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From Wikipedia
In physics and science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass, length, time, electric charge, and temperature) which describe it; for example, speed has the dimension length / time, and may be measured in meters per second, miles per hour, or other units. Dimensional analysis is based on the fact that a physical law must be independent of the units used to measure the physical variables. A straightforward practical consequence is that any meaningful equation (and any inequality and inequation) must have the same dimensions in the left and right sides. Checking this is the basic way of performing dimensional analysis.
Dimensional analysis is routinely used to check the plausibility of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units, or their dimensions if any.
The basic principle of dimensional analysis was known to Isaac Newton (1686) who referred to it as the "Great Principle of Similitude". James Clerk Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. The 19thcentury French mathematician Joseph Fourier made important contributions based on the idea that physical laws like should be independent of the units employed to measure the physical variables. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham Ï€ theorem. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.
Introduction
Definition
The dimensions of a physical quantity are associated with combinations of mass, length, time, electric charge, and temperature, represented by sansserif symbols M, L, T, Q, and Î˜, respectively, each raised to rational powers.
The term dimension is more abstract than scaleunit: mass is a dimension, while kilograms are a scale unit (choice of standard) in the mass dimension.
As examples, the dimension of the physical quantity speed is distance/time (L/T or LT^{âˆ’1}), and the dimension of the physical quantity force is "mass Ã— acceleration" or "massÃ—(distance/time)/time" (ML/T^{2} or MLT^{âˆ’2}). In principle, other dimensions of physical quantity could be defined as "fundamental" (such as momentum or energy or electric current) in lieu of some of those shown above. Most physicists do not recognize temperature, Î˜, as a fundamental dimension of physical quantity since it essentially expresses the energy per particle per degree of freedom, which can be expressed in terms of energy (or mass, length, and time). Still others do not recognize electric charge, Q, as a separate fundamental dimension of physical quantity, since it has been expressed in terms of mass, length, and time in unit systems such as the cgs system. There are also physicists that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity.
The unit of a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of meters, feet, inches, miles or micrometres; but any length always has a dimension of L, independent of what units are arbitrarily chosen to measure it. Two different units of the same physical quantity have conversion factors that relate them. For example: 1 in = 2.54 cm; then (2.54 cm/in) is the conversion factor, and is itself dimensionless and equal to one. Therefore multiplying by that conversion factor does not change a quantity. Dimensional symbols do not have conversion factors.
Mathematical properties
Dimensional symbols, such as L, form a group: The identity is defined as L^{0} = 1, and the inverse to L is 1/L or L^{âˆ’1}. L rais
Dimensional modeling (DM) is the name of a set of techniques and concepts used in data warehouse design. It is considered to be different from entityrelationship modeling (ER). Dimensional Modeling does not necessarily involve a relational database. The same modeling approach, at the logical level, can be used for any physical form, such as multidimensional database or even flat files. According to Dr. Kimball, DM is a design technique for databases intended to support enduser queries in a data warehouse. It is oriented around understandability and performance. According to him, although transactionoriented ER is very useful for the transaction capture, it should be avoided for enduser delivery.
Dimensional modeling always uses the concepts of facts (measures), and dimensions (context). Facts are typically (but not always) numeric values that can be aggregated, and dimensions are groups of hierarchies and descriptors that define the facts. For example, sales amount is a fact; timestamp, product, register#, store#, etc. are elements of dimensions. Dimensional models are built by business process area, e.g. store sales, inventory, claims, etc. Because the different business process areas share some but not all dimensions, efficiency in design, operation, and consistency, is achieved using conformed dimensions, i.e. using one copy of the shared dimension across subject areas. The term "conformed dimensions" was originated by Ralph Kimball.
Dimensional modeling structure:
The dimensional model is built on a starlike schema, with dimensions surrounding the fact table. To build the schema, the following design model is used:
 Choose the business process
 Declare the Grain
 Identify the dimensions
 Identify the Fact
 CHOOSE THE BUSINESS PROCESS:
The process of dimensional modeling builds on a 4step design method that helps to ensure the usability of the dimensional model and the use of the data warehouse. The basics in the design build on the actual business process which the data warehouse should cover. Therefore the first step in the model is to describe the business process which the model builds on. This could for instance be a sales situation in a retail store. To describe the business process, one can choose to do this in plain text or use basic Business Process Modeling Notation (BPMN) or other design guides like the Unified Modeling Language (UML).
 DECLARING THE GRAIN:
After describing the Business Process, the next step in the design is to declare the grain of the model. The grain of the model is the exact description of what the dimensional model should be focusing on. This could for instance be â€œAn individual line item on a customer slip from a retail storeâ€�. To clarify what the grain means, you should pick the central process and describe it with one sentence. Furthermore the grain (sentence) is what you are going to build your dimensions and fact table from. You might find it necessary to go back to this step to alter the grain due to new information gained on what your model is supposed to be able to deliver.
 IDENTIFY THE DIMENSION:
The third step in the design process is to define the dimensions of the model. The dimensions must be defined within the grain from the second step of the 4step process. Dimensions are the foundation of the fact table, and is where the data for the fact table is collected. Typically dimensions are nouns like date, store, inventory etc. These dimensions are where all the data is stored. For example, the date dimension could contain data such as year, month and weekday.
 IDENTIFY THE FACT:
After defining the dimensions, the next step in the process is to make keys for the fact table. This step is to identify the numeric facts that will populate each fact table row. This step is closely related to the business users of the system, since this is where they get access to data stored in the data warehouse. Therefore most of the fact table rows are numerical, additive figures such as quantity or cost per unit, etc.
Dimensional normalization or snowflaking removes redundant attributes, which are known in the normal flatten denormalized dimensions. Dimensions are strictly joined together in sub dimensions.
Snowflaking has an influence on the data structure that differs from many philosophies of data warehouses. Singe data (fact) table surrounded by multiple descriptive (dimension) tables
Developers often don't normalize dimensions due to several facts:
 Normalization makes the data structure more complex
 Performance can be slower, due to the many joins between tables
 The space savings are minimal
 The use of bitmap indexes can't be done
 Query Performance, 3NF databases suffer from performance problems when aggregating or retrieving many dimensional values that analysis may require. If you are only going to do operational reports then you may be able to get by with 3NF because your operational user will be looking for very fine grain data.
There are some good arguments, why normalization can be useful. It can be an advantage when part of hierarchies, is common to more than one dimension. When the same dimension can be reusable, for example a geographic dimension can be reusable because both the customer and supplier dimensions use it. With the possibility of having many characteristics of a product, snowflaking or normalizing is the possible way to handle such a problem
Benefits of dimensional modeling
Benefits of the dimensional modeling are following:
 Understandability  Compared to normalized model the dimensional model is easier to understand and more intuitive. In dimensional models information is grouped into coherent business categories or dimensions which make it easier to read and interpret. Simplicity allows also software to efficiently navigate databases. But in normalized models data is divided into many discrete entities and even the simple business process might result in dozens of tables that might be joined together in complex way.
 Query performance  Dimensional models are more denormalized and optimized for data querying while normalized models seek to eliminate data redundancies and are optimized for transaction loading and updating. The predictable framework of a dimensional model allows the database to make strong assumptions about the data that aid in performance. Each dimension is a equivalent entry point into the fact table and this symmetrical structure allows effectively handle complex queries. Query optimization for star join databases is simple, predictable, and controllable.
 Extensibility  dimensional model is extensible to accommodate unexpected new data. Existing tables can be change in place either by simply adding new data rows in the table or executing SQL alter table. No queries or other applications that sits on top of the Warehouse needs to be reprogrammed to accommodate the change. Old queries and applications continue to run without yielding different results. But in norm
A chemical formula or molecular formula is a way of expressing information about the atoms that constitute a particular chemical compound.
The chemical formula identifies each constituent element by its chemical symbol and indicates the number of atoms of each element found in each discrete molecule of that compound. If a molecule contains more than one atom of a particular element, this quantity is indicated using a subscript after the chemical symbol (although 18thcentury books often used superscripts) and also can be combined by more chemical elements. For example, methane, a small molecule consisting of one carbon atom and four hydrogen atoms, has the chemical formula CH_{4}. The sugar molecule glucose has six carbon atoms, twelve hydrogen atoms and six oxygen atoms, so its chemical formula is C_{6}H_{12}O_{6}.
Chemical formulas may be used in chemical equations to describe chemical reactions. For ionic compounds and other nonmolecular substances an empirical formula may be used, in which the subscripts indicate the ratio of the elements.
The 19thcentury Swedish chemist JÃ¶ns Jakob Berzelius worked out this system for writing chemical formulas.
Molecular geometry and structural formulas
The connectivity of a molecule often has a strong influence on its physical and chemical properties and behavior. Two molecules composed of the same numbers of the same types of atoms (i.e. a pair of isomers) might have completely different chemical and/or physical properties if the atoms are connected differently or in different positions. In such cases, a structural formula can be useful, as it illustrates which atoms are bonded to which other ones. From the connectivity, it is often possible to deduce the approximate shape of the molecule.
A chemical formula supplies information about the types and spatial arrangement of bonds in the chemical, though it does not necessarily specify the exact isomer. For example ethane consists of two carbon atoms singlebonded to each other, with each carbon atom having three hydrogen atoms bonded to it. Its chemical formula can be rendered as CH_{3}CH_{3}. In ethylene there is a double bond between the carbon atoms (and thus each carbon only has two hydrogens), therefore the chemical formula may be written: CH_{2}CH_{2}, and the fact that there is a double bond between the carbons is implicit because carbon has a valence of four. However, a more explicit method is to write H_{2}C=CH_{2} or less commonly H_{2}C::CH_{2}. The two lines (or two pairs of dots) indicate that a double bond connects the atoms on either side of them.
A triple bond may be expressed with three lines or pairs of dots, and if there may be ambiguity, a single line or pair of dots may be used to indicate a single bond.
Molecules with multiple functional groups that are the same may be expressed by enclosing the repeated group in round brackets. For example isobutane may be written (CH_{3})_{3}CH. This semistructural formula implies a different connectivity from other molecules that can be formed using the same atoms in the same proportions (isomers). The formula (CH_{3})_{3}CH implies a central carbon atom attached to one hydrogen atom and three CH_{3} groups. The same number of atoms of each element (10 hydrogens and 4 carbons, or C_{4}H_{10}) may be used to make a straight chain molecule, butane: CH_{3}CH_{2}CH_{2}CH_{3}.
The alkene but2ene has two isomers which the chemical formula CH_{3}CH=CHCH_{3} does not identify. The relative position of the two methyl groups must be indicated by additional notation denoting whether the methyl groups are on the same side of the double bond (cis or Z) or on the opposite sides from each other (trans or E).
Polymers
For polymers, parentheses are placed around the repeating unit. For example, a hydrocarbon molecule that is described as CH_{3}(CH_{2})_{50}CH_{3}, is a molecule with fifty repeating units. If the number of repeating units is unknown or variable, the letter n may be used to indicate this formula: CH_{3}(CH_{2})_{n}CH_{3}.
Ions
For ions, the charge on a particular atom may be denoted with a righthand superscript. For example Na^{+}, or Cu^{2+}. The total charge on a charged molecule or a polyatomic ion may also be shown in this way. For example: hydronium, H_{3}O^{+} or sulfate, SO_{4}^{2âˆ’}.
For more complex ions, brackets [ ] are often used to enclose the ionic formula, as in [B_{12}H_{12}]^{2âˆ’}, which is found in compounds such as Cs_{2}[B_{12}H_{12}]. Parentheses ( ) can be nested inside brackets to indicate a repeating unit, as in [Co(NH_{3})_{6}]^{3+}. Here (NH_{3})_{6} indicates that the ion contains six NH<
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Answers:You are correct about the initial velocity of the ball thrown straight up because: V = Vo + at 0 = Vo + (9.81)(1) (Velocity is zero at the highest point of the throw) Vo = 9.81 m/s You should also know that as long as you throw the ball with the same initial vertical velocity as the one thrown straight up, the ball will reach the same height regardless of how far it goes horizontally. Now you can draw a right angle triangle. This right angle triangle should represent the initial velocity of the ball. You should probably know that a velocity is a vector quantity at this point and that they can be broken down into it's components, x and y or horizontal and vertical. Assuming that's 32 degrees from the horizontal and knowing that your initial vertical velocity must be the same as the one thrown straight up, then you know that the y or vertical component of your initial velocity must be 9.81 m/s. Using the sine function, you can find the initial velocity (solve for the hypotenuse). Note that solving for the x or horizontal is pointless. sin = opp / hyp or (ycomponent of the initial velocity) / (initial velocity) initial velocity = 9.81 / sin(32) = 18.51 m/s
Answers:*One dimensional motion There are four variables which put together in an equation can describe this motion. These are Initial Velocity (u); Final Velocity (v), Acceleration (a), Distance Traveled (s) and Time elapsed (t). The equations which tell us the relationship between these variables are as given below. v = u + at v2 = u2 + 2 as click for calculator s = ut + 1/2 at2 average velocity = (v + u)/2 *Newton's laws of motion Through Newton's second law, which states: The acceleration of a body is directly proportional to the net unbalanced force and inversely proportional to the body's mass, a relationship is established between Force (F), Mass (m) and acceleration (a). This is of course a wonderful relation and of immense usefulness. F = m x a *Momentum (p) is the quantity of motion in a body. A heavy body moving at a fast velocity is difficult to stop. A light body at a slow speed, on the other hand can be stopped easily. So momentum has to do with both mass and velocity. p = mv *Impulse is the change in the momentum of a body caused over a very short time. Let m be the mass and v and u the final and initial velocities of a body. Impulse = Ft = mv  mu ===================== Work and energy As we know from the law of conservation of energy: energy is always conserved. *Work is the product of force and the distance over which it moves. Imagine you are pushing a heavy box across the room. The further you move the more work you do! If W is work, F the force and x the distance then. W = Fx *Energy comes in many shapes. The ones we see over here are kinetic energy (KE) and potential energy (PE) Transitional KE = mv2 Rotational KE = Iw2 here I is the moment of inertia of the object (a simple manner in which one can understand moment of inertia is to consider it to be similar to mass in transitional KE) a w is angular velocity Gravitational PE = mgh where h is the height of the object Elastic PE = k L 2 where k is the spring constant ( it gives how much a spring will stretch for a unit force) and L is the length of the spring. *Power Power (P) is work( W) done in unit time (t). P = W/t as work and energy (E) are same it follows power is also energy consumed or generated per unit time. P = E/t In measuring power Horsepower is a unit which is in common use. However in physics we use Watt. So the first thing to do in solving any problem related to power is to convert horsepower to Watts. 1 horsepower (hp) = 746 Watts *Circular motion a = v2 / r F = ma = mv2/r *Newton's law of universal gravitation About fifty years after Kepler announced the laws now named after him, Isaac Newton showed that every particle in the Universe attracts every other with a force which is proportional to the products of their masses and inversely proportional to the square of their separation. Hence: If F is the force due to gravity, g the acceleration due to gravity, G the Universal Gravitational Constant (6.67x1011 N.m2/kg2), m the mass and r the distance between two objects. Then F = G m1 m2 / r2 *Acceleration due to gravity outside the Earth It can be shown that the acceleration due to gravity outside of a spherical shell of uniform density is the same as it would be if the entire mass of the shell were to be concentrated at its center. Using this we can express the acceleration due to gravity (g') at a radius (r) outside the earth in terms of the Earth's radius (re) and the acceleration due to gravity at the Earth's surface (g) g' = (re2 / r2) g *Acceleration due to gravity inside the Earth Here let r represent the radius of the point inside the earth. The formula for finding out the acceleration due to gravity at this point becomes: g' = ( r / re )g In both the above formulas, as expected, g' becomes equal to g when r = re. *Density The mass of a substance contained in unit volume is its density (D). D = m/V Measuring of densities of substances is easier if we compare them with the density of some other substance of know density. Water is used for this purpose. The ratio of the density of the substance to that of water is called the Specific Gravity (SG) of the substance. SG = Dsubstance / Dwater The density of water is 1000 kg/m3 *Pressure Pressure (P) is Force (F) per unit area (A) P = F/A Electricity According to Ohm's Law electric potential difference(V) is directly proportional to the product of the current(I) times the resistance(R). V = I R The relationship between power (P) and current and voltage is P = I V Using the equations above we can also write P = V2 / R and P = I2 R
Answers:Here is a site with the equations of motion that are used for projectiles. Aloha
Answers:First of all, the process of going up/down are symmetric: Initially Vx = Vy (45 degree) Finally Vx remains the same, Vy changes the sign, Vy = Vx thus during time t (the time when grasshopper reaches the ground again) Vx t = 1 m (distance traveled) g t = Vy  (Vy) = 2 Vx (change of velocity) So by eliminate Vx, we have 0.5 g t^2 = 1, or t = sqrt(2/g) = 0.447 s, then using Vx t = 1 and Vx = Vy Vx = Vy = 2.236 m/s. V = sqrt( Vx^2 + Vy^2) = 3.16 m (this is how much I got, slightly different from your figure 3.3) The maximum height is Vy (t/2)  (1/2) g (t/2)^2 = 0.25 m
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