Circular Motion

$(\frac{d\vec{p}}{dt})\parallel = \vec{F}_{net}\parallel$ is the rate of change of the amgnitude of the momentum.

$(\frac{d\vec{p}}{dt})\perp = \vec{F}_{net}\perp$ is the rate of change of the direction of the momentum, which is numerically equal to the sideways (perpendicular) component of net force.  Other associations:$$|(\frac{d\vec{p}}{dt})\perp| = |\vec{p}|\frac{|\vec{p}|}{R}$$$$|(\frac{d\vec{p}}{dt})\perp| = |\vec{p}||\frac{d\hat{p}}{dt}| = |\vec{p}|\frac{|\vec{p}|}{R} = \frac{\gamma mv^2}{R}$$$$|\vec{a}\perp| = |(\frac{d\vec{v}}{dt})\perp| = |\vec{v}||\frac{d\hat{v}}{dt}| = |\vec{v}|\frac{|\vec{v}|}{R} = \frac{v^2}{R}$$$$\frac{d\vec{p}}{dt}\leftarrow\vec{F}_{net}$$

Speed of Sound in a Solid

The Speed of Sound in a Solid can be determined through the equation:$$v = \omega d = \sqrt{\frac{k_{s,i}}{m_a}d}$$

$v$ is the speed of sound in m/s.
$k_{s,i}$ is the stiffness of the interatomic bond.
$m_a$ is the mass of one atom.
$d$ is the length of the interatomic bond.

Friction

"Friction is the dissipation of kinetic energy into internal energy in the form of agitation of atoms throughout the objects.  We say that friction is a "dissipative" process."

Friction can either speed up an object or slow it down, depending on its motion.  For instance, on an airport conveyor belt your bags initially have no momentum, but with the addition of the movement of the conveyor belt and friction forces opposing the bag staying in its zero position, the bag begins to move in the direction of the conveyor belt.  We can model this sliding friction with the equation:$$F_{KF} = \mu_k F_N$$$F_N$ is the "normal force," or what is causing the objects to contact each other.
$\mu_k$ is the coefficient of kinetic friction, a value that will range depending on the material.
The speed of an object does not effect the sliding friction force.

 If a force is applied to an object that does not exceed the friction force, $F_{friction}$, we say that you are being counteracted by the static friction.  In order for movement to occur, the force applied to an object must be greater than or equal to the frictional force in the direction of the application.$$F_{SF} <= \mu_s F_N$$$\mu_s$ is the coefficient of static friction.  $\mu_s$ will more than likely be larger than $\mu_k$ because once an object is already moving, it requires only as much force as it already has to continue its movement.  You can feel this effect when you manage to slide a heavy object across the floor, and although it is difficult initially it becomes much easier once it is moving.

Any force of at least $F_{SF}$ in value can be applied to an object, but once it is exceeded, the equations governing its frictional status changes to sliding, $F_{KF}$.

Young's Modulus

"Young's modulus ($Y$) is the ratio of stress to strain for a particular material.  young's modulus is a property of the material, and does not depend on the size or shape of an object.  The stiffer the material, the larger is Young's modulus."

The strain on a material is the fractional stretch, or the change in length divided by the original length.  This gives a ratio defined as $$strain = \frac{\Delta L}{L}$$The stress on a material is the tension force per unit of area.  This eliminates the dependency of thickness of the material by using both the tension force ($F_T$) and cross-sectional area of a wire ($A$).$$stress = \frac{F_T}{A}$$These ratios can be related down to the atomic level, the stress ($\frac{F_T}{A}$) being the force that each chain of atomic bonds must exert, and the strain ($\frac{\Delta L}{L}$) being the stretch of the interatomic bond.  The stiffer the material, the larger the modulus.  We write young's modulus as$$\frac{F_T}{A} = Y\frac{\Delta L}{L}$$$$Y = \frac{stress}{strain} = \frac{\frac{F_T}{A}}{\frac{\Delta L}{L}}$$This is dependent upon the fact that too large a stress will result in the material coming apart and breaking, a process known as "yielding."  The Young's modulus in terms of atomic quantities can be defined as:$$Y = \frac{\frac{k_{s,i}s}{d^2}}{\frac{s}{d}} = \frac{k_{s,i}}{d}$$$d$ represents the relaxed length of an interatomic bond, and the diameter of one atom.  The cross-sectional area of an atom is $d^2$, considering we are viewing the atom as occupying a cube of space in the crystal lattice versus its approximately spherical shape.
The stretch of the interatomic bond is $s$, or the $\Delta L$.
The interatomic force is $k_{s,i}$.

Tension Forces

Tension Forces are just forces that compensate for other forces.  For instance, when a ball is hanging motionless on a wire, the net force on the ball must be zero.  The gravity of the Earth is pulling it down, therefore there must be another force in the opposite direction compensating for this.

The force exerted by an object such as a wire or string is called a tension.$$\Delta \vec{p} = \vec{F}_{net}\Delta t$$$$0 = (\vec{F}_T - \vec{F}_{ext})\Delta t$$

Properties of Atoms

• All matter consists of atoms, the typical radius of an atom is about $r = 1 \times 10^-10 meters$.
• Atoms attract each other when they are close, but not too close.
• Atoms repel each other when they get to close together.
• Atoms move even at very low temperatures.  This applies to solids, liquids, and gases.

Center of Mass

The center of mass is the center of a system of masses, located at a point that is central in relation to the masses of all of the objects within the system.  To describe the position of the center of mass, we use the equation$$\vec{r}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{r}_i}{\sum\limits_{i=0}^n m_i}$$$$\vec{r}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{r}_i}{M_{total}}$$

The velocity of the center of mass is the same thing, except finding velocity instead of position.$$\vec{v}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{v}_i}{M_{total}}$$

If the velocity is very small compared to the speed of light, $\gamma = 1$ we can say that the total velocities and masses of the system are equal based off of the equation for momentum.  This helps us find the momentum of the center of mass:$$\vec{p} = \gamma\times m\vec{v}$$$$\vec{p}_{sys} = M_{total}\vec{v}_{CM}$$$$\vec{p}_{sys} = \sum\limits_{i=0}^n m_i \vec{v}_i$$

Conservation of Momentum

The concept of conservation of momentum is simple to understand by looking at the Momentum Principle.  The Momentum Principle predicts how much the momentum of a system will change based on external forces, or forces in the surroundings.  Momentum gained by the system is transferred from the surroundings, so we say the momentum is conserved.$$\Delta \vec{p}_{sys} = -\Delta \vec{p}_{surr}$$$$\Delta \vec{p}_{sys} + \Delta \vec{p}_{surr} = \vec{0}$$Basically, all force is equal and opposite.

Coulomb's Law

To quantify electric interactions we use an equation known as Coulomb's Law $$\vec{F}_{elec\space on \space 2 \space by \space 1} = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{|\vec{r}|^2}\hat{r}$$
$\vec{r} = \vec{r}_2 - \vec{r}_1$ is the position of 2 relative to 1.
$\frac{1}{4\pi\epsilon_0}$ is a universal constant, equivalent to $9\times10^9 \frac{Nm^2}{C^2}$
The charges $q_1$ and $q_2$ are measured in units of Coulombs, abbreviated $C$.

Like gravity, it is proportional to the inverse square of the distance between the center of its objects.  The universal constant is much larger than that of the gravitational constant, meaning that the electric interaction is much stronger than gravitational interaction.

Fundamental Interactions

Gravitational
Responsible for attraction between objects that have mass.  Every object with mass exerts a gravitational force on objects within its gravitational range.  A small mass on Earth may exert a gravitational force on a couple objects in its near vicinity, while the Earth exerts a gravitational force on all the objects around Earth.

Electromagnetic
Responsible for attraction or repulsion between objects that have electric charge.  Particles with the same charge (protons/electrons) repulse each other, while opposite charges attract.  Electric forces are what keep atoms bonded together, and what cause reactions.

Strong
Responsible for interactions between objects made of quarks, such as protons and neutrons.  They are held together in the nucleus despite the repulsion between protons.

Weak
Affects all kinds of elementary interaction, but is weaker than the strong and electromagnetic interactions.  Basically is in everything, just really weak - hence the name. 

Gravitational Force

The gravitational force is between at least two objects.  It:

• Acts along a line connecting the two objects.
• Is proportional to the masses.
• Is inversely proportional to the square of the distance between the centers of the two objects.

It can be modeled in several different ways.  The approximate gravitational force is $$|\vec{F}_{grav}| = mg$$However, there is a much more accurate, albeit complicated, way of determining the forces of gravity.  To determine the force of gravity from one object upon another, you simply use the equation:$$\vec{F}_{grav \space on \space 2 \space by \space 1} = -G\frac{m_1 m_2}{|\vec{r}|^2} \hat{r}$$
$\vec{r} = \vec{r}_2 - \vec{r}_1$ extends from the center of object 1 to object 2.
$G = 6.7 \times 10^-11 \frac{N m^2}{kg^2}$ and is known as the gravitational constant ($G$).

To calculate the magnitude of gravitational field near an object's surface, use the equation:$$g = G\frac{M_E}{R_E^2}$$

Spring Force

The spring force was determined through experimentation to be accurately modeled by the equation:$$|\vec{F}_{spring}| = -k_s (|\vec{L}| - L_i)\hat{L}$$$$\vec{F} = -k_s s \hat{L}$$It's easy to determine from this equation:$$s = |\vec{L}| - L_i$$
$k_s$ is the spring constant, or stiffness, of the spring.  It dictates the stretching capabilities of the spring.
$L_i$ is the length of the relaxed spring (no compression/extension).
$\vec{L}$ extends from the point of attachment of the spring to the mass at the other end (the total, final length).
The stretch of $s$ can be negative or positive (compression or extension).

Momentum Update Formula

The Momentum Update Formula is simply a rearrangement of the Momentum Principle:$$\Delta\vec{p} = \vec{p}_f - \vec{p}_i = \vec{F}_{net}\Delta t$$$$\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$$If you know the initial momentum ($\vec{p}_i$) of an object and the net force ($\vec{F}_{net}$) over a period of time ($\Delta t$) short enough to make it approximately constant, you can predict the final momentum ($\vec{p}_f$).

Impulse

Impulse is the product of the force ($\vec{F}$) and change in time ($\Delta t$).  $$Impulse = \vec{F}\Delta t$$The impulse differs from the Momentum Principle in that it is distinct in forces.  The change in momentum ($\Delta\vec{p}$) is equal to the sum of the impulses applied, or:  $$\Delta\vec{p} = \sum\limits_{i=1}^n \vec{F}_i\Delta t$$

Force

The concept of force was created to quantify the interaction between 2 objects.  Force is regarded as a vector because it has a magnitude and is exerted in a specific direction.  Force is most easily measured by utilizing a spring - when we stand on a spring scale, it is merely measuring the force of our weight due to gravity pressing against it.  By measuring how much a spring compresses when a force is applied we can obtain a very accurate measurement - an equation found (link to post) here.

Force is largely measured in newtons (N).  1 N is approximately the downward gravitational force of the Earth on a small apple (Newton formulated the idea of gravity by watching an apple fall from a tree, what a coincidence).

Force is the derivative of momentum, described as "the instantaneous time rate of change of the momentum of an object is equal to the net force acting on the object," or more simply, "the derivative of the momentum with respect to time is equal to the net force acting on the object."$$\frac{d\vec{p}}{dt} = \vec{F}_{net}$$

The net force ($\vec{F}_{net}$) is widely used in physics to describe all the forces acting on a system:  $$\vec{F}_{net} =  \sum\limits_{i=1}^n \vec{F}_i$$

Momentum Principle

The Momentum Principle is Newton's Second Law - a fundamental principle of physics.  It is used to predict the behavior of objects, restating Newton's First Law in a form that is measurable and causal.  $$\Delta\vec{p} = \vec{F}_{net}\Delta t$$This equation simply states that:  The change in momentum ($\Delta\vec{p}$) is equal to the net force ($\vec{F}_{net}$) times the elapsed time ($\Delta t$).

This is only effective if the net force is nearly constant.  So, either the net force must be constant, or the measurement used in increments of time that are small enough to model nearly constant force.

For instance, if we had a net force that changed at $t=3$ and $t=7$, respectively, we could make the measurement from $t=0$, $\Delta\vec{p} = \vec{F}_{net}\times3$, and then again from $t=3$ as $\Delta\vec{p} = \vec{F}_{net}\times4$ to conclude at $t=7$.

System vs. Surroundings

In physics, a system is a collection of objects that interacts with objects making up the surroundings.  The system is observed and recorded, while the surroundings are what creates change in the system.  The momentum of a system can only be changed by interactions with its surroundings.

Momentum

In real life, it seems obvious that stopping a baseball traveling at high speed is much easier than, say, a refrigerator hurdling towards you at high speed.  The mass, or weight, of the refrigerator is much greater than the baseball!  Everyone knows that it becomes increasingly difficult to move or change something as it becomes heavier, but how are we to describe this phenomenon using physics?

In order to represent the combination of an object's mass ($m$) and velocity ($\vec{v}$), a vector quantity called momentum, $\vec{p}$, is defined.  Notice that mass is a scalar and velocity a vector.  In our visible, daily lives, this equation exists simply as $\vec{p} = m \times \vec{v}$.  However there is another important factor that we must account for.

Certain experiments have shown that as particles travel closer and closer to the speed of light ($c = 3 \times 10^8$ m/s), the amount of interaction required in order to create an increase in velocity becomes increasingly large.  This is Einstein's "relativistic" definition of momentum, defined below.

In order to model the disparity between low speed velocity and velocity that approaches the speed of light, we introduce the scalar proportionality factor, gamma $\gamma$, which is equal to:  $$\gamma = \frac{1}{\sqrt{1-(\frac{|\vec{v}|}{c})^2}}$$
Using this proportionality factor, we can now write the correct equation to model momentum at all speeds:  $$\vec{p} = \frac{1}{\sqrt{1-(\frac{|\vec{v}|}{c})^2}} m \vec{v}$$ $$\vec{p} = \gamma \times m \vec{v}$$

Generally, gamma is approximately exactly 1 at low speeds, and is only effective at very high speeds.  We can simplify our momentum equations in low speed situations for the sake of ease down to:  $$\vec{p} = 1 \times m \vec{v} = m \vec{v}$$

Momentum is the integral of the net force ($\vec{F}_{net}$)$$\vec{p} = \int \vec{F}_{net}$$Using the definition of acceleration and its relation to velocity ($\vec{a} = \frac{\delta \vec{v}}{\delta t}$), we can define the approximate rate of change of momentum as$$\frac{\delta \vec{p}}{\delta t} = m\vec{a}$$$$\vec{F}_{net} = (\frac{\delta \vec{p}}{\delta t} = m\frac{\delta \vec{v}}{\delta t}) = m\vec{a}$$

Acceleration

Acceleration is the time rate of change of velocity, just as velocity is the time rate of change of position.

Instantaneous acceleration can be calculated as the time rate of change (derivative) of velocity:  $$\vec{a} = \frac{\delta \vec{v}}{\delta t}$$
Average acceleration can be calculated as a change in velocity:  $$\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$$

Position Update Formula

The position update formula simply states that given the initial position ($\vec{r}_i$) of an object, we can predict the final position ($\vec{r}_f$) using the average velocity ($\vec{v}_{avg}$) and change in time ($\Delta t$).

$$\Delta \vec{r} = \vec{v}_{avg} \Delta t$$
$$\vec{r}_f - \vec{r}_i = \vec{v}_{avg}(t_f - t_i)$$

Or just simply

$$\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t$$

Instantaneous Velocity

Instantaneous velocity measures the precise velocity of the ball at a given position.  We can approximate the instantaneous velocity by using the average velocity ($\vec{v}_{avg}$) from one position to another ($\Delta \vec{r}$) over a time interval ($\Delta t$), but it will be unlikely to be accurate.  In order to increase the accuracy of the measurement of the instantaneous velocity, we must make the time interval and position interval increasingly small.  The smaller the position and time interval, the more precise a value we will obtain for the instantaneous velocity:
$$\vec{v} = \lim_{\Delta t \to 0}\frac{\Delta \vec{r}}{\Delta t}$$

Otherwise known in calculus as

$$\vec{v} = \frac{\delta \vec{r}}{\delta t}$$

 As the values become smaller, they become more immediate, or "instantaneous."  The more complicated explanation of this equation:

$$\vec{v} = \frac{\delta\vec{r}}{\delta t} = \frac{\delta}{\delta t}<x, y, z> = <\frac{\delta x}{\delta t}, \frac{\delta y}{\delta t}, \frac{\delta z}{\delta t}> = <v_x, v_y, v_z>$$

This gives us the definition of velocity as being the "time rate of change of position:  $\vec{v} = \frac{\delta \vec{r}}{\delta t}$."

Average Velocity

Velocity is uses to describe the speed and direction of an object.  If we know the object's speed at a certain time as well as the direction it faces, then we can predict where it will be in the future.

A common way to determine the velocity of an object is to find its average velocity ($\vec{v}_{avg}$) - the distance ($\Delta \vec{r}$) it has traveled divided by the elapsed time ($\Delta t$) it took to travel.

$$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$$

This equation can be rearranged to determine the distance traveled, $\Delta \vec{r} = \vec{v}_{avg}\times \Delta t$, and time elapsed, $\Delta t = \frac{\Delta \vec{r}}{\vec{v}_{avg}}$.

In a situation where the rate of change of velocity is not constant, we instead use the equation:$$\vec{v}_{avg} = \frac{\sum\limits_{i=1}^n \vec{v}_i}{n}$$

Euclidean Vectors

Euclidian Vectors are objects/measurements that have magnitude and direction in a space.  A vector is usually represented as $\vec{r}$ where $r$ is any letter.

Magnitude is equivalent to length of the vector - a quantity specifying how long it is from beginning to end when graphically represented in its respective coordinate system.  A magnitude is usually represented using double lines, $|\vec{r}|$, and is always positive.  It is a scalar, a singular value that is not a vector.

The direction is a separate vector that specifies the direction that a vector faces.  The direction of a vector is called a unit vector, and is represented as $\hat{r}$.  It is a different kind of vector that exists purely as a supplement.  For instance, in 3-dimensional space a magnitude could extend any which direction within that space because it is a single number.  It has no context, it is a number .  With the addition of a unit vector, it becomes a posited, measurable quantity that extends as far as the magnitude in the direction of the unit vector.

A vector $\vec{r}$ is the product of the magnitude $|\vec{r}|$ and unit vector $\hat{r}$:

$$\vec{r} = |\vec{r}|\times\hat{r}$$

Each vector is made up of components that each represent a dimension within the Euclidean space it resides.  For instance, in a Cartesian coordinate system with 3-dimensions X, Y, and Z:

 $$\vec{r} = <r_x, r_y, r_z>$$

To find the magnitude of a vector, square each component, sum them together, and then take the square root of the result.

$$|\vec{r}| = \sqrt{r_x^2 + r_y^2 + r_z^2}$$

To find the unit vector (direction) of a vector, divide the vector by its magnitude.  Each component will be divided individually to find the resulting components of the unit vector.  The resulting components should never be more than 1.

$$\hat{r} = \frac{\vec{r}}{|\vec{r}|} = \frac{<r_x, r_y, r_z>}{|\vec{r}|}$$