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Preface

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About this guide
Section One	Units	Linear motion	Force	Momentum	Normal force and friction	Work	Energy
Section Two	Torque	Circular Motion	Fluids	Fields	Gravity	Waves	Wave Overtones	Standing Waves	Sound
Section Three	Thermodynamics	Electricity	Magnetism	Optics
Apendicies	Physics constants	Frictional coefficients	Greek alphabet	Logs	Vectors and scalars

About this guide

Physics Study Guide

This guide is meant as a supplement to a year long freshman level physics course with a trigonometry prerequisite. Some ideas from calculus are included in the book but are not necessary to understand the content. The overview of equations and definitions and eventually sample problem solutions are pertinent to an introductory, college-level physics course suitable for pre-meds. This is not a stand alone textbook rather the intent is to help the student and any other interested person quickly familiarize themselves with concepts and terminology so as to use the appropriate equations to get the desired answers to physics problems.

Contributing

Everyone is encouraged to contribute to the guide. Be bold in your edits! If you have a question about how we do things here look at the Style Guide or post your question on the talk page.

Authors

Karl Wick

Adon Metcalfe

Brendan Abbott

Tristan Sabel

Fromund Hock

Martin Smith-Martinez

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License."

Reviews

Our first review of the Physics Study Guide is in, by email to the author:

Thanks karl!
it's very helpful!!!!

Interwiki

Physics Study Guide

This guide is meant as a supplement to a year long freshman level physics course with a trigonometry prerequisite suitable for pre-meds. This is not a stand alone textbook. The intent is to help the student and any other interested person quickly familiarize themselves with concepts and terminology.

Physics

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Section One

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

The SI System of Measurement

Simple Units

Time

It is defined as the duration between two events. In the international system of measurement (S.I.) the second (s) is the basic unit of time and it is defined as the time it takes a cesium (Cs) atom to perform 9 192 631 770 complete oscillations. The Earth revolves around its own axis in 86400 seconds; this time is known as 1 day, and the 86400th part of one day is known as a second.

Distance

In the international system of measurement (S.I.) the meter (m) is the basic unit of distance and is defined as the distance traveled by light in 1/(299 792 458)th of a second.

Mass

In the international system of measurement (S.I.) the kilogram (kg) is the basic unit of mass and is defined as the mass of a platinum-iridium alloy cylinder kept at the Bureau International des Poids et Mesures in Paris France. See Wikipedia article.

Current

In the international system of measurement (S.I.) the ampere (A) is the basic measure of electrical current. It is defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre (m) apart in vacuum, would produce between these conductors a force equal to 2×10^-7 newton (N) per metre of length.

Unit of Thermodynamic Temperature

The kelvin (K), unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

Unit of Amount of Substance

1. The mole (mol) is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

Luminous Intensity

The candela (cd) is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 10¹² hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

Derived Units

Charge

The SI unit of charge is the coulomb (C). It is equal to 1 A*s (ampere times second).

Velocity

The SI unit for velocity is in m/s or meters per second.

Force

The SI unit of force is the newton (N), named after Sir Isaac Newton. It is equal to $1 \ \mathrm{kg} \cdot \mathrm{m}/\mathrm{s}^2$ .

Energy

The SI unit of energy is the joule (J). The joule has base units of kg·m²/s² = N·m. A joule is defined as the work done or energy required to exert a force of one newton for a distance of one metre. See Wikipedia article.

Prefixes

Prefix	yotta	zetta	exa	peta	tera	giga	mega	kilo	hecto	deca		deci	centi	milli	micro	nano	pico	femto	atto	zepto	yocto
Symbol	Y	Z	E	P	T	G	M	k	h	da		d	c	m	µ	n	p	f	a	z	y
10ⁿ	10²⁴	10²¹	10¹⁸	10¹⁵	10¹²	10⁹	10⁶	10³	10²	10¹	10⁰	10^-1	10^-2	10^-3	10^-6	10^-9	10^-12	10^-15	10^-18	10^-21	10^-24
1000ⁿ	1000⁸	1000⁷	1000⁶	1000⁵	1000⁴	1000³	1000²	1000¹						1000^-1	1000^-2	1000^-3	1000^-4	1000^-5	1000^-6	1000^-7	1000^-8

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

Kinematics

Kinematics is the description of motion. The motion of a point particle is fully described using three terms - displacement, velocity, and acceleration. For real objects (which are not mathematical points), translational kinematics describes the motion of an object's center of mass through space, while angular kinematics describes how an object rotates about its center of mass. In this section, we focus only on translational kinematics.

Displacement, velocity, and acceleration are defined as follows.

Position

Wiktionary defines "vector" as "a quantity that has both magnitude and direction, typically written as a column of scalars". That is, a number that has a direction assigned to it.

In physics, a vector often describes the motion of an object. For example, Warty the Woodchuck goes 35 feet toward a hole in the ground.

We can divide vectors into parts called "components" that each describe a part of the vector. Usually a vector is divided into x and y components.

One dimensional coordinate system

Two dimensional coordinate system

Three dimensional coordinate system

Displacement

$\Delta \vec{x}\equiv\vec x_f- \vec x_i\,$

Displacement answers the question, "Has the object moved?"

Note the $\equiv$ symbol. This symbol is a sort of "super equals" symbol, indicating that not only does $\vec x_f- \vec x_i$ EQUAL the displacement $\Delta\vec{x}$ , but more importantly displacement is OPERATIONALLY DEFINED by $\vec x_f- \vec x_i$ .

We say that $\vec x_f- \vec x_i$ operationally defines displacement, because $\vec x_f- \vec x_i$ gives a step by step procedure for determining displacement. Namely ...

Measure where the object is initially.
Measure where the object is at some later time.
Determine the difference of these two position values.

Be sure to note that DISPLACEMENT is NOT the same as DISTANCE traveled.

For example, imagine traveling one time along the circumference of a circle. If you end where you started, your displacement is zero, even though you have clearly traveled some distance. In fact, displacement is an average distance traveled. On your trip along the circle, your north and south motion averaged out, as did your east and west motion.

Clearly we are losing some important information. The key to regaining this information is to use smaller displacement intervals. For example, instead of calculating your displacement for your trip along the circle in one large step, consider dividing the circle into 16 equal segments. Calculate the distance you traveled along each of these segments, and then add all your results together. Now your total traveled distance is not zero, but something approximating the circumference of the circle. Is your approximation good enough? Ultimately, that depends on the level of accuracy you need in a particular application, but luckily you can always use finer resolution. For example, we could break your trip into 32 equal segments for a better approximation.

Returning to your trip around the circle, you know the true distance is simply the circumference of the circle. The problem is that we often face a practical limitation for determining the true distance traveled. (The traveled path may have too many twists and turns, for example.) Luckily, we can always determine displacement, and by carefully choosing small enough displacement steps, we can use displacement to obtain a pretty good approximation for the true distance traveled. (The mathematics of calculus provides a formal methodology for formally estimating a "true value" through the use of successively better approximations.) In the rest of this discussion, I will replace $Δ$ with $δ$ to indicate that small enough displacement steps have been used to provide a good enough approximation for the true distance traveled.

Velocity

$\vec v_{av}\equiv \frac{\Delta\vec{x}}{\Delta t}$

[Δ, delta, upper-case Greek D, is a prefix conventionally used to denote a difference.] Velocity answers the question "Is the object moving now, and if so - how quickly?"

Once again we have an operational definition: we are told what steps to follow to calculate velocity.

Note that this is a definition for average velocity. The displacement Δx is the vector sum of the smaller displacements which it contains, and some of these may subtract out. By contrast, the distance traveled is the scalar sum of the smaller distances, all of which are non-negative (they are the magnitudes of the displacements). Thus the distance traveled can be larger than the magnitude of the displacement, as in the example of travel on a circle, above. Consequently, the average velocity may be small (or zero, or negative) while the speed is positive.

If we are careful to use very small displacement steps, so that they come pretty close to approximating the true distance traveled, then we can write the definition for INSTANTANEOUS velocity as

$\vec v_{inst}\equiv \frac{\vec{\delta x}}{\delta t}$

[δ is the lower-case delta.] Or with the idea of limits from calculus, we have ...

$\vec v_{inst}\equiv \frac{d \vec x}{dt}$

[d, like Δ and δ, is merely a prefix; however, its use definitely specifies that this is a sufficiently small difference so that the error--due to stepping (instead of smoothly changing) the quantity--becomes negligible.]RLittauer (talk) 19:56, 22 December 2007 (UTC)

Acceleration

$\vec a_{av}\equiv \frac{\vec{v_f}-\vec{v_i}}{t_f-t_i}\equiv \frac{\Delta\vec{v}}{\Delta t}$

Acceleration answers the question "Is the object's velocity changing, and if so - how quickly?"

Once again we have an operational definition. We are told what steps to follow to calculate acceleration.

Again, also note that technically we have a definition for AVERAGE acceleration. As for displacement, if we are careful to use a series of small velocity changes, then we can write the definition for INSTANTANEOUS acceleration as

$\vec a_{inst}\equiv \frac{\delta\vec{v}}{\delta t}$

Or with the help of calculus, we have ...

$\vec a_{inst}\equiv \frac{d \vec v}{dt} = \frac{d^2\vec x}{dt^2}$

Vectors

Notice that the definitions given above for displacement, velocity, and acceleration included little arrows over many of the terms. The little arrow reminds us that direction is an important part of displacement, velocity, the change in velocity, and acceleration. These quantities are VECTORS. By convention, the little arrow always points right when placed over a letter. So for example, $\vec v$ just reminds us that velocity is a vector, and does NOT imply that this particular velocity is rightward. Why do we need vectors? As a simple example, consider velocity. It is not enough to know how fast one is moving. We also need to know which direction we are moving. Less trivially, consider how many different ways an object could be experiencing an acceleration (a change in its velocity). Ultimately there are three distinct ways an object could accelerate.

The object could be speeding up.
The object could be slowing down.
The object could be traveling at constant speed, while changing its direction of motion.

(More general accelerations are simply combinations of 1 and 3 or 2 and 3).

Importantly, a change in the direction of motion is just as much an acceleration as is speeding up or slowing down. In classical mechanics, no direction is associated with time (you cannot point to next Tuesday). So the definition of $\vec a_{av}$ tells us that acceleration will point wherever the CHANGE in velocity $\Delta\vec{v}$ points. Understanding that the direction of $\Delta \vec{v}$ determines the direction of $\vec a$ leads to three non-mathematical but very powerful rules of thumb.

If the velocity and acceleration of an object point in the same direction, the object's speed is increasing.
If the velocity and acceleration of an object point in opposite directions, the object's speed is decreasing.
If the velocity and acceleration of an object are perpendicular to each other, the object's speed stays constant, while the object's direction of motion changes.

(Again, more general motion is simply a combination of 1 and 3 or 2 and 3.)

Using these three simple rules will dramatically help your intuition of what is happening in a particular problem. In fact, much of the first semester of college physics is simply the application of these three rules in different formats.

Equations of motion : Constant acceleration

A particle is said to move with constant acceleration if its velocity changes by equal amounts in equal intervals of time, no matter how small the intervals may be.

$\frac{d \vec a}{dt} = 0\ \mathrm{m\ s^{-2}}$

Since acceleration is a vector constant acceleration means that both direction and magnitude of this vector don't change during the motion. This means that average and instantaneous acceleration are equal. We can use that to derive an equation for velocity as a function of time by integrating the constant acceleration.

$\boldsymbol{v}(t)=\boldsymbol{v}(0)+\int\limits_{0}^{t}\boldsymbol{a}\ dt$

Giving the following equation for velocity as a function of time.

$\boldsymbol{v}(t)=\boldsymbol{v}_0+\boldsymbol{a}t$

To derive the equation for position we simply integrate the equation for velocity.

$\boldsymbol{x}(t)=\boldsymbol{x}(0)+\int\limits_{0}^{t}\boldsymbol{v}(t)\ dt$

Integrating again gives the equation for position.

$\boldsymbol{x}(t)=\boldsymbol{x}_0+\boldsymbol{v}_0t+\frac{1}{2}\boldsymbol{a}t^2$

The following are the 'Equations of Motion'. They are simple and obvious equations if you think over them for a while.

Equations of Motion
Equation	Description
$\vec{x}=\vec{x}_0 + \vec{v}_0 t+\frac{\vec{a}t^2}{2} \$	Position as a function of time
$\vec v = \vec v_0 + \vec a t \$	Velocity as a function of time
The following equations can be derived from the two equations above by combining them and eliminating variables.
$v^2 = v_0^2 + 2\vec{a}\cdot(\vec{x}-\vec{x}_0) \$	Eliminating time (Very useful, see the section on Energy)
$\vec{x}=\vec{x}_0+\frac{\vec{v}_0t+\vec{v}t}{2}$	Eliminating acceleration

Key to Symbols
Symbol	Description
$\vec{v}$	velocity at time t
$\vec{v_0}$	initial velocity
$\vec{a}$	acceleration (constant)
$t\$	time taken during the motion
$\vec{x}$	position at time t
$\vec{x_0}$	initial position

Acceleration in One Dimension

(Needs content)...

Acceleration in Two Dimensions

(Needs content)

Acceleration in Three Dimensions

(Needs content)

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

Force

A net force on a body causes a body to accelerate. The amount of that acceleration depends on the body's inertia (or its tendency to resist changes in motion), which is measured as its mass. When Isaac Newton formulated Newtonian mechanics, he discovered three fundamental laws of motion.

Later, Albert Einstein proved that these laws are just a convenient approximation. These laws, however, greatly simplify calculations and are used when studying objects at velocities that are small compared with the speed of light.

Friction

It is the force that opposes relative motion or tendency of relative motion between two surfaces in contact represented by f. When two surfaces move relative to each other or they have a tendency to move relative to each other, at the point (or surface) of contact, there appears a force which opposes this relative motion or tendency of relative motion between two surface in contact. It acts on both the surfaces in contact with equal magnitude and opposite direction (Newton's 3rd law). Friction force tries to stop relative motion between two surfaces in contact, if it is there, and when two surfaces in contact are at rest relative to each other, the friction force tries to maintain this relative rest. Friction force can assume the magnitude (below a certain maximum magnitude called limiting static friction) required to maintain relative rest between two surfaces in contact. Because of this friction force is called a self adjusting force.

Earlier, it was believed that friction was caused due to the roughness of the two surfaces in contact with each other. However, modern theory stipulates that the cause of friction is the Coulombic force between the atoms present in the surface of the regions in contact with each other.

Formula: Limiting Friction = (Friction Coefficient)(Normal reaction)

Static Friction = the friction force that keeps an object at relative rest.

Kinetic Friction = sliding friction

Newton's First Law of Motion

(The Law of Inertia)

A static object with no net force acting on it remains at rest or if in movement it will maintain a constant velocity

This means, essentially, that acceleration does not occur without the presence of a force. The object tends to maintain its state of motion. If it is at rest, it remains at rest and if it is moving with a velocity then it keeps moving with the same velocity. This tendency of the object to maintain its state of motion is greater for larger mass. The "mass" is, therefore, a measure of the inertia of the object

In a state of equilibrium, where the object is at rest or proceeding at a constant velocity, the net force in every direction must be equal to 0.

At a constant velocity (including zero velocity), the sum of forces is 0. If the sum of forces does not equal zero, the object will accelerate (change velocity over time).

It is important to note, that this law is applicable only in non-accelerated coordinate systems. It is so, because the perception of force in accelerated systems are different. A body under balanced force system in one frame of reference, for example a person standing in an accelerating lift, is acted upon by a net force in the earth's frame of reference.

Inertia is the tendency of an object to maintain its velocity i.e. to resist acceleration.

Inertia is not a force.
Inertia varies directly with mass.

Newton's Second Law of Motion

The time rate of change in momentum is proportional to the applied force and takes place in the direction of the force.

'The acceleration of an object is proportional to the force acting upon it.

These two statements mean the same thing, and is represented in the following basic form (the system of measurement is chosen such that constant of proportionality is 1) :

$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m\vec{v})$

The product of mass and velocity i.e. mv is called the momentum. The net force on a particle is ,thus, equal to rate change of momentum of the particle with time. Generally mass of the object under consideration is constant and thus can be taken out of the derivative.

$\vec{F} = m \frac{\mathrm{d}}{\mathrm{d}t}(\vec{v})= m\vec{a}$

Force is equal to mass times acceleration. This version of Newton's Second Law of Motion assumes that the mass of the body does not change with time, and as such, does not represent a general mathematical form of the Law. Consequently, this equation cannot, for example, be applied to the motion of a rocket, which loses its mass (the lost mass is ejected at the rear of the rocket) with the passage of time.

An example: If we want to find out the downward force of gravity on an object on Earth, we can use the following formula:

$\|\vec{F}\| = m\|\vec{g}\|$

Hence, if we replace m with whatever mass is appropriate, and multiply it by 9.806 65 m/s², it will give the force in newtons that the earth's gravity has on the object in question(in other words, the body's weight).

Newton's Third Law of Motion

Forces occur in pairs equal in magnitude and opposite in direction

This means that for every force applied on a body A by a body B, body B receives an equal force in the exact opposite direction. This is because forces can only be applied by a body on another body. It is important to note here that the pair of forces act on two different bodies, affecting their state of motion. This is to emphasize that pair of equal forces do not cancel out.

There are no spontaneous forces.

It is very important to note that the forces in a "Newton 3 pair", described above, can never act on the same body. One acts on A, the other on B. A common error is to imagine that the force of gravity on a stationary object and the "contact force" upwards of the table supporting the object are equal by Newton's third law. This is not true. They may be equal - but because of the second law (their sum must be zero because the object is not accelerating), not because of the third.

The "Newton 3 pair" of the force of gravity (= earth's pull) on the object is the force of the object attracting the earth, pulling it upwards. The "Newton 3 pair" of the table pushing it up is that it, in its turn, pushes the table down.

Equations

To find Displacement

$\Delta x=\vec{v}_{i}(\mathrm t) + \frac{1}{2}(\vec{a})(\mathrm t)^{2}$

To find Final Velocity

$\vec{v}_{f}=\vec{v}_{i}(\mathrm t) + (\vec{a})(\mathrm t)$

To find Final Velocity

$(\vec{v}_{f})^{2}=(\vec{v}_{i})^{2} + 2(\vec{a})(\Delta x)$

To find Force when mass is changing

$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m\vec{v})$

To find Force when mass is a constant

$\vec{F} = m \frac{\mathrm{d}}{\mathrm{d}t}(\vec{v})= m\vec{a}$

Variables
$\vec{F}$ Force (N) $m\$ Mass (kg) $\vec{a}$ Acceleration (m/s²) $\vec{p}$ Momentum (kg m/s) $t\$ time (s) $\vec{T}$ Tension (N) $\vec{g}$ Acceleration due to gravity near the earth's surface ( $\\|\vec{g}\\|=9.806\ 65\mathrm{m/s^2}$ see Physics Constants)

Definitions
Force (F): Force is equal to rate change of momentum with time. (Newton’s second law). A vector. Units: newtons (N) The newton (N): defined as the force it takes to accelerate one kilogram one metre per second squared (U.S. meter per second squared), that is, the push it takes to speed up one kilogram from rest to a velocity of 1 m/s in 1 second $1 \mathrm{N} = 1 \mathrm{kg}\cdot\mathrm{m}/\mathrm{s}^2$ Mass (m) : Also called inertia. The tendency of an object to resist change in its motion, and its response to a gravitational field. A scalar. Units: kilograms (kg) Acceleration (a): Change in velocity (Δv) divided by time (t). A vector. Units: meters per second squared (U.S. meters per seconds squared) (m/s²) Momentum (p): Mass times velocity. Expresses the motion of a body and its resistance to changing that motion. A vector. Units: kg m/s

Definitions

Force (F): Force is equal to rate change of momentum with time. (Newton’s second law). A vector. Units: newtons (N)

The newton (N): defined as the force it takes to accelerate one kilogram one metre per second squared (U.S. meter per second squared), that is, the push it takes to speed up one kilogram from rest to a velocity of 1 m/s in 1 second $1 \mathrm{N} = 1 \mathrm{kg}\cdot\mathrm{m}/\mathrm{s}^2$

Mass (m) : Also called inertia. The tendency of an object to resist change in its motion, and its response to a gravitational field. A scalar. Units: kilograms (kg)

Acceleration (a): Change in velocity (Δv) divided by time (t). A vector. Units: meters per second squared (U.S. meters per seconds squared) (m/s²)

Momentum (p): Mass times velocity. Expresses the motion of a body and its resistance to changing that motion. A vector. Units: kg m/s

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

Momentum

Linear momentum

$\vec{p} = m\vec{v}$

Momentum is equal to mass times velocity.

Angular momentum

$\vec{L} = \vec{r}\times\vec{p} = m\vec{r}\times\vec{v}$

Angular momentum of an object revolving around an external axis O is equal to the cross-product of the position vector with respect to O and its linear momentum.

$\vec{L} = I\vec{\omega}$

Angular momentum of a rotating object is equal to the moment of inertia times angular velocity.

Force and linear momentum, torque and angular momentum

$\vec{F} = \frac{\Delta\vec{p}}{\Delta t}$

is equal to the change in linear momentum over the change in time.

$\vec{\tau} = \frac{\Delta\vec{L}}{\Delta t}$

Net torque is equal to the change in angular momentum over the change in time.

Conservation of momentum

$\vec{L}_i = \vec{L}_f$

Let us prove this law.

We'll take two particles, say, a and b. Their momentums are $\vec p_a$ and $\vec p_b$ .They are moving opposite to each other along the x-axis and they collide. Now force is given by:

$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$

According to Newton's third law,the forces on each particle are equal and opposite.So,

$\frac{\mathrm{d}\vec{p}_a}{\mathrm{d}t} =- \frac{\mathrm{d} \vec{p}_b}{\mathrm{d}t}$

Rearranging,

$\frac{d (\vec{p}_a + \vec{p}_b)}{\mathrm{d}t} =0$

This means that the sum of the momentums does not change with time. Therefore, the law is proved.

Variables

p: momentum, (kg·m/s)
m: mass, (kg)
v: velocity (m/s)
L: angular momentum, (kg·m²/s)
I: moment of inertia, (kg·m²)
ω: angular velocity (rad/s)
α: angular acceleration (rad/s²)
F: force (N)
t: time (s)
r: position vector (m)

Bold denotes a vector quantity.
Italics denotes a scalar quantity.

Definition of terms

Momentum (p): Mass times velocity. (kg·m/s)

Mass (m) : A quantity that describes how much material exists, or how the material responds in a gravitational field. Mass is a measure of inertia. (kg)

Velocity (v): Displacement divided by time (m/s)

Angular momentum (L): A vector quantity that represents the tendency of an object in circular or rotational motion to remain in this motion. (kg·m²/s)

Moment of inertia (I): A scalar property of a rotating object. This quantity depends on the mass of the object and how it is distributed. The equation that defines this is different for differently shaped objects. (kg·m²)

Angular speed (ω): A scalar measure of the rotation of an object. Instantaneous velocity divided by radius of motion (rad/s)

Angular velocity (ω): A vector measure of the rotation of an object. Instantaneous velocity divided by radius of motion, in the direction of the axis of rotation. (rad/s)

Force (F): mass times acceleration, a vector. Units: newtons (N)

Time (t) : (s)

Isolated system: A system in which there are no external forces acting on the system.

Position vector (r): a vector from a specific origin with a magnitude of the distance from the origin to the position being measured in the direction of that position. (m)

Calculus-based Momentum

$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$

Force is equal to the derivative of linear momentum with respect to time.

$\vec{\tau} = \frac{\mathrm{d}\vec{L}}{\mathrm{d}t}$

Torque is equal to the derivative of angular momentum with respect to time.

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

The Normal Force

Why is it that we stay steady in our chairs when we sit down? According to the first law of motion, if an object is translationally in equilibrium (velocity is constant), the sum of all the forces acting on the object must be equal to zero. For a person sitting on a chair, it can thus be postulated that a normal force is present balancing the gravitational force that pulls the sitting person down. However, it should be noted that only some of the normal force can cancel the other forces to zero like in the case of a sitting person. In Physics, the term normal as a modifier of the force implies that this force is acting perpendicular to the surface at the point of contact of the two objects in question. Imagine a person leaning on a vertical wall. Since the person does not stumble or fall, he/she must be in equilibrium. Thus, the component of his/her weight along the horizontal is balanced or countered (opposite direction) by an equal amount of force -- this force is the normal force on the wall. So, on a slope, the normal force would not point upwards as on a horizontal surface but rather perpendicular to the slope surface.

The normal force can be provided by any one of the four fundamental forces, but is typically provided by electromagnetism since microscopically, it is the repulsion of electrons that enables interaction between surfaces of matter. There is no easy way to calculate the normal force, other than by assuming first that there is a normal force acting on a body in contact with a surface (direction perpendicular to the surface). If the object is not accelerating (for the case of uniform circular motion, the object is accelerating) then somehow, the magnitude of the normal force can be solved. In most cases, the magnitude of the normal force can be solved together with other unknowns in a given problem.

Sometimes, the problem does not warrant the knowledge of the normal force(s). It is in this regard that other formalisms (e.g. Lagrange method of undertermined coefficients) can be used to eventually solve the physical problem.

Friction

When there is relative motion between two surfaces, there is a resistance to the motion. This force is called friction. Friction is the reason why people could not accept Newton's first law of Motion, that an object tends to keep its state of motion. Friction acts opposite to the direction of the original force. The frictional force is equal to the frictional coefficient times the normal force.

Friction is caused due to attractive forces between the molecules near the surfaces of the objects. If two steel plates are made really flat and polished and cleaned and made to touch in a vacuum, it bonds together. It would look as if the steel was just one piece. The bonds are formed as in a normal steel piece. This is called cold welding. And this is the main cause of friction.

The above equation is an empirical one--in general, the frictional coefficient is not constant. However, for a large variety of contact surfaces, there is a well characterized value. This kind of friction is called Coulomb friction. There is a separate coefficient for both static and kinetic friction. This is because once an object is pushed on, it will suddenly jerk once you apply enough force and it begins to move.

Also, the frictional coefficient varies greatly depending on what two substances are in contact, and the temperature and smoothness of the two substances. For example, the frictional coefficients of glass on glass are very high. When you have similar materials, in most cases you don't have Coulomb friction.

For static friction, the force of friction actually increases proportionally to the force applied, keeping the body immobile. Once, however, the force exceeds the maximum frictional force, the body will begin to move. The maximum frictional force is calculated as follows:

$\left|\vec{F}_f\right| \leq \mu_s\left|\vec{N}\right|$

The static frictional force is less than or equal to the coefficient of static friction times the normal force. Once the frictional force equals the coefficient of static friction times the normal force, the object will break away and begin to move.

Once it is moving, the frictional force then obeys:

$\left|\vec{F}_f\right| = \mu_k\left|\vec{N}\right|$

The kinetic frictional force is equal to the coefficient of kinetic friction times the normal force. As stated before, this always opposes the direction of motion.

Variables

Symbol	Units	Definition
$\vec{F}_f$	$\mathrm{N}\$	Force of friction
$\mu\$	none	Coefficient of friction

Definition of Terms

Normal force (N): The force on an object perpendicular to the surface it rests on utilized in order to account for the body's lack of movement. Units: newtons (N)

Force of friction (F_f): The force placed on a moving object opposite its direction of motion due to the inherent roughness of all surfaces. Units: newtons (N)

Coefficient of friction (μ): The coefficient that determines the amount of friction. This varies tremendously based on the surfaces in contact. There are no units for the coefficient of either static or kinetic friction

It's important to note, that in real life we often have to deal with viscose and turbulent friction - they appear when you move the body through the matter.

Viscose friction is proportional to velocity and takes place at approximately low speeds. Turbulent friction is proportional to $V 2$ and takes place at higher velocities.

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

Work

Work is equal to the scalar product of force and displacement.

$W = \vec{F}\cdot\vec{d}$

The scalar product of two vectors is defined as the product of their lengths with the cosine of the angle between them. Work is equal to force times displacement times the cosine of the angle between the directions of force and displacement.

$W =\|\vec{F}\|\ \|\vec{d}\|\cos\theta$

Work is equal to change in kinetic energy plus change in potential energy for example the potenitial energy due to gravity.

$W = \Delta\mathrm{KE}+ \Delta\mathrm{PE}_g\$

Work is equal to average power times time.

$W = Pt\$

The Work done by a force taking something from point 1 to point 2 is

$W_{1,2}=\int_{\vec{x}_1}^{\vec{x}_2}\vec{F}\cdot d\vec{l}$

Work is in fact just a transfer of energy. When we 'do work' on an object, we transfer some of our energy to it. This means that the work done on an object is its increase in energy. Actually, the kinetic energy and potential energy is measured by calculating the amount of work done on an object. The gravitational potential energy (there are many types of potential energies) is measure as 'mgh'. mg is the weight/force. And h is the distance. The product is nothing but the work done. Even kinetic energy is a simple deduction from the laws of linear motion. Try substituing for v^2 in the formula for kinetic energy.

Variables

W: Work (J)
F: Force (N)
d: Displacement (m)

Definition of terms

Work (W): Force times distance. Units: joules (J)

Force (F): mass times acceleration (Newton’s classic definition). A vector. Units: newtons (N)

When work is applied to an object or a system it adds or removes kinetic energy to or from that object or system. More precisely, a net force in one direction, when applied to an object moving opposite or in the same direction as the force, kinetic energy will be added or removed to or from that object. Note that work and energy are measured in the same unit, the joule (J).

Advanced work topics

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

Energy

Kinetic energy is equal to one-half of mass times the square of velocity.

$\mathrm{KE}=\frac{1}{2}m\|\vec{v}\|^2$

Kinetic energy is equal to one-half of moment of inertia times the square of angular velocity.

$\mathrm{KE}=\frac{1}{2}I\omega^2$

Potential energy due to gravity is equal to the product of mass, acceleration due to gravity, and height (elevation) of the object.

$\mathrm{PE}_g= -m\vec{g}\cdot\vec{x}=mgy$

Potential energy due to spring deformation is equal to one-half the product of the spring constant times the square of the change in length of the spring.

$\mathrm{PE}_e=\frac{1}{2}k\|\vec{x}-\vec{x}_e\|^2=\frac{1}{2}k\|\Delta\vec{x}\|^2$

Definition of terms

Energy: a theoretically indefinable quantity that describes potential to do work. SI unit for energy is the joule (J). Also common is the calorie (cal).

The joule: defined as the energy needed to push with the force of one newton over the distance of one meter. Equivalent to one newton-meter (N·m) or one watt-second (W·s).

1 joule = 1 J = 1 newton • 1 meter = 1 watt • 1 second

Energy comes in many varieties, including Kinetic energy, Potential energy, and Heat energy.

Kinetic energy (K): The energy that an object has due to its motion. Half of velocity squared times mass. Units: joules (J)

Potential energy due to gravity (U_G): The energy that an object has stored in it by elevation from a mass, such as raised above the surface of the earth. This energy is released when the object becomes free to move. Mass times height time acceleration due to gravity. Units: joules (J)

Potential energy due to spring compression (U_E): Energy stored in spring when it is compressed. Units: joules (J)

Heat energy (Q): Units: joules (J)

Spring compression (D_x): The difference in length between the spring at rest and the spring when stretched or compressed. Units: meters (m)

Spring constant (k): a constant specific to each spring, which describes its “springiness”, or how much work is needed to compress the spring. Units: newtons per meter (N/m)

Change in spring length (Δx): The distance between the at-rest length of the spring minus the compressed or extended length of the spring. Units: meters (m)

Moment of inertia (I): Describes mass and its distribution. (kg•m²)

Angular momentum (ω): Angular velocity times mass (inertia). (rad/s)

Section Two

Rotational Kinematics

Rotational Kinematics concerns with the description of spinning bodies. The orientation of any object can be described by three angular quantities called the Euler angles.

A two dimensional polar co-ordinate system

A three dimentional spherical-polar co-ordinate system

Euler angles which uniquely describe the orientation of an object in space

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones	Standing Waves	Sound
Thermodynamics	Electricity	Magnetism	Optics
Physical Constants	Frictional Coefficients	Greek Alphabet	Logarithms	Vectors and Scalars	Other Topics

Torque and Circular Motion

Circular motion is the motion of a particle at a set distance (called radius) from a point. For circular motion, there needs to be a force that makes the particle turn. This force is called the 'centripetal force.' Please note that the centripetal force is not a new type of force-it is just a force causing rotational motion. To make this clearer, let us study the following examples:

If Stan ties a piece of thread to a small pebble and rotates it in a horizontal circle above his head, the circular motion of the pebble is caused by the tension force in the thread.
In the case of the motion of the planets around the sun (which is roughly circular), the force is provided by the gravitational force exerted by the sun on the planets.

Thus, we see that the centripetal force acting on a body is always provided by some other type of force -- centripetal force, thus, is simply a name to indicate the force that provides this circular motion. This centripetal force is always acting inward toward the center. You will know this if you swing an object in a circular motion. If you notice carefully, you will see that you have to continously pull inward. We know that an opposite force should exist for this centripetal force(by Newton's 3rd Law of Motion). This is the centrifugal force, which exists only if we study the body from a non-inertial frame of reference(an accelerating frame of reference,such as in circular motion). This is a so-called 'pseudo-force', which is used to make the Newton's law applicable to the person who is inside a non-inertial frame . The centrifugal force is equal and opposite to the centripetal force. It is caused due to inertia of a body.

$\omega_{\mathrm{avg} }=\frac{ \omega_1 +\omega_f }{2} =\frac{\theta }{t}$

Average angular velocity is equal to one-half of the sum of initial and final angular velocities assuming constant acceleration, and is also equal to the angle gone through divided by the time taken.

$\alpha =\frac{ \Delta \omega }{t}$

Angular acceleration is equal to change in angular velocity divided by time taken.

Angular momentum

$\mathbf{l} = \mathbf{r}\times\mathbf{p} = m(\mathbf{r}\times\mathbf{v})$

Angular momentum of an object revolving around an external axis O is equal to the cross-product of the position vector with respect to O and its linear momentum.

$\mathbf{L} = I\boldsymbol{\omega}$

Angular momentum of a rotating object is equal to the moment of inertia times angular velocity.

$L = I ω$

$\tau =I\alpha =\frac{\Delta L}{t}$

Torque is equal to moment of inertia times angular acceleration, which is also equal to the change in angular momentum divided by time taken.

$K_R =\frac{1}{2} I\omega^2$

Rotational Kinetic Energy is equal to one-half of the product of moment of inertia and the angular velocity squared.

IT IS USEFUL TO NOTE THAT

The equations for rotational motion are analogous to those for linear motion-just look at those listed above. When studying rotational dynamics, remember:

the place of force is taken by torque

the place of mass is taken by moment of inertia

the place of displacement is taken by angle

the place of linear velocity, momentum, acceleration, etc. is taken by their angular counterparts.

Variables

τ: torque, (N·m)
I: moment of inertia, (kg·m²)
α: angular acceleration, (rad/s²)
L: angular momentum, (kg·m²/s)
t: time (s)
K_r: rotational kinetic energy, (J = kg·m²/s²)
ω: angular velocity, (rad/s)

Definition of terms

Torque (τ): Force times distance. A vector. (N·m)

Moment of inertia (I): Describes the object's resistance to torque - the rotational analog to inertial mass. (kg·m²)

Angular momentum (L): (kg·m²/s)

Angular velocity (ω): (rad/s)

Angular acceleration (α): (rad/s²)

Time (t): (s)

Physics Study Guide (Print Version)

Units	Linear Motion	Force	Momentum	Normal Force and Friction	Work	Energy
Torque & Circular Motion	Fluids	Fields	Gravity	Waves	Wave overtones