:

### Chapter 3: Forces from Motion

The **kinematic chain** connects position, velocity, and acceleration
by derivative relationships:

#### Net-Force Diagrams

A **net-force diagram** is similar to a **free-body diagram** but more
vividly displays the quantitative relationship between these
forces. Net force is the vector sum of all external forces acting
on an object. Here are the steps to creating a net-force diagram:

- Draw a free-body diagram first
- Copy the force arrows and arrange them in sequence by drawing each new force arrow with its tail end starting at the tip of the previously drawn arrow
- Draw the arrow that represents the vector sum of these force arrows.

#### Third-Law and Second-Law Pairs

A pair of forces fitting the following conditions are **third-law
partners**:

- Each force in the pair must act on a
*different*object - Both forces must reflect the
*same interaction*between the objects.

Two forces that act on the same object *cannot* be third-law
partners.

Forces are called **second-law partners** if:

- Both forces act on the
*same*object - The forces oppose each other along a certain axis
- No other forces have any component along that axis
- The component of the object's acceleration along that axis is zero.

#### Graphs of One-Dimensional Motion

When constructing a graph of \(a_x(t)\) from one of \(v_x(t)\), the
*value* of the object's x-acceleration at every instant of time
will correspond to the *slope* of \(v_x(t)\) when the latter is
plotted versus \(t\) on the graph.

When constructing a graph of \(v_x(t)\) from one of \(x(t)\), the
*value* of \(v_x\) at every instant of time will correspond to the
*slope* of a graph of \(x(t)\) versus \(t\).

*Always* use the symbols \(v_x\), and \(a_x\) to refer to signed
numbers and *always* use \(r\), \(v\), and \(a\) to refer to the
(nonnegative) magnitudes of the corresponding vectors.

### Chapter 4: Motion from Forces

The kinematic chain of relationships link an object's position to
its velocity and its acceleration. We can reverse the chain by
taking *antiderivatives*. This is called the **reverse kinematic
chain**:

There are lots of different approaches to finding the antiderivatives of acceleration.

- Graphical Antiderivatives
Use the reverse of the

**slope method**seen in N3. Note that there is a certain ambiguity in this process from constants. - Integrals for One-Dimensional Motion
\(v_x(t) = \int a_x(t) dt + C_1 \qquad and \qquad x(t) = \int v_x(t) dt + C_2\)

Note that we can determine the \(C_1\) if we know the object's initial velocity \(v_x(t)\). We need only substitute t = 0 into the expression for $v_x(t) and choose the value of \(C_2\) that gives the correct initial velocity \(v_x(0)\). We can similarly determine \(C_2\) by choosing it so that we get the correct value of \(x(0)\).

- Constant x-acceleration equations
\(v_x(t) = a_xt + v_0x \qquad and \qquad x(t) = \frac{1}{2}a_xt^2 + v_{0x}t + x_0\)

When functions are really *vector* functions, we can find
antiderivativesd by considering each component separately. This
means that we can treat vector functions (symbolically) as if they
were scalar functions:

#### Constructing Trajectory Diagrams

To draw a trajectory diagram, first *center* the object's initial
velocity arrow on its initial position. Then use the known
acceleration vector to construct future position points from the
initial arrow's two endpoints.

### Chapter 6: Linearly Constrained Motion

A **free-particle diagram** is like a free-body diagram, but all of
the force arrow's tails start at the center of mass. Also, the
diagram should contain a set of reference frame axes.

If an object moves in a straight line at a constant velocity, its acceleration is zero, implying that the net force on the object must be zero.

#### TODO Static and Kinetic Friction Forces

\(F_{KF}\) is usually less than the maximum value of \(F_{SF}\).

\begin{equation} mag(\vec{F}_{SF}) \leq mag(\vec{F}_{SF,max}) \approx \mu_s mag(\vec{F}_N) \end{equation} \begin{equation} mag(\vec{F}_{KF}) \approx \mu_k mag(\vec{F}_N) \end{equation}#### TODO Drag Forces

Where \(C\) is the drag coefficient, \(p\) the density, and \(A\) the cross section. Note that the density of air is \(1.2\:kg/m^3\).

### Chapter 7: Coupled Objects

A pair of **coupled objects** consists of two objects that are
constrained by some kind of connection so that their motions are
linked in some well-defined manner.

For cases where forces are acting on more than one object, we have a notation convention for force symbols. We add to the basic symbol for the type of force involved a pair of superscripts, the first indicating the object on which this force is exerted and the second (in parentheses) indicating the other object involved in the interaction that gives rise to this force. For example: \(\vec{F}^{A(B)}\). Note that this really helps recognize third-law partners.

#### TODO Strings, Real, and Ideal

See page 118. Talk about working with real strings that have a
mass and **ideal strings** that don't.

#### TODO Pulleys

see page 122. Talk about **ideal pulleys** that have a very small
mass and are nearly frictionless.

### TODO Chapter 10: Projectile Motion

In physics we define an object's *weight* \(\vec{F}_g\) to be the
force that gravity exerts on that object. The weight force itself
is given by \(\vec{F}_g = m\vec{g}\). If the *only* force on an
object is its weight \(\vec{F}_g\), we say that it is **freely
falling**.

As long as (1) an object remains "sufficiently close" to the
surface of the earth and (2) its trajectory is "sufficiently short"
that the curvature of the earth is not significant, the magnitude
and direction of \(\vec{g}\) (and thus of the object's weight
\(\vec{F}_g\)) will be approximately constant. If in addition (3) the
object is "not significantly affected" by other forces except
possibly air drag, then we call it a **projectile** and its motion
**projectile motion**. When (4) we *also* neglect drag, we say that
the object's motion is **simple projectile motion**.

#### Simple Projective Motion

Newton's second law implies that object's in **simple projectile
motion** will have constant acceleration \(\vec{a} = \vec{g}\).

*The object must be in simple projectile motion for these
equations to apply*.

How do we determine when the object has reached its maximum altitude and when it reaches the ground? Well the peak is when \(v_z(t)=0\) and the object hits the ground at \(z(t) = 0\).

#### Drag and Terminal Speed

Since the force from drag \(\vec{F}_D\) is proportional, as a
falling object's downward speed increases, the grag force also
increases, decreasing the net force on the object and thus its
downward acceleration. When the net force is essentially zero, the
object no longer accelerates, but continues to fall with constant
downward velocity. We call the magnitude of this final constant
downward velocity the object's **terminal speed** \(v_Tf\).

Note that this equation is created by the equation for the force from drag solved for v. The \(F_D\) has been substituted for \(mg\) since at terminal velocity drag is equal to weight (\(F_g=F_D\)).

An object falling from rest at time \(t = 0\) will behave as a simple projectile for times \(t << t_T \equiv v_T / g\), but will essentially reach its terminal spped after a time \(t \approx 2t_T\).

## Unit R: The Laws of Physics are Frame-Independent

### Chapter 1: The Principle of Relativity

Think of being on a smooth sailing plane. If you drop something it
acts as if the plane is sitting in the loading dock. The laws of
physics are the same inside a laboratory moving at a constant
velocity as they are in a laboratory at rest. This is an unpolished
statement of what we call the **principle of relativity**. This idea
is the foundation for Einstein's **special theory of relativity**.

An **event** is any physical occurrence that we can consider to
happen at a definite place in space and at a definite instant in
time. We can quantify when and where the event occurs by four
numbers: three numbers that specify the location of the event in
some three-dimensional spatial coordinate system and one number
that specifies what time the event occurred. We call these four
numbers the event's **spacetime** coordinates. The motion of *any*
particle can be mathematically described to arbitrary accuracy by
specifying the spacetime coordinates of a sufficiently large number
of events suitably distributed along its path.

We need to add the *time* of an event to our definition of
reference frames. Imagine that we attach a clock to every lattice
intersection. We can then define the *time* coordinate of an event
to be the time displayed on the lattice clock nearest the event
(relative to some specified time \(t = 0\)) and the event's three
*spatial* coordinates to be the lattice coordinates of that nearest
clock, specified in the usual way by stating the distances along
the lattice directions that one has to travel (from some specified
spatial origin) to the clock. We can determine these four numbers
to whatever precision we want by sufficiently decreasing the
lattice spacing and the time between clock ticks.

Why is it important to have a clock at *every* lattice
intersection? The point is to make sure that there is a clock
essentially *at* the location of any event to be measured. If we
attempt to read the time of an event by using a clock located a
substantial distance away, we need to make assumptions about how
long it took the information that the event has occurred to *reach*
that distant clock. For example, if we read the time when the
*sound* from an event reaches the distant clock, we should correct
that value by subtracting the time that it takes sound to travel
from the event to the clock; byt to do this, we have to know the
speed of sound in our lattice. We can avoid this problem if we
require that an event's time coordinate be measured by a clock
essentially *present* at the event.

We must *synchronize* the clocks. This will be discussed later.

An **operational definition** of a physical quantity defines that
quantity by describing how the quantity can be *measured*. A
**reference frame** is defined to be a rigid cubical lattice of
appropriately synchronized clocks *or its functional
equivalent*. The **spacetime coordinates** of an event in a given
refgerence frame are defined to be an ordered set of four numbers,
the first specifying the *time* of the event as registed by the
nearest clock in the lattice, followed by three that specify the
spatial coordinates of that clock in the lattice. An **observer** is
defined to be a (possibly hypothetical) person who interprets
measurements made in a reference frame.

A reference frame may be moving or at rest, accelerating, or even
rotating about some axis. The beauty of the definition of spacetime
coordinates given above is that measurements of the coordinates of
events can be carried out in a reference frame no matter how it is
moving. However, not all reference frames are equally useful for
doing physics. We can divide reference frames into two general
classes: **inertial frames** and **noninertial frames**. An *inertial
frame* is one in which an isolated object is always and everywhere
observed to move at a constant velocity. In a *noninertial frame*
such an object is observed to move with a nonconstant velocity in
at least some situations.

Any inertial reference frame will be observed to move at a
*constant velocity* relative to *any other* inertial reference
frame. Conversely, a rigid, nonrotating reference frame that moves
at a constant with respect to nay other inertial reference frame
must *itself* by inertial.

Consider two inertial reference frames, which we will call the
**Home Frame** and the **Other Frame**. An isolated object at rest in
the Other Frame must move at a constant velocity with respect to
the Home Frame, so the whole Other Frame must move at the same
constant velocity relative to the Home Frame.

We cannot physically distinguish a reference frame "moving at a
constant veloicty" from one "at rest". The principle of relativity
specifically states that a reference frame moving at a constant
velocity is *physically equivalent* to a frame at rest. Our final
polished statement of **The Principle of Relativity** is a follows:
*The laws of physics are the same in all inertial reference
frames*.

In **standard orientation** we have the Home Frame's \(x\), \(y\), and
\(z\) axes point in the same directions as the corresponding axes in
the Other Frame. We conventionally distinguish the Home Frame and
the Other Frame axes by referring to the Home Frame axes as \(x\),
\(y\), and \(z\) and the Other Frame axes as \(x'\), \(y'\), and \(z'\). It
also is conventional to define the origin event (the event that
defines \(t = 0\) in both frames) to be the instant at which the
spatial origin of one frame passes the origin of the
other. Finally, we conventionally choose the common \(x\) axis so
that the Other Frame moves in the \(+x\) direction with respect to
the Home Frame.

Suppose the Other Frame moves at a constant velocity \(\vec{\beta}\). The relationship between the object's position vectors in the two frames a given time is as follows:

\begin{equation} t' = t \qquad x' = x - \beta t \qquad y' = y \qquad z' = z \end{equation}
Physicists call these four equations the **galilean transformation
equations**. They allow us to find the position of the object at a
given time \(t'\) in the Other Frame if we know its position at time
\(t = t'\) in the Home Frame (*assuming*, or course, that time is
universal and absolute).

If we take the time derivative of both sides of each of the last three equations, we get:

\begin{equation} v_x' = v_x - \beta \qquad v_y' = v_y \qquad v_z' = v_z \end{equation}
These equations tell us how to find the velocity of an object in
the Other Frame, given its velocity in the Home Frame: we call
these equations the **galilean velocity transformation equations**.

If we take the time derivative again, we get:

\begin{equation} a_x' = a_x \qquad a_y' = a_y \qquad a_z' = a_z \end{equation}
This tells us that *observers in both inertial frames agree about
an object's acceleration at a given time*, even thought they may
well disagree about the object position and velocity components at
that time!

When we say that the "laws of physics are the same" in different inertial frames we realize that observers in different inertial frames may disagree abut the values of various quantities (particularly positions and velocities) by each oberver will agree that the same basic equations will be found to describe the laws of physics in all inertial reference frames.

### Chapter 2: Synchronizing Clocks

In 1878, James Clerk Maxwell published a set of equations (now
called **Maxwell's equations**) that summarized the laws of
electromagnetism in compact and elegant form. These equations
predicted that light waves must travel at a specific speed \(c =
3.00 \times 10^8 m/s\). Einstein argued, to make Maxwell's equations
consistent with the principle of relativity, that *the speed of
light is a frame-independent quantity*.

If time is not "universal and absolute", what is it? Einstein said
that we must define what we mean by "time" *operationally* within
each inertial frame by specifying a concrete and specific procedure
for synchronizing that frame's clocks that is consistent with both
the principle of relativity and the laws of electromagnetism. Since
the speed of light has to be \(c\) in every inertial frame anyway,
let us in face synchronize the clocks in our inertial reference
frame by *assuming* that light always has the same speed of \(c\)!
Imagine that we have a master clock at the spatial origin of our
reference frame. At exactly \(t = 0\), we send a light flash from
that clock that ripples out to the other clocks in the frame.

#### SR Units

In this unit distance will be measured in a new unit, called a
*light-second* or just *second*. A **light-second** or **second** is
defined to be the distance that light travels in 1s of time. There
is exactly 299,792,458 m in 1 light-second by definition.

Agreeing to measure distance in seconds allows us to state the definition of clock synchronization in an inertial frame in a particularly nice and concise manner: Two clocks in an inertial reference frame are defined to be synchronized if the time interval (in seconds) registered by the clocks for a light flash to travel between them is equal to their separation (in light-seconds).

In this chapter we will use a slightly modified version of SI
units, called **SR units**, where distance is measured in *seconds*
(i.e., light-seconds) instead of in meters.

In SR units, the light-second is considered to be *equivalent* to
the second of time, and both units are simply referred to as
*seconeds*. This means that these units can be canceled if one
appears in the numerator of an expression and the other in the
denominator. For example, in SI units, velocity has units of
meters per second; but in SR units it has units of seconds per
second = unitless(!). Thus an object that travels 0.5 light
seconds in 1.0 s has a speed in SR units of 0.5s/1.0s = 0.5 (no
units!).

#### Spacetime Diagrams

We can conveniently depict the coordinates of events by using a
special kind of graph called a **spacetime diagram**.

Note that the point marked \(O\) in the figure also represents an
event. This event occurs at time \(t = 0\) and at position \(x =
0\). We call this event the **origin event** of the diagram.

If we need to draw a spacetime diagram of an event \(A\) that occurs in space somewhere in the \(xy\) place, we must add another axis to the spacetime diagram (seen in the image to the right).

On a spacetime diagram, an event is represented by a
point. The motion of any object is represented by an infinite set
of infinitesimally separated points, which is a *curve*.

In drawing spacetime diagrams, it is also convenient and conventional to use the same-size scale on both axes. If this is done, a flash of light always has a slope of either 1 (if the flash is moving in the \(+x\) direction) or -1 (if the flash is moving in the \(-x\) direction), since light travels 1.0s of distance in 1.0s of time by definition in every inertial reference frame. It is also conventional to draw the worldline of a flash of light with a dashed line instead of a solid line.

#### The Radar Method

If we are willing to confine our attention to events occurring only along the x axis (and thus to objects moving only along that axis), it is possible to determine the spacetime coordinates of an event with a single master clock and some light flashes: we don't need ton construct a lattice at all! The method is analogous to locating an airplace by using radar.

After sending a light flash out, the values of the emission and reception times \(t_A\) and \(t_B\) are sufficient to determine both the location and the time that event \(E\) occurred!

\begin{equation} t_E = \frac{1}{2}(t_B + t_A) \qquad x_E = \frac{1}{2}(t_B - t_A) \end{equation}This equation expresses an event \(E\)'s spacetime coordinates \(t_E\) and \(x_E\) in terms of the time \(t_A\) at which a light flash leaves a given frame's origin and the time \(t_B\) when it returns after being reflected by the event.