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Physicists usually define symmetry as invariance under transformation Basically, an object or system is called symmetric when you observe it from a different perspective And it remains unchanged Triangle Rotation Triangle
Most often we think of symmetry in terms of patterns or shapes
We encounter symmetry of translation, reflection, and rotation all around us… Translational symmetry Rotational symmetry Reflectivesymmetry
The most obvious kind of isometry is called a translation, and amounts to just pushing an object in a straight line to a new location: There is not a fixed point
another very familiar kind of isometry is rotation: Translations have no fixed points at all, while rotations have exactly one (called a pivot point, around which everything rotates doesn't move at all.
Reflections are isometries that have infinitely many fixed points.
The least well-known kind of isometry is usually called a glide reflection. It is a kind of cross between a reflection and a translation:
if we consider both orientation and fixed point behavior, each type of isometry has a unique character:
What kind of symmetrydo you find with a Checkerboard?
The laws of nature are also symmetrical With respect to translation Whether you perform an experiment in New York or Los Angeles, at the other edge of the Milky Way or in a galaxy a billion light-years from here, you will be able to describe the results using the same laws. With respect to rotation The laws look precisely the same whether we make measurements from the bottom, top, sides, etc. - physics has no preferred direction in space.
With respect to reflection The laws of physics are the same in a right-handed system of coordinates as in a left-handed system With respect to time The laws work exactly the same in experiment today as they did on an experiment performed yesterday or last year.
One of Einstein’s main goals in his explanation of general relativity was to formulate a theory in which the laws of nature would look precisely the same to all observers. In other words, the laws had to be symmetrical under any change in our point of view in space and time
Conservation Laws the word conserve does not have its usual meaning of trying not to waste something. The word “conserve” in physics means that a particular measurable property of a closed system does not change with time Thus, a conserved quantity is something that you wouldn't be able to get rid of even if you wanted to.
Since the total momentum in the system stayed the same, the momentum was “conserved”
Conservation of Angular Momentum: The total angular momentum of the system is constant. Conservation of Energy: the total energy of the system is constant. Conservation of Momentum: the mass times the velocity of the center of mass is constant.
Symmetry & the Laws of Physics The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system
Noether’s theorem states that each symmetry of a physical system implies that some physical property of that system is conserved. Conversely… Each conserved quantity has a corresponding symmetry
Everyone is familiar with energy but no one knows exactly what energy actually is For our purposes we will define Energy asthe measure of the ability to generate motion. A system that has energy has the ability to do work (motion in action). Energy is measured in the same units (joules) as work because energy is transferred during the action of work.
The SI unit of energy is the joule, J, (rhymes with cool), named after the British physicist James Joule. One Joule is the amount of energy required in order to heat 0.24 g of water by 1 °C. (The number 0.24 is not worth memorizing.)
Energy appears in different forms such as mass, heat, or motion Energy can travel in different ways such as light, sound, or electricity The workings of our universe—including all of our Technology—can be viewed from the perspective of energy flowing from one place to another and changing back and forth from one form to the next
Physics Fundamental Principle: Conservation of energy The total energy of a closed system always remains constant. Energy cannot be created or destroyed, but only transferred from one system to another
What is a system? A ‘system’ just means a group of objects that can be treated as a single unit. Example: our solar system
The law of Conservation of energy has many implications such as… Your car will not run forever on one tank of gas
Don’t we create energy at a power plant? No, we simply transform energy at our power plants
Energy in the form of mass Einstein showed that mass itself could be converted to and from energy, according to his celebrated equation E = mc2, in which c is the speed of light. Thus we can view mass as simply another form of energy, and it is valid to measure it in units of joules. The mass of a 15-gram pencil corresponds to about 1.3 × 1015 J.
We get almost all of our energy from the sun The sun exchanges mass for energy through a nuclear reaction
Cosmic rays, however, are continually striking you and your surroundings and converting part of their energy of motion into the mass of newly created particles. A single high-energy cosmic ray can create a shower of millions of previously nonexistent particles when it strikes the atmosphere.
Mass can be defined from two different perspectives: 1) Mass is the measure of the amount of matter that a body contains 2) Mass is a measure of the inertial property of that body, that is, of its resistance to change of motion (Inertia).
Physics Fundamental Principle: Conservation of mass the total mass of a closed system always remains constant
In physics we study matter (and energy!) by measuring it Thus, many properties of matter are expressed quantitatively (associated with numbers) In order to understand what the numbers refer to we need to include “units”
Physics uses a select groupof the SI units knownas MKS (meter, kilogram, second) system
Writing a number in scientific notation:1)Put the decimal after the first digit and drop the zeroes2) Count the number of decimal places moved in step 13) Write as a product of the number (step 1) and 10raised to the power of the count (step 2) The Andromeda Galaxy (the closest one to our Milky Way galaxy) contains at least 200,000,000,000 stars. So we would write 200,000,000,000 in scientific notation as: 2.0 x 1011 This number is read as follows: "two point zero times ten to the eleventh."
Now we try a number that is very small. Change 0.000000902 to scientific notation The decimal must be moved behind the 9 The coefficient will be 9.02 The decimal moves seven spaces to the right, making the exponent -7 Answer equals 9.02 x 10-7
Examples Write each of the following numbers in scientific notation: (a) 93,000,000 (b) .00005144 (c) -33,452.8
Changing numbers from scientific notation to standard notation. Change 6.03 x 107 to standard notation. we can simply move the decimal seven places to the right because the exponent is 7. So, 6.03 x 107 = 60 300 000
Now let us try one with a negative exponent. Ex.2 Change 5.3 x 10-4 to standard notation. The exponent tells us to move the decimal four places to the left. so, 5.3 x 10-4 = 0.00053
Express in standard form: 1. 1.09 x 103 2. 4.22715 x 108
Scientific certainty Precision Accuracy Precision indicates how close together or how repeatable the results are. Accuracy indicates how close a measurement is to the accepted value. Accurate Inaccurate
PRECISION AND ACCURACY -- Quiz Consider the data obtained for the length of an object as measured by three students. The length is known to be 14.54 cm. Which of the conclusions summarizes the data? Trial #1 #2 #3 #4 #5 Student A 14.8 14.1 14.5 14.6 14.2Student B 14.8 14.2 14.6 14.5 14.8Student C 14.6 14.5 14.5 14.4 14.6 a) Student A has done the most precise work and student C the most accurate.b) Student C has done the most precise and accurate work.c) Student C has done the most precise work and student A the most accurate.d) Student C has done the most precise work and student B the most accurate.e) Student B has done the most precise work and student C the most accurate.
Significant figures All digits of a measured quantity are called significant figures
Scientific uncertainty all measurements contain some uncertainty. Such data is reported in significant figures to inform the reader of the uncertainty of the measurement. We record all significant figures unto the first uncertain number.
Dimensional analysis (also known as the factor-label method or unit-factor method) Whatever you measure, you have to use units Why is this important?
Example 2: If you are going 50 miles per hour, how many feet per second are you traveling?
We will use dimensional analysis on every physics problem we attempt to solve this year Remember the “joule”, the unit for energy?