1.4 Dimensional Analysis - University Physics Volume 1 _ OpenStax.pdf

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5/21/20201.4 Dimensional Analysis - University Physics Volume 1 | OpenStax1/5Learning ObjectivesBy the end of this section, you will be able to:Find the dimensions of a mathematical expression involving physical quantities.Determine whether an equation involving physical quantities is dimensionally consistent.The dimensionof any physical quantity expresses its dependence on the base quantities as a product of symbols (or powers of symbols)representing the base quantities. Table 1.3 lists the base quantities and the symbols used for their dimension. For example, a measurement oflength is said to have dimension L or L, a measurement of mass has dimension M or M, and a measurement of time has dimension T or T. Likeunits, dimensions obey the rules of algebra. Thus, area is the product of two lengths and so has dimension L, or length squared. Similarly, volumeis the product of three lengths and has dimension L, or length cubed. Speed has dimension length over time, L/T or LT. Volumetric mass densityhas dimension M/Lor ML, or mass over length cubed. In general, the dimension of any physical quantity can be written as for some powers and g. We can write the dimensions of a length in this form with and the remaining six powers all set equalto zero: Any quantity with a dimension that can be written so that all seven powers are zero (that is, its dimension is ) is called dimensionless(or sometimes “of dimension 1,” because anything raised to the zero power is one). Physicists oftencall dimensionless quantities pure numbers.Base QuantitySymbol for DimensionLengthLMassMTimeTCurrentIThermodynamic temperatureΘAmount of substanceNLuminous intensityJTable 1.3Base Quantities and Their DimensionsPhysicists often use square brackets around the symbol for a physical quantity to represent the dimensions of that quantity. For example, if isthe radius of a cylinder and is its height, then we write and to indicate the dimensions of the radius and height are both those11123–13–3
5/21/20201.4 Dimensional Analysis - University Physics Volume 1 | OpenStax2/5of length, or L. Similarly, if we use the symbol for the surface area of a cylinder and for its volume, then [A] = Land [V] = L. If we use thesymbol for the mass of the cylinder and for the density of the material from which the cylinder is made, then and

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