Part I - Introductory Concepts and Basic Mathematical Tools Used in Physics
Chapter 1 - Introductory Concepts
Science: Search for relationships that explain and predict the behavior of the observable Universe.
Hypothesis: Possible solution to a problem, or possible explanation to a phenomenon, that goes beyond the known facts.
Theory: Reasonable explanation of a series of observed phenomena, which often involves an imaginary model that helps to create a picture of how the observed phenomena could be produced.
Law: Statement that describes a natural phenomenon; it is a concise general statement about how nature behaves.
Principle: Statement about the behavior of nature that is less general than a law.
Physics: Science that deals with the relationships between matter, energy, space, and time, and their structure. It is the study and analysis of all the natural phenomena that have mathematical structure.
Matter: Everything, which occupies space, can be perceived by one or more senses (directly or indirectly), and constitutes a physical body; it is everything that has the attributes of inertia and interaction.
Basic Interactions in Nature:
1.- Gravitational: Interaction between two objects that occurs because they both have mass. The gravitational interaction is due exclusively to the existence of mass, is always attractive, operates over very long distances, and is the weakest of all the four basic interactions observed in nature.
2.- Electromagnetic: Ability of an object to interact with another object because they are both electrically charged. The electromagnetic interaction is due exclusively to the existence of the electric charge. Since two types of electric charge have been observed in nature, it manifests in two different ways: attraction or repulsion. It is about 10E40 times greater than the gravitational interaction; it operates over all distances, but falls off with the square of the distance between the objects.
3.- Strong: Interaction that holds the particles in the atomic nucleus together. It occurs between a type of elementary particles called hadrons (like the proton, the neutron, and the pion), and is about 100 times greater than the electromagnetic interaction. It operates at very short range (about 10E–15 m).
4.- Weak: Interaction that occurs between a type of elementary particles called leptons (like the electron, the muon, and the neutrino), that is about 10E10 times weaker than the electromagnetic interaction.
Energy: Ability to cause changes in matter.
Time: The continuous, forward flow of events.
Space: A limited extension that may extend in one, two, or three dimensions; it extends with no bounds in all directions and is the field of physical objects, events, and their order and relationship.
Inertia: Tendency of matter to resist to changes in its state of motion.
Interaction: Capacity of matter to "detect" the presence or existence of other pieces of matter.
Chapter 2 - Basic Mathematical Tools Used in Physics
Unit: Value or quantity in terms of which other values or quantities are expressed. A unit is fixed by definition.
Scientific Notation: Way of expressing a number as the product of two factors. One factor is a power of 10, and the other is a number greater than or equal to 1, but less than 10.
Order of Magnitude: Power of 10 that better represents a number.
Direct Measurement: Measurement done by comparing directly a physical dimension with a standard.
Indirect Measurement: Measurement done by performing calculations using directly obtained quantities, or indirectly obtained quantities previously obtained.
Significant Figures: Digits that are known with certainty plus the first uncertain digit in a measurement. In an experimental measurement, they are all the numbers that can be read directly from the instrument's scale plus one doubtful or estimated digit.
Chapter 3 - Fundamentals of Error Theory and Basic Statistical Analysis
3.1 - Error Theory
Positive Error: Errors that tend to make observations always too high.
Negative Errors: Errors that tend to make observations always too low.
Systematic Errors: Errors that tend to make all observations made always positive or always negative. Their causes are known, so they may be estimated and corrected. They are always associated with a particular instrument or technique. Avoiding systematic errors depends on the skills of the observer to detect, prevent and correct them. They are usually grouped in 3 different categories: instrumental, personal, and external.
(1) Instrumental: Produced by faults in the apparatus used. To reduce or correct them, the instruments should be checked with accurate standards and the necessary corrections applied to the observations.
(2) Personal: Depend upon the particular way in which the observer takes the data. They may be minimized by taking observations under various conditions and by using several observers working independently.
(3) External: Usually caused by conditions over which the observer has no control, but that can be estimated. Therefore, they cannot be eliminated, but the necessary corrections may be applied.
Random Errors: Errors that result from unknown and unpredictable variations in experimental situations, sometimes beyond the control of the observer. Usually due to a large number of factors, each of which adds its own small contribution to the total error. Positive and negative errors are equally probable, so random errors are subject to the laws of chance. Random errors can be minimized by improving and refining experimental techniques and by repeating the measurement a large number of times, so erroneous readings become statistically insignificant.
Accuracy: Measures the correctness of an experimental result; this is, how close the experimental result comes to the true value. The accuracy of an experimental value depends on systematic errors. When a measurement has no systematic error we say it is exact.
Precision: Measures the reliability of an experimental result; this is, it measures the magnitude of the uncertainty of the result (how close different outcomes are). When random errors are minimized, we say that the result is precise.
3.2 - Statistical Quantification of Errors
Most Probable Value: Also known as average value. For a list of N experimental measurements, it is obtained by adding all the measurements and dividing the result by the quantity of measurements that were added.
Deviation: Quantity that tells how much a particular measurement scatters from the mean. A deviation may be positive or negative, so the sum of the deviations is expected to be (ideally) 0.
Average (or Mean) Deviation: Quantity that measures the dispersion of a set of experimental measurements from the mean (a measure of precision).
Variance: Average of the squares of the deviations of a set of measurements; it is used to avoid the problem of negative deviations and absolute values.
Standard Deviation: Quantity that gives the absolute error of a set of measurements. It describes the precision of the mean of a set of measurements. Operationally defined as the square root of the variance. For a small number of measurements, it can be statistically shown that a better value for the standard deviation is obtained when the sum of the squares of the deviations is divided by N-1 instead of N.
Accepted or "True" Value: Most accurate value of a quantity that is usually obtained through sophisticated experiments or mathematical methods. This is the value found on textbooks and handbooks.
Absolute Difference: Absolute value of the difference between an experimental value and the accepted value of a quantity.
Fractional Error: Ratio of the absolute difference to the accepted value.
Percent Error: Most common way of expressing the fractional error, given by the fractional error X 100%.
Percent Difference: Ratio of the absolute difference between two experimental values to their average. When there are more than two measurements, the PD is found by dividing the absolute value of the difference of the extreme values by the average of all the measurements.
Chapter 4.- Graphical Analysis
Direct Proportion y = kx
Linear Function y = yo + kx
Potential Function y = kxn
Exponential Function y = yoekx
Chapter 5 - Vector Analysis
Vector Analysis: Mathematical formalism that tells us how to use mathematical objects called vector, which have direct application in physics and engineering.
Scalar: Quantity that has only magnitude; it requires only a number to completely define it.
Vector: Quantity that has two characteristics: magnitude and direction.
Vector Algebra: Mathematical theory developed for vectors. It is the group of rules that can be applied to the set of mathematical objects called vectors.
Scalar Product: Process for multiplying two vectors in such a way that a scalar is obtained. Operationally, it is defined as the product of the magnitude of the two vectors and the cosine of the angle between them.
Vector Product: Process for multiplying two vectors in such a way that they produce a new vector. The magnitude of the new vector is the product of the magnitudes of the two vectors multiplied and the sine of the angle between them. The direction of the resulting vector is perpendicular to the plane formed by the vectors that were multiplied, as given by the right-hand-rule.
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