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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|
|Apendicies||Physics constants||Frictional coefficients||Greek alphabet||Logs||Vectors and scalars|
About this guide
'I have a friend who's an artist and he's some times taken a view which I don't agree with very well. He'll hold up a flower and say, "look how beautiful it is," and I'll agree, I think. And he says, "you see, I as an artist can see how beautiful this is, but you as a scientist, oh, take this all apart and it becomes a dull thing." And I think he's kind of nutty. First of all, the beauty that he sees is available to other people and to me, too, I believe, although I might not be quite as refined aesthetically as he is. But I can appreciate the beauty of a flower. At the same time, I see much more about the flower that he sees. I could imagine the cells in there, the complicated actions inside which also have a beauty. I mean, it's not just beauty at this dimension of one centimeter: there is also beauty at a smaller dimension, the inner structure...also the processes. The fact that the colors in the flower are evolved in order to attract insects to pollinate it is interesting -- it means that insects can see the color. It adds a question -- does this aesthetic sense also exist in the lower forms that are...why is it aesthetic, all kinds of interesting questions which a science knowledge only adds to the excitement and mystery and the awe of a flower. It only adds. I don't understand how it subtracts.' ... Richard Feynman
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.
|Karl Wick||Adon Metcalfe||Brendan Abbott||Tristan Sabel||Fromund Hock||Martin Hoecker-Martinez|
The SI System of Measurement
These are basic units upon which most units depends.
Time 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 with respect to the Sun; this is known as 1 day, and the 86400th part of one day is known as a second.
In the international system of measurement (S.I.) the metre (m) ('meter' in the US) is the basic unit of length and is defined as the distance travelled by light in a vacuum in 1/299,792,458 second. This definition establishes that the speed of light in a vacuum is precisely 299,792,458 metres per second.
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 specific platinum-iridium alloy cylinder kept at the Bureau International des Poids et Mesures in Sèvres, France. A duplicate of the Sèvres cylinder is kept at the National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland. See Wikipedia article.
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.
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. (A steradian (sr) is the SI unit of solid angle, equal to the angle at the centre of a sphere subtended by a part of the surface equal in area to the square of the radius.)
These are units obtained by combining two or more fundamental units.
The SI unit of charge is the coulomb (C). It is equal to ampere times second:
The SI unit for velocity is in m/s or metres per second.
The SI unit of force is the newton (), named after Sir Isaac Newton. It is equal to .
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.
The SI unit of pressure is the pascal (Pa). The pascal has base units of or . See Wikipedia article.
The SI units are not always convenient to use, even with the larger (and smaller) prefixes. For astronomy, the following units are prevalent:
The Julian year is defined by the IAU as exactly 365.25 days, a day being exactly 60*60*24 = 86,400 SI seconds. The Julian year is therefore equal to 31,557,600 seconds.
The Astronomical Unit (au or ua), often used for measuring distances in the Solar system, is the average distance from the Earth to the Sun. It is 149,597,870,691 m, ± 30 m, as currently defined.
The light year (ly) is defined as the distance light travels in a homogeneous isotopic non-attenuating medium (a vacuum) in one Julian year. Due to the word "year", the light year is often mistaken for a unit of time in popular culture. It is, however, a unit of length (distance), and is equal to exactly 9,460,730,472,580,800 m.
The parsec (pc), or "parallax second", is the distance of an object that appears to move two arc-seconds against the background stars as the Earth moves around the sun, or by definition one arc-second of parallax angle. This angle is measured in reference to a line connecting the object and the Sun, and thus the apparent motion is one arc-second on either side of this "central" position. The parsec is approximately 3.26156 ly.
Kinematics is the description of motion. The motion of a point particle is fully described using three terms - position, 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 centre of mass. In this section, we focus only on translational kinematics. Position, displacement, velocity, and acceleration are defined as follows.
"Position" is a relative term that describes the location of an object RELATIVE to some chosen stationary point that is usually described as the "origin".
A vector is 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 10 meters towards a hole in the ground.
We can divide vectors into parts called "components", of which the vector is a sum. For example, a two-dimensional vector is divided into x and y components.
Displacement answers the question, "Has the object moved?"
Note the symbol. This symbol is a sort of "super equals" symbol, indicating that not only does EQUAL the displacement , but more importantly displacement is operationally defined by .
We say that operationally defines displacement, because gives a step by step procedure for determining displacement.
- Measure where the object is initially.
- Measure where the object is at some later time.
- Determine the difference between these two position values.
Be sure to note that displacement is not the same as distance travelled.
For example, imagine travelling one time along the circumference of a circle. If you end where you started, your displacement is zero, even though you have clearly travelled some distance. In fact, displacement is an average distance travelled. 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 travelled along each of these segments, and then add all your results together. Now your total travelled 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 travelled. (The travelled 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 travelled. (The mathematics of calculus provides a formal methodology for estimating a "true value" through the use of successively better approximations.) In the rest of this discussion, I will replace with to indicate that small enough displacement steps have been used to provide a good enough approximation for the true distance travelled.
[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 travelled is the scalar sum of the smaller distances, all of which are non-negative (they are the magnitudes of the displacements). Thus the distance travelled 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 travelled, then we can write the definition for instantaneous velocity as
[δ is the lower-case delta.] Or with the idea of limits from calculus, we have:
[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.]
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:
Or with the help of calculus, we have:
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, and acceleration. These quantities are vectors. By convention, the little arrow always points right when placed over a letter. So for example, 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 tells us that acceleration will point wherever the change in velocity points. Understanding that the direction of determines the direction of 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 initial speed stays constant (in that initial direction), while the speed of the object in the direction of the acceleration increases. Think of a bullet fired horizontally in a vertical gravitational field. Since velocity in the one direction remains constant, and the velocity in the other direction increases, the overall velocity (absolute velocity) also increases.
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
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.
Giving the following equation for velocity as a function of time.
To derive the equation for position we simply integrate the equation for velocity.
Integrating again gives the equation for position.
The following are the equations of motion:
|Position as a function of time|
|Velocity as a function of time|
The following equations can be derived from the two equations above by combining them and eliminating variables.
|Eliminating time (very useful, see the section on Energy)|
|velocity (at time t)|
|time (taken during the motion)|
|position (at time t)|
Acceleration in One Dimension
Acceleration in Two Dimensions
Acceleration in Three Dimensions
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What does force in motion mean?
Force means strength and power. Motion means movement. That’s why we need forces and motions in our life. We need calculation when we want to know how fast things go, travel and other things which have force and motion. ...
How do we calculate the speed?
If you want to calculate the average speed, distance travelled or time taken you need to use this formula and remember it:
This is an easy formula to use, you can find the distance travelled, time taken or average speed, you need at least 2 values to find the whole answer.
Is velocity the same thing as speed?
Velocity is a vector quantity that refers to "the rate at which an object changes its position", whereas speed is a scalar quantity, which cannot be negative. Imagine a kid moving rapidly, one step forward and one step back, always returning to the original starting position. While this might result in a frenzy activity, it would result in a zero velocity, because the kid always returns to the original position, the motion would never result in a change in position, in other words would be zero.
Speed is measured in the same physical units of measurement as velocity, but does not contain an element of direction. Speed is thus the magnitude component of velocity. Velocity contains both the magnitude and direction components. You can think of velocity as the displacement/duration, whereas speed can be though as distance/duration.
When a car is speeding up we say that it is accelerating, when it slows down we say it is decelerating.
How do we calculate it?
When we want to calculate it, the method goes like that: A lorry driver brakes hard, and slows from 25 m/s to 5 m/s in 5 seconds. What was the vehicle's acceleration?
What is initial velocity and final velocity? Initial velocity is the beginning before motion starts or in the middle of the motion, final velocity is when the motion stops.
There is another way to calculate it and it is like that This equations which are written is the primary ones, which means that when you don’t have lets say final velocity, how will you calculate the equation?
This is the way you are going to calculate.
When you want to know how fast an athletic person is running, what you need is a stopwatch in your hand, then when the person starts to run, you start the stopwatch and when the person who is sprinting stops at the end point, you stop the watch and see how fast he ran, and if you want to see if the athlete is wasting his energy, while he is running look at his movement, and you will know by that if he is wasting his energy or not.
This athletic person is running, and while he is running the scientist could know if he was wasting his energy if they want by the stop watch and looking at his momentum.
Take a slope, a trolley, some tapes and a stop watch, then put the tapes on the slope and take the trolley on the slope, and the stopwatch in your hand, as soon as you release the trolley, start timing the trolley at how fast it will move, when the trolley stops at the end then stop the timing. After wards, after seeing the timing , record it, then you let the slope a little bit high, and you will see, how little by little it will decelerate.
Isaac Newton was an English physicist, mathematician (described in his own day as a "natural philosopher") , astronomer and alchemist. Newton is one of the most influential scientists of all time, and he is known, among other things, for contributing to development of classical mechanics and for inventing, independently from Gottfried Leibniz, calculus.
Newton's laws of motion
Newton is also known by his three laws of motion, which describe the relationship between a body and the forces acting upon it, and its motion in response to said forces.
- First Law (also known as the law of inertia) states that every body continues in its state of rest or state of uniform motion unless compelled to change that state by being subject to an external force. The moment of inertia is defined as the tendency of matter to resist any change in its state of motion or state of rest.
- Second Law The vector sum of the external forces on an object is equal to the mass of that object multiplied by the acceleration vector of the object, or algebraically .
- Third Law states that when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
Some useful symbols seen and that we will see:
|Change in velocity|
A force is any interaction that tends to change the motion of an object. In other words, a force can cause an object with mass to change its velocity. Force can also be described by intuitive concepts such as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol .
How to calculate the force?
When we want to calculate the force, and we have the mass and acceleration, we can simply use the simple formula stated in the Newton's second law above, that is , where is the mass (or the amount of matter in a body), and is the acceleration. Note that the Newton’s second law is defined as a numerical measure of inertia.
What is inertia?
Inertia is the tendency of a body to maintain its state of rest or uniform motion, unless acted upon by an external force.
Robert Hooke was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work.
Hooke's law is a principle of physics that states that the force needed to extend or compress a spring by some distance is proportional to that distance, or algebraically , where is a constant factor characteristic of the spring, its stiffness.
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.
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 surfaces in contact. It acts on both the surfaces in contact with equal magnitude and opposite directions (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) :
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.
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:
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.
To find Displacement
To find Final Velocity
To find Final Velocity
To find Force when mass is changing
To find Force when mass is a constant
Momentum is equal to mass times velocity.
Angular momentum of an object revolving around an external axis is equal to the cross-product of the position vector with respect to and its linear momentum.
Angular momentum of a rotating object is equal to the moment of inertia times angular velocity.
Force and linear momentum, torque and angular momentum
Net force is equal to the change in linear momentum over the change in time.
Net torque is equal to the change in angular momentum over the change in time.
Conservation of momentum
Let us prove this law.
We'll take two particles . Their momentums are . They are moving opposite to each other along the -axis and they collide. Now force is given by:
According to Newton's third law, the forces on each particle are equal and opposite.So,
This means that the sum of the momentums does not change with time. Therefore, the law is proved.
p: momentum, (kg·m/s)
Definition of terms
Momentum (p): Mass times velocity. (kg·m/s)
Force is equal to the derivative of linear momentum with respect to time.
Torque is equal to the derivative of angular momentum with respect to time.
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.
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:
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:
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.
|Force of friction|
|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)
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 and takes place at higher velocities.
Work is equal to the scalar product of force and displacement.
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.
Work is equal to change in kinetic energy plus change in potential energy for example the potential energy due to gravity.
Work is equal to average power times time.
The Work done by a force taking something from point 1 to point 2 is
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 measured 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 substituting for v^2 in the formula for kinetic energy.
W: Work (J)
Definition of terms
Work (W): Force times distance. Units: joules (J)
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).
Kinetic energy is simply the capacity to do work by virtue of motion.
(Translational) kinetic energy is equal to one-half of mass times the square of velocity.
(Rotational) kinetic energy is equal to one-half of moment of inertia times the square of angular velocity.
Total kinetic energy is simply the sum of the translational and rotational kinetic energies. In most cases, these energies are separately dealt with. It is easy to remember the rotational kinetic energy if you think of the moment of inertia I as the rotational mass. However, you should note that this substitution is not universal but rather a rule of thumb.
Potential energy is simply the capacity to do work by virtue of position (or arrangement) relative to some zero-energy reference position (or arrangement).
Potential energy due to gravity is equal to the product of mass, acceleration due to gravity, and height (elevation) of the object.
Note that this is simply the vertical displacement multiplied by the weight of the object. The reference position is usually the level ground but the initial position like the rooftop or treetop can also be used. 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.
The reference point of spring deformation is normally when the spring is "relaxed," i.e. the net force exerted by the spring is zero. It will be easy to remember that the one-half factor is inserted to compensate for finite '"change in length" since one would want to think of the product of force and change in length directly. Since the force actually varies with , it is instructive to need a "correction factor" during integration.
K: Kinetic energy (J)
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).
Energy comes in many varieties, including Kinetic energy, Potential energy, and Heat energy.
Uniform Circular Motion
Speed and frequency
Uniform circular motion assumes that an object is moving (1) in circular motion, and (2) at constant speed ; then
where is the radius of the circular path, and is the time period for one revolution.
Any object travelling on a circle will return to its original starting point in the period of one revolution, . At this point the object has travelled a distance . If is the time that it takes to travel distance then the object's speed is
Uniform circular motion can be explicitly described in terms of polar coordinates through angular frequency, :
where is the angular coordinate of the object (see the diagram on the right-hand side for reference).
Since the speed in uniform circular motion is constant, it follows that
From that fact, a number of useful relations follow:
The equations that relate how changes with time are analogous to those of linear motion at constant speed. In particular,
The angle at , , is commonly referred to as phase.
Velocity, centripetal acceleration and force
The position of an object in a plane can be converted from polar to cartesian coordinates through the equations
Expressing as a function of time gives equations for the cartesian coordinates as a function of time in uniform circular motion:
Differentiation with respect to time gives the components of the velocity vector:
Velocity in circular motion is a vector tangential to the trajectory of the object. Furthermore, even though the speed is constant the velocity vector changes direction over time. Further differentiation leads to the components of the acceleration (which are just the rate of change of the velocity components):
The acceleration vector is perpendicular to the velocity and oriented towards the centre of the circular trajectory. For that reason, acceleration in circular motion is referred to as centripetal acceleration.
The absolute value of centripetal acceleration may be readily obtained by
For centripetal acceleration, and therefore circular motion, to be maintained a centripetal force must act on the object. From Newton's Second Law it follows directly that the force will be given by
the components being
and the absolute value
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 Stone 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 continuously 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. e.g. If a driver suddenly turns the car to the left, you go towards the right side of the car because of centrifugal force. The centrifugal force is equal and opposite to the centripetal force. It is caused due to inertia of a body.
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.
Angular acceleration is equal to change in angular velocity divided by time taken.
Angular momentum of an object revolving around an external axis is equal to the cross-product of the position vector with respect to and its linear momentum.
Angular momentum of a rotating object is equal to the moment of inertia times angular velocity.
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.
Definition of terms
Torque (): Force times distance. A vector.
Moment of inertia (): Describes the object's resistance to torque — the rotational analog to inertial mass.
Angular momentum ():
Angular velocity ():
Angular acceleration ():
Rotational kinetic energy ():
Buoyancy is the force due to pressure differences on the top and bottom of an object under a fluid (gas or liquid).
Net force = buoyant force - force due to gravity on the object
Fluid flow is a complex phenomenon. An ideal fluid may be described as:
- The fluid flow is steady i.e its velocity at each point is constant with time.
- The fluid is incompressible. This condition applies well to liquids and in certain circumstances to gases.
- The fluid flow is non-viscous. Internal friction is neglected. An object moving through this fluid does not experience a retarding force. We relax this condition in the discussion of Stokes' Law.
- The fluid flow is irrotational. There is no angular momentum of the fluid about any point. A very small wheel placed at an arbitrary point in the fluid does not rotate about its center. Note that if turbulence is present, the wheel would most likely rotate and its flow is then not irrotational.
As the fluid moves through a pipe of varying cross-section and elevation, the pressure will change along the pipe. The Swiss physicist Daniel Bernoulli (1700-1782) first derived an expression relating the pressure to fluid speed and height. This result is a consequence of conservation of energy and applies to ideal fluids as described above.
Consider an ideal fluid flowing in a pipe of varying cross-section. A fluid in a section of length moves to the section of length in time . The relation given by Bernoulli is:
- is pressure at cross-section
- is height of cross-section
- is density
- is velocity of fluid at cross-section
In words, the Bernoulli relation may be stated as: As we move along a streamline the sum of the pressure (), the kinetic energy per unit volume and the potential energy per unit volume remains a constant.
(To be concluded)
A field is one of the more difficult concepts to grasp in physics. A field is an area or region in which an influence or force is effective regardless of the presence or absence of a material medium. Simply put, a field is a collection of vectors often representing the force an object would feel if it were placed at any particular point in space. With gravity, the field is measured in newtons, as it depends solely on the mass of an object, but with electricity, it is measured in newtons per coulomb, as the force on an electrical charge depends on the amount of that charge. Typically these fields are calculated based on canceling out the effect of a body in the point in space that the field is desired. As a result, a field is a vector, and as such, it can (and should) be added when calculating the field created by TWO objects at one point in space.
Fields are typically illustrated through the use of what are called field lines or lines of force. Given a source that exerts a force on points around it, sample lines are drawn representing the direction of the field at points in space around the force-exerting source.
There are three major categories of fields:
- Uniform fields are fields that have the same value at any point in space. As a result, the lines of force are parallel.
- Spherical fields are fields that have an origin at a particular point in space and vary at varying distances from that point.
- Complex fields are fields that are difficult to work with mathematically (except under simple cases, such as fields created by two point object), but field lines can still typically be drawn. Dipoles are a specific kind of complex field.
Magnetism also has a field, measured in Tesla, and it also has field lines, but its use is more complicated than simple "force" fields. Secondly, it also only appears in a two-pole form, and as such, is difficult to calculate easily.
The particles that form these magnetic fields and lines of force are called electrons and not magnetons. A magneton is a quantity in magnetism.
Definition of terms
Field: A collection of vectors that often represents the force that an object would feel if it were placed in any point in space.
Newtonian Gravity (simplified gravitation) is an apparent force (a.k.a. pseudoforce) that simulates the attraction of one mass to another mass. Unlike the three fundamental (real) forces of electromagnetism and the strong and weak nuclear forces, gravity is purely attractive. As a force it is measured in newtons. The distance between two objects is measured between their centers of mass.
Gravitational force is equal to the product of the universal gravitational constant and the masses of the two objects, divided by the square of the distance between their centers of mass.
The value of the gravitational field which is equivalent to the acceleration due to gravity caused by an object at a point in space is equal to the first equation about gravitational force, with the effect of the second mass taken out.
Gravitational potential energy of a body to infinity is equal to the universal gravitational constant times the mass of a body from which the gravitational field is being created times the mass of the body whose potential energy is being measured over the distance between the two centers of mass. Therefore, the difference in potential energy between two points is the difference of the potential energy from the position of the center of mass to infinity at both points. Near the earth's surface, this approximates:
Potential energy due to gravity near the earth's surface is equal to the product of mass, acceleration due to gravity, and height (elevation) of the object.
If the potential energy from the body's center of mass to infinity is known, however, it is possible to calculate the escape velocity, or the velocity necessary to escape the gravitational field of an object. This can be derived based on utilizing the law of conservation of energy and the equation to calculate kinetic energy as follows:
F: force (N)
Definition of terms
Universal constant of gravitation (G): This is a constant that is the same everywhere in the known universe and can be used to calculate gravitational attraction and acceleration due to gravity.
A black hole is a geometrically defined region of space time exhibiting such large centripetal gravitational effects that nothing such as particles and electromagnetic radiation such as light may escape from inside of it. That is the escape velocity upon the event horizon is equivalent to the speed of light. General relativity is a metric theory of gravitation generalizing space time and Newton's law of universal gravitational attraction as a geometric property of space time.
Wave is defined as the movement of any periodic motion like a spring, a pendulum, a water wave, an electric wave, a sound wave, a light wave, etc.
Any periodic wave that has amplitude varied with time, phase sinusoidally can be expressed mathematically as
- R(t , θ) = R Sin (ωt + θ)
- Minimum wave height (trough) at angle 0, π, 2π, ...
- F(R,t,θ) = 0 at θ = nπ
- Maximum wave height (peak or crest) at π/2, 3π/2, ...
- F(R,t,θ) = R at θ = (2n+1)π/2
- Wavelength (distance between two crests) λ = 2π.
- λ = 2π - A circle or a wave
- 2λ = 2(2π) - Two circles or two waves
- kλ = k2π - Circle k or k amount of waves
- Wave Number,
- Velocity (or Angular Velocity),
- ω = 2πf
- Time Frequency,
- f = 1 / t
- t = 1 / f
Wave speed is equal to the frequency times the wavelength. It can be understood as how frequently a certain distance (the wavelength in this case) is traversed.
Frequency is equal to speed divided by wavelength.
Period is equal to the inverse of frequency.
λ: wavelength (m)
Definition of terms
Wavelength (λ): The length of one wave, or the distance from a point on one wave to the same point on the next wave. Units: meters (m). In light, λ tells us the color.
The wave’s extremes, its peaks and valleys, are called antinodes. At the middle of the wave are points that do not move, called nodes.
Examples of waves: Water waves, sound waves, light waves, seismic waves, shock waves, electromagnetic waves …
A wave is said to oscillate, which means to move back and forth in a regular, repeating way. This fluctuation can be between extremes of position, force, or quantity.
Different types of waves have different types of oscillations.
Longitudinal waves: Oscillation is parallel to the direction of the wave. Examples: sound waves, waves in a spring.
Transverse waves: Oscillation is perpendicular to direction of the wave. Example: light
When waves overlap each other it is called interference. This is divided into constructive and destructive interference.
Constructive interference: the waves line up perfectly and add to each others’ strength.
Destructive interference: the two waves cancel each other out, resulting in no wave.
In real life, waves usually give a mishmash of constructive and destructive interference and quickly die out. However, at certain wavelengths standing waves form, resulting in resonance. These are waves that bounce back into themselves in a strengthening way, reaching maximum amplitude.
Resonance is a special case of forced vibration when the frequency of the impressed periodic force is equal to the natural frequency of the body so that it vibrates with increased amplitude, spontaneously.
For resonance in a taut string, the first harmonic is determined for a wave form with one antinode and two nodes. That is, the two ends of the string are nodes because they do not vibrate while the middle of the string is an antinode because it experiences the greatest change in amplitude. This means that one half of a full wavelength is represented by the length of the resonating structure.
The frequency of the first harmonic is equal to wave speed divided by twice the length of the string. (Recall that wave speed is equal to wavelength times frequency.)
The wavelength of the first harmonic is equal to double the length of the string.
The "nth" wavelength is equal to the fundamental wavelength divided by n.
Harmonics for a taut string*
|Harmonic number||Overtone number||F =||λ =|
|F1||First harmonic||---||F1 = v/2L||λ1 = 2L|
|F2||Second harmonic||First overtone||F2 = 2F1||λ2 =λ1/2|
|F3||Third harmonic||Second overtone||F3 = 3F1||λ3 = λ1/3|
|Fn||Nth harmonic||(Nth - 1) overtone||Fn = nF1||λn = λ1/n|
* or any wave system with two identical ends, such as a pipe with two open or closed ends. In the case of a pipe with two open ends, there are two antinodes at the ends of the pipe and a single node in the middle of the pipe, but the mathematics work out identically.
Definition of terms
Frequency (F): Units: (1/s), hertz (Hz)
The first overtone is the first allowed harmonic above the fundamental frequency (F1).
In the case of a system with two different ends (as in the case of a tube open at one end), the closed end is a node and the open end is an antinode. The first resonant frequency has only a quarter of a wave in the tube. This means that the first harmonic is characterized by a wavelength four times the length of the tube.
The wavelength of the first harmonic is equal to four times the length of the string.
The "nth" wavelength is equal to the fundamental wavelength divided by n.
Note that "n" must be odd in this case as only odd harmonics will resonate in this situation.
Harmonics for a system with two different ends*
|Harmonic number||Overtone number||F =||λ =|
|F1||First harmonic||---||F1 = v/4L||λ1 = 4L|
|F2||Third harmonic||First overtone||F2 = 3F1||λ2 = λ1/3|
|F3||Fifth harmonic||Second overtone||F3 = 5F1||λ3 = λ1/5|
|Fn||Nth harmonic†||(Nth - 1)/2 overtone||F(n-1)/2 = nF1||λn = λ1/n|
* such as a pipe with one end open and one end closed
†In this case only the odd harmonics resonate, so n is an odd integer.
Vs: velocity of sound
- dependent on qualities of the medium transmitting the sound, (the air) such as its density, temperature, and “springiness.” A complicated equation, we concentrate only on temperature.
- increases as temperature increases (molecules move faster.)
- is higher for liquids and solids than for gasses (molecules are closer together.)
- for “room air” is 340 meters per second (m/s).
- Speed of sound is 343 meters per second at 20 degrees C. Based on the material sound is passing through and the temperature, the speed of sound changes.
Wave speed is equal to the square root of tension divided by the linear density of the string.
Linear density of the string is equal to the mass divided by the length of the string.
The fundamental wavelength is equal to two times the length of the string.
λ: wavelength (m)
Definition of terms
Tension (F): (not frequency) in the string (t is used for time in these equations). Units: newtons (N)
Fundamental frequency: the frequency when the wavelength is the longest allowed, this gives us the lowest sound that we can get from the system.
In a string, the length of the string is half of the largest wavelength that can create a standing wave, called its fundamental wavelength.
Sound is defined as mechanical sinosodial vibratory longitudinal impulse waves which oscillate the pressure of a transmitting medium by means of adiabatic compression and decompression consequently resulting in the increase in the angular momentum and hence rotational kinetic energy of the particles present within the transmitting medium producing frequencies audible within hearing range, that is between the threshold of audibility and the threshold of pain on a Fletchford Munson equal loudness contour diagram.
When two glasses collide, we hear a sound. When we pluck a guitar string, we hear a sound.
Different sounds are generated from different sources. Generally speaking, the collision of two objects results in a sound.
Sound does not exist in a vacuum; it travels through the materials of a medium. Sound is a longitudinal wave in which the mechanical vibration constituting the wave occurs along the direction of the wave's propagation.
The velocity of sound waves depends on the temperature and the pressure of the medium. For example, sound travels at different speeds in air and water. We can therefore define sound as a mechanical disturbance produced by the collision of two or more physical quantities from a state of equilibrium that propagates through an elastic material medium.
The amplitude is the magnitude of sound pressure change within a sound wave. Sound amplitude can be measured in pascals (Pa), though its more common to refer to the sound (pressure) level as Sound intensity(dB,dBSPL,dB(SPL)), and the perceived sound level as Loudness(dBA, dB(A)). Sound intensity is flow of sound energy per unit time through a fixed area. It has units of watts per square meter. The reference Intensity is defined as the minimum Intensity that is audible to the human ear, it is equal to 10-12 W/m2, or one picowatt per square meter. When the intensity is quoted in decibels this reference value is used. Loudness is sound intensity altered according to the frequency response of the human ear and is measured in a unit called the A-weighted decibel (dB(A), also used to be called phon).
The decibel is not, as is commonly believed, the unit of sound. Sound is measured in terms of pressure. However, the decibel is used to express the pressure as very large variations of pressure are commonly encountered. The decibel is a dimensionless quantity and is used to express the ratio of one power quantity to another. The definition of the decibel is , where x is a squared quantity, ie pressure squared, volts squared etc. The decibel is useful to define relative changes. For instance, the required sound decrease for new cars might be 3 dB, this means, compared to the old car the new car must be 3 dB quieter. The absolute level of the car, in this case, does not matter.
Definition of terms
Intensity (I): the amount of energy transferred through 1 m2 each second. Units: watts per square meter
Lowest audible sound: I = 0 dB = 10-12 W/m2 (A sound with dB < 0 is
inaudible to a human.)
Sample equation: Change in sound intensity
Δβ = β2 - β1
= 10 log(I2/I0) - 10 log(I1/I0)
= 10 [log(I2/I0) - log(I1/I0)]
= 10 log[(I2/I0)/(I1/I0)]
= 10 log(I2/I1)
where log is the base-10 logarithm.
f' is the observed frequency, f is the actual frequency, v is the speed of sound (), T is temperature in degrees Celsius is the speed of the observer, and is the speed of the source. If the observer is approaching the source, use the top operator (the +) in the numerator, and if the source is approaching the observer, use the top operator (the -) in the denominator. If the observer is moving away from the source, use the bottom operator (the -) in the numerator, and if the source is moving away from the observer, use the bottom operator (the +) in the denominator.
A. An ambulance, which is emitting a 400 Hz siren, is moving at a speed of 30 m/s towards a stationary observer. The speed of sound in this case is 339 m/s.
B. An M551 Sheridan, moving at 10 m/s is following a Renault FT-17 which is moving in the same direction at 5 m/s and emitting a 30 Hz tone. The speed of sound in this case is 342 m/s.
Thermodynamics deals with the movement of heat and its conversion to mechanical and electrical energy among others.
Laws of Thermodynamics
The First Law is a statement of conservation of energy law:
The First Law can be expressed as the change in internal energy of a system () equals the amount of energy added to a system (Q), such as heat, minus the work expended by the system on its surroundings (W).
If Q is positive, the system has gained energy (by heating).
If W is positive, the system has lost energy from doing work on its surroundings.
As written the equations have a problem in that neither Q or W are state functions or quantities which can be known by direct measurement without knowing the history of the system.
In a gas, the first law can be written in terms of state functions as
After the first law of Thermodynamics had been named, physicists realised that there was another more fundamental law, which they termed the 'zero-th'.
This is that:
If two bodies are at the same temperature, there is no resultant heat flow between them.
An alternate form of the 'zero-th' law can be described:
If two bodies are in thermal equilibrium with a third, all are in thermal equilibrium with each other.
This second statement, in turn, gives rise to a definition of Temperature (T):
Temperature is the only thing that is the same between two otherwise unlike bodies that are in thermal equilibrium with each other.
This law states that heat will never of itself flow from a cold object to a hot object.
where is the Boltzmann constant () and is the partition function, i. e. the number of all possible states in the system.
This was the statistical definition of entropy, there is also a "macroscopic" definition:
where T is the temperature and dQ is the increment in energy of the system.
The third law states that a temperature of absolute zero cannot be reached.
There are several different scales used to measure temperature. Those you will most often come across in physics are degrees Celsius and kelvins.
Celsius temperatures use the symbol Θ. The symbol for degrees Celsius is °C. Kelvin temperatures use the symbol T. The symbol for kelvins is K.
The Celsius Scale
The Celsius scale is based on the melting and boiling points of water.
The temperature for freezing water is 0 °C. This is called the freezing point
The temperature of boiling water is 100 °C. This is called the steam point.
The Celsius scale is sometimes known as 'Centigrade', but the CGPM chose degrees Celsius from among the three names then in use way back in 1948, and centesimal and centigrade should no longer be used. See Wikipedia for more details.
The Kelvin Scale
The Kelvin scale is based on a more fundamental temperature than the melting point of ice. This is absolute zero (equivalent to −273.15 °C), the lowest possible temperature anything could be cooled to—where the kinetic energy of any system is at its minimum. The Kelvin scale was developed from an observation on how the pressure and volume of a sample of gas changes with temperature- PV/T is a constant. If the temperature ( T)was reduced, then the pressure ( P) exerted by Volume (V) the Gas would also reduce, in direct proportion. This is a simple experiment and can be carried out in most school labs. Gases were assumed to exert no pressure at -273 degree Celsius. ( In fact all gases will have condensed into liquids or solids at a somewhat higher temperature)
Although the Kelvin scale starts at a different point to Celsius, its units are of exactly the same size.
Temperature in kelvins (K) = Temperature in degrees Celsius (°C) + 273.15
Specific Latent Heat
Energy is needed to break bonds when a substance changes state. This energy is sometimes called the latent heat. Temperature remains constant during changes of state.
To calculate the energy needed for a change of state, the following equation is used:
Heat transferred, ΔQ (J) = Mass, m (kg) x specific latent heat capacity, L (J/kg)
The specific latent heat, L, is the energy needed to change the state of 1 kg of the substance without changing the temperature.
The latent heat of fusion refers to melting. The latent heat of vapourisation refers to boiling.
Specific Heat Capacity
The specific heat capacity is the energy needed to raise the temperature of a given mass by a certain temperature.
The change in temperature of a substance being heated or cooled depends on the mass of the substance and on how much energy is put in. However, it also depends on the properties of that given substance. How this affects temperature variation is expressed by the substance's specific heat capacity (c). This is measured in J/(kg·K) in SI units.
Change in internal energy, ΔU (J) = mass, m (kg) x specific heat capacity, c (J/(kg·K)) x temperature change, ΔT (K)
The force resulting from two nearby charges is equal to k times charge one times charge two divided by the square of the distance between the charges.
The electric field created by a charge is equal to the force generated divided by the charge.
Electric field is equal to a constant, “k”, times the charge divided by the square of the distance between the charge and the point in question.
Electric potential energy is equal to a constant, “k” multiplied by the two charges and divided by the distance between the charges.
F: Force (N)
Electricity acts as if all matter were divided into four categories:
- Superconductors, which allow current to flow with no resistance. (However these have only been produced in relatively extreme laboratory conditions, such as at temperatures approaching absolute zero)
- Conductors, which allow electric current to flow with little resistance.
- Semiconductors, which allow some electric current to flow but with significant resistance.
- Insulators, which do not allow electric current to flow.
Charges are positive (+) or negative (-). Any two like charges repel each other, and opposite charges attract each other.
A charge in an electrical field feels a force. The charge is not a vector, but force is a vector, and so is the electric field. If a charge is positive, then force and the electric field point in the same direction. If the charge is negative, then the electric field and force vectors point in opposite directions.
A point charge in space causes an electric field. The field is stronger closer to the point and weaker farther away.
Electricity is made of subatomic particles called Electrons and so are Electric Fields and Magnetic Fields. One must also note that electrical fields come under the category of spherical fields as the inverse square law may be applied to the electrical field. This means that the electrical force, exhibited by the electrical field emitted by the subatomic electron charge (-), acting upon a body is inversely proportional to the distance between the center point of the electric field (subatomic electron) and the body on which the electric force is acting upon.
An electric circuit is composed of conducting wires (through which an electric current flows through), a key or switch which is utilized to open and close the circuit, components which transfer electrical energy to a form of energy required by the component and an electromotive source (such as a voltaic cell). A voltaic cell is an electromotive source in which are present two plates, zinc and copper, placed in dilute sulphuric acid. Whence the circuit is closed the zinc reacts with the sulphuric acid to produce zinc sulphate. The electromotive force which discharges the electrical energy in the electric current is considered to be originated on the surface of the zinc plate in the voltaic cell. However, depending upon the cell, closing the circuit gives rise to polarization, accumulation of hydrogen bubbles on the surface of the copper plate which seriously interferes with the movement of electricity and reduces the magnitude of the electromotive force. For this reason Leclanché cells are utilized. Consisting of similar characteristics as that of the voltaic cell however a Mage difference is present. Instead of the use of copper plates, a carbon plate is used. For this reason, magneze dioxide may be placed on the carbon to react to form a compound which whence in contact with hydrogen bubbles will turn the hydrogen into water, hence increasing the size of the electromotive force produced by the cell. The resistance encountered in conducting wires: Inversely proportional to the diameter of the conducting wire. Directly proportional to the length of the conducting wire. Varies with different substances. Varies with temperature of the conducting wire.
In order to maintain a constant flow of an electric current a constant expenditure of chemical or mechanical energy is required. An electric current is accompanied by an electric field and a magnetic field. A device employed into determining the presence of an electric current is known as a galvanoscope. The conducting wire through which the electric current flows through is he led over and parallel to the galvanoscope the magnetoscope preset inside of the galvanoscope being deflected in the opposite direction to which the electric current flows in. So with the aid of a galvanoscope one may not only deduce the magnetic properties of an electric current, the exhibition of a magnetic field, but the direction in which the current flows through. An electromotive force may also be generators by a dynamo. A rotating magnet present inside of a helix. The magnetic properties of electric currents may be used to construct magnets. An electomagnet is commonly described as a mass of iron on which is placed a helix/solenoid through which flows an electric current. The magnetic field emitted by the electric current is increased if the solenoid is placed around a magnetic mass of iron or any other substance possessing magnetic properties, that is the magnetic field of the iron is added to that of the electic current producing a more powerful magnetic field. Conductors may be arranged in two variants. Series and parallel circuits. In series, the current passes through each conductor in turn, where Ohm's law changes to I = nE/(nr + R), where I is the current intensity, n is the number of cells arranged in series in the circuit, E is the electromotive force applied to the circuit, r is the internal resistance ( the resistance the current that is produced in the cell experiences whence passing from the zinc plate to the copper or carbon plate through the sulphuric acid ) and R is the external resistance.
- For a good introduction to Gauss' Law and Ampere's Law, check out this website
The magnetic force exerted on a moving particle in a magnetic field is the cross product of the magnetic field and the velocity of the particle, multiplied by the charge of the particle.
Because the magnetic force is perpendicular to the particle's velocity, this causes uniform circular motion. That motion can be explained by the following
The radius of this circle is directly proportional to the mass and the velocity of the particle and inversely proportional to the charge of the particle and the field strength of the magnetic field.
The period and frequency of this motion (referred to as the cyclotron period and frequency) can be derived as well.
The magnetic field created by charge flowing through a straight wire is equal to a constant, , multiplied by the current flowing through the wire and divide by the distance from the wire.
The magnetic field created by a magnetic dipole (at distances much greater than the size of the dipole) is approximately equal to a constant, , multiplied by the dipole moment divided by the cube of the distance from the dipole. EDIT: This forumla is incomplete. The field from a dipole is a vector that depends not only on the distance from the dipole, but also the angle relative to the orientation of the magnetic moment. This is because of the vector nature of the magnetic moment and its associated magnetic field. The field component pointing in the same directions as the magnetic moment is the above formula multiplied by (3*(Cos[theta])^2-1).
The magnetic field created by an ideal solenoid is equal to a constant, , times the number of turns of the solenoid times the current flowing through the solenoid.
The magnetic field created by an ideal toroid is equal to a constant, , times the number of turns of the toroid times the current flowing through the toroid divided by the circumference of the toroid.
The magnetic force between two wires is equal to a constant, , times the current in one wire times the current in the other wire times the length of the wires divided by the distance between the wires.
The torque on a current loop in a magnetic field is equal to the cross product of the magnetic field and the area enclosed by the current loop (the area vector is perpendicular to the current loop).
The dipole moment of a current loop is equal to the current in the loop times the area of the loop times the number of turns of the loop.
The magnetic potential energy is the opposite of the dot product of the magnetic field and the dipole moment.
F: Force (N)
Electronics is the application of electromagnetic (and quantum) theory to construct devices that can perform useful tasks, from as simple as electrical heaters or light bulbs to as complex as the Large Hadron Collider.
Print version;component of electronics
To discuss electronics we need the basic concepts from electricity: charge, current which is flow of charge, and potential which is the potential energy difference between two places. Please make sure these concepts are familiar before continuing.
The interest of electronics is circuits. A circuit consists of wires' that connect components. Typical components are resistors, voltage sources. In order to maintain a constant electrical current a continued exhaustion of chemical or mechanical energy is required.
A voltaic cell is a common electromotive power source for a circuit. It consists primarily of two plates, a positive copper plate and a negative zinc plate. One must note that the plates are placed in dilute sulfuric acid. Whence a key is switched or the circuit is closed the zinc reacts with the sulfuric in which it is placed in. One may discern that the electromotive force which is then applied to the circuit is believed to be converted from chemical to electrical energy upon the surface of the zinc cell. Whence the E.M.F is converted from chemical to electrical energy the electromotive force travels through the sulfuric acid. However disadvantages occur whence utilizing a voltaic or galvanic cell. Whence the circuit is closed, simultaneously hydrogen bubbles accumulate upon the copper plate surface hence decreasing the total electromotive power source applied to the circuit in question. The resistance with which the electromotive source encounters as it flows form the zinc plate to the copper plate through the dilute sulfuric acid is distinguished as internal resistance. Hence the voltaic cell uses diluted rather than concentrated sulfuric acid so as to reduce the internal resistance present inside of the component which applies the electromotive force to the circuit.
A circuit can be open, when there is a break so that no current can flow, or it can be closed, so that current can flow. These definitions allow us to discuss electronics efficiently.
Direct current and alternating Current
When the Electrons moves through a particular medium in straight without oscillating on their mean position then it is Direct Current. This DC is used in almost every electronical component. Alternating Current on the other hand as the name suggest, the Electrons alter back and forth and hence produces much electrical energy. AC is mainly used to reduce the wastage of electrons and passing much energy at one time.
If 'V' is Potential Difference applied at two ends of conductor and 'I' is Current flowing through the conductor then 'I' is directly proportional to its 'V'. V = I x R
Kirchoff's laws generally hold for direct current (DC) circuits, but fail when dealing with changing electric current and voltage such as alternating current (AC) or signal processing in combination with capacitors, inductors, and antennas.
Kirchoff's current law
The sum of all the currents entering and leaving any point in a circuit is equal to zero.
It is based on the assumption that current flows only in conductors, and that whenever current flows into one end of a conductor it immediately flows out the other end.
Kirchoff's voltage law
The sum of all the voltages around the circuit loop is equal to zero.
It is based on the assumption that there is no fluctuating magnetic field linking the closed circuit loop.
p=work done/time taken p=I*V (current * Voltage) Other Equations for P can also be derived using Ohm's Law, such as P = IR^2, because V=IR, which can be plugged in to get P= I * (IR) and another equation being P = V^2/R.
Resistors in series
Resistors in parallel
1/R = 1/R1+1/R2+1/R3+......
Superposition of sources
Current is the rate of flow of charge.
= Current [amperes - A]
= Charge [coulombs - C]
= Time [seconds - s]
Voltage is equal to current multiplied by resistance
Power is equal to the product of voltage and current
Electronics is the flow of current through semiconductor devices like silicon and germanium.
Semiconductor devices are those which behave like conductors at higher temperature.
Transistor, diode, SCR are some electronic devices.
Light is that range of electromagnetic energy that is visible to the human eye, the visible colors. The optical radiation includes not only the visible range, but a broader range of invisible electromagnetic radiation that could be influenced in its radiation behavior in a similar way as the visible radiation, but needs often other transmitters or receivers for this radiation. Dependent on the kind of experimental question light - optical radiation behaves as a wave or a particle named lightwave or photon. The birth or death of photons needs electrons - electromagnetic charges, that change their energy.
The speed of light is fastest in the vacuum.
In a wave we have to distinguish between the speed of transport of energy or the speed of the transport of on phase state of a wave of a defined frequency. In vacuum the speed of waves of any photon energy - wavelength is the same, but the transmission speed through material is dependent on wavelength - photon energy. At the time the measurement of the speed of light in vacuum reached the uncertainty of the unit of length, the meter, this basic unit got in 1960 a new definition, based on the unit of time. Taking the best known measurement values it was defined without any uncertainties of length, that the speed of light is 299,792,458 meters per second. For this reason the only uncertainty in the speed of light is the uncertainty of the realization of the unit of time, the second. (If you like to get the standard of length, cooperate with the watchmaker).
However, when electromagnetic radiation enters a medium with refractive index, , its speed would become
where is the speed of light in the medium.
Refraction occurs when light travels from one medium into another (i.e. from air into water). Refraction is the changing of direction of light due to the changing speed of light. Refraction occurs toward the normal when light travels from a medium into a denser medium. Example when light travels from air into a block of glass, light is refracted towards the normal. The ratio between the sine of the angle of the incident ray and sine of the angle of the refracted ray is the same as the ratios of the indexes of refraction.
This is known as Snell's Law - an easy way to remember this is that 'Snell' is 'lens' backwards.
Mirrors and lenses
- is the focal length.
- is negative in convex mirror and concave lens.
- is positive in concave mirror and convex lens.
- is the distance from the image to the mirror or lens
- For a mirror, it is positive if the image appears in front of the mirror. It is negative if the image appears behind.
- For a lens, it is positive if the image appears on the opposite side of the lens as the light source. It is negative if the image appears on the same side of the lens as the light source.
- is the distance from the object to the mirror or the lens (always positive). The only case, when it is negative, is the case, when you don't have a real object, but you do have an imaginary object - a converging set of rays from another optical system.
- an easy way to remember the formula is to memorize "if I Do I Die", which stands for
- is the magnification.
- If it is positive the image is upright
- If it is negative the image is inverted
- is the image height.
- is the object height.
- is the distance from the image to the mirror or lens (also often )
- For a mirror, it is positive if the image appears in front of the mirror. It is negative if the image appears behind.
- For a lens, it is positive if the image appears on the opposite side of the lens as the light source. It is negative if the image appears on the same side of the lens as the light source.
- is the distance from the object to the mirror or lens (also often )
Commonly Used Physical Constants
|Speed of light (in vacuum)||(exact)|
|Newtonian Gravitaional Constant|
|Mass of the electron|
|Mass of the proton|
|Fine structure constant||dimensionless|
|Molar gass constant|
|Standard acceleration of gravity||defined|
To Be Merged Into Table
This list is prepared in the format
- Constant (symbol) : value
- Coulomb's Law Constant (k) : 1/(4 π ε0) = 9.0 × 109 N·m2/C2
- Faraday constant (F) : 96,485 C·mol−1
- Mass of a neutron (mn) : 1.67495 × 10−27 kg
- Mass of Earth : 5.98 × 1024 kg
- Mass of the Moon : 7.35 × 1022 kg
- Mean radius of Earth : 6.37 × 106 m
- Mean radius of the Moon : 1.74 × 106 m
- Dirac's Constant () : = 1.05457148 × 10−34 J·s
- Speed of sound in air at STP : 3.31 × 102 m/s
- Unified Atomic Mass Unit (u) : 1.66 × 10−27 kg
Approximate Coefficients of Friction
|Rubber on concrete (dry)||0.68||0.90|
|Rubber on concrete (wet)||0.58||-.--|
|Rubber on asphalt (dry)||0.7||0.85|
|Rubber on asphalt (wet)||0.53||-.--|
|Rubber on ice||0.15||-.--|
|Waxed ski on snow||0.05||0.14|
|Wood on wood||0.30||0.42|
|Steel on steel||0.57||0.74|
|Copper on steel||0.36||0.53|
|Teflon on teflon||0.04||-.--|
About the Common uses in Physics
While these are indeed common usages, it should be pointed out that there are many other usages and that other letters are used for the same purpose. The reason is quite simple: there are only so many symbols in the Greek and Latin alphabets, and scientists and mathematicians generally do not use symbols from other languages. It is a common trap to associate a symbol exclusively with some particular meaning, rather than learning and understanding the physics and relations behind it.
|Capital||Lower case||Name||Common use in Physics|
Alpha particle (helium nucleus)
Fine Structure Constant
|beta||Beta particle — high energy electron|
|gamma||Gamma ray (high energy EM wave)|
Ratio of heat capacities (in an ideal gas)
Relativistic correction factor
|delta||Δ="Change in" |
δ="Infinitesimal change in (), also used to denote the Dirac delta function (reference needed)"
Strain (Direct e.g. tensile or compression)
|zeta||(no common use)|
|theta||Angle (°, rad)|
|iota||The lower case is rarely used, while is sometimes used for the identity matrix or the moment of inertia. Note that is not to be confused with the Roman character (which has a dot and is much more widely used in mathematics and physics).|
Eigenvalue of a matrix
|mu||Coefficient of friction|
|omicron||(no common use)|
|pi||Product symbol |
Stress (Direct e.g. tensile, compression)
Tau particle (a lepton)
|upsilon||mass to light ratio|
Angle (°, rad)
|chi||Rabi frequency (lasers)|
|omega||Ohms (unit of electrical resistance)|
ω Angular velocity
Review of logs
Been a while since you used logs? Here is a quick refresher for you.
The log (short for logarithm) of a number N is the exponent used to raise a certain "base" number B to get N. In short, means that .
Typically, logs use base 10. An increase of "1" in a base 10 log is equivalent to an increase by a power of 10 in normal notation. In logs, "3" is 100 times the size of "1". If the log is written without an explicit base, 10 is (usually) implied.
|therefore: log(10–12) = –12 |
|also: log(1000) = 3|
Another common base for logs is the trancendental number , which is approximately 2.7182818.... Since , these can be more convenient than . Often, the notation is used instead of .
The following properties of logs are true regardless of whether the base is 10, , or some other number.
logA + logB = log(AB)
Adding the log of A to the log of B will give the same result as taking the log of the product A times B.
Subtracting the log of B from the log of A will give the same result as taking the log of the quotient A divided by B.
The log of (A to the Bth power) is equal to the product (B times the log of A).
A few examples:
log(2) + log(3) = log(6)
log(30) – log(2) = log(15)
log(8) = log(23) = 3log(2) Vectors are quantities that are characterized by having both a numerical quantity (called the "magnitude" and denoted as |v|) and a direction. Velocity is an example of a vector; it describes the time rated change in position with a numerical quantity (meters per second) as well as indicating the direction of movement.
The definition of a vector is any quantity that adds according to the parallelogram law (there are some physical quantities that have magnitude and direction that are not vectors).
Scalars are quantities in physics that have no direction. Mass is a scalar; it can describe the quantity of matter with units (kilograms) but does not describe any direction.
Multiplying vectors and scalars
- A scalar times a scalar gives a scalar result.
- A vector scalar-multiplied by a vector gives a scalar result (called the dot-product).
- A vector cross-multiplied by a vector gives a vector result (called the cross-product).
- A vector times a scalar gives a vector result.
Frequently Asked Questions about Vectors
When are scalar and vector compositions essentially the same?
Answer: when multiple vectors are in same direction then we can just add the magnitudes.so, the scalar and vector composition will be same as we do not add the directions.
What is a "dot-product"? (work when force not parallel to displacement)
Answer: Let's take gravity as our force. If you jump out of an airplane and fall you will pick up speed. (for simplicity's sake, let's ignore air drag). To work out the kinetic energy at any point you simply multiply the value of the force caused by gravity by the distance moved in the direction of the force. For example, a 180 N boy falling a distance of 10 m will have 1800 J of extra kinetic energy. We say that the man has had 1800 J of work done on him by the force of gravity.
Notice that energy is not a vector. It has a value but no direction. Gravity and displacement are vectors. They have a value plus a direction. (In this case, their directions are down and down respectively) The reason we can get a scalar energy from vectors gravity and displacement is because, in this case, they happen to point in the same direction. Gravity acts downwards and displacement is also downwards.
When two vectors point in the same direction, you can get the scalar product by just multiplying the value of the two vectors together and ignoring the direction.
But what happens if they don't point in the same direction?
Consider a man walking up a hill. Obviously it takes energy to do this because you are going against the force of gravity. The steeper the hill, the more energy it takes every step to climb it. This is something we all know unless we live on a salt lake.
In a situation like this we can still work out the work done. In the diagram, the green lines represent the displacement. To find out how much work against gravity the man does, we work out the projection of the displacement along the line of action of the force of gravity. In this case it's just the y component of the man's displacement. This is where the cos θ comes in. θ is merely the angle between the velocity vector and the force vector.
If the two forces do not point in the same direction, you can still get the scalar product by multiplying the projection of one force in the direction of the other force. Thus:
There is another method of defining the dot product which relies on components.
What is a "cross-product"? (Force on a charged particle in a magnetic field)
Answer: Suppose there is a charged particle moving in a constant magnetic field. According to the laws of electromagnetism, the particle is acted upon by a force called the Lorentz force. If this particle is moving from left to right at 30 m/s and the field is 30 Tesla pointing straight down perpendicular to the particle, the particle will actually curve in a circle spiraling out of the plane of the two with an acceleration of its charge in coulombs times 900 newtons per coulomb! This is because the calculation of the Lorentz force involves a cross-product.when cross product can replace the sin0 can take place during multiplication. A cross product can be calculated simply using the angle between the two vectors and your right hand. If the forces point parallel or 180° from each other, it's simple: the cross-product does not exist. If they are exactly perpendicular, the cross-product has a magnitude of the product of the two magnitudes. For all others in between however, the following formula is used:
But if the result is a vector, then what is the direction? That too is fairly simple, utilizing a method called the "right-hand rule".
The right-hand rule works as follows: Place your right-hand flat along the first of the two vectors with the palm facing the second vector and your thumb sticking out perpendicular to your hand. Then proceed to curl your hand towards the second vector. The direction that your thumb points is the direction that cross-product vector points! Though this definition is easy to explain visually it is slightly more complicated to calculate than the dot product.
How to draw vectors that are in or out of the plane of the page (or board)
Answer: Vectors in the plane of the page are drawn as arrows on the page. A vector that goes into the plane of the screen is typically drawn as circles with an inscribed X. A vector that comes out of the plane of the screen is typically drawn as circles with dots at their centers. The X is meant to represent the fletching on the back of an arrow or dart while the dot is meant to represent the tip of the arrow.
GNU Free Documentation License
|As of July 15, 2009 Wikibooks has moved to a dual-licensing system that supersedes the previous GFDL only licensing. In short, this means that text licensed under the GFDL only can no longer be imported to Wikibooks, retroactive to 1 November 2008. Additionally, Wikibooks text might or might not now be exportable under the GFDL depending on whether or not any content was added and not removed since July 15.|
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If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.
If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.
You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License.
However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.
Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it.
10. FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document.
"Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site.
"CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization.
"Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document.
An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008.
The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.
How to use this License for your documents
To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:
- Copyright (c) YEAR YOUR NAME.
- Permission is granted to copy, distribute and/or modify this document
- under the terms of the GNU Free Documentation License, Version 1.3
- 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".
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this:
- with the Invariant Sections being LIST THEIR TITLES, with the
- Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.
If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.
If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.