Dimensional AnalysisUnits and TypesPowers of UnitsMissing UnitsDimensional ExpressionsBase QuantitiesChecking a ResultLimitationsDimensional AnalysisDimensional AnalysisUnits and TypesUnits and TypesUnits are meters, seconds, feet, tons, etc.Units are meters, seconds, feet, tons, etc.Types of units are length, mass, force, volume, etc.Types of units are length, mass, force, volume, etc.The type of unit of a value is called the dimension.The type of unit of a value is called the dimension.•A value in square meters has dimensions of an area.A value in square meters has dimensions of an area.•A value in kilometers per hour has dimensions of a velocity.A value in kilometers per hour has dimensions of a velocity.Powers of UnitsPowers of UnitsIt is useful to convert the dimensions of units into It is useful to convert the dimensions of units into fundamental dimensions.fundamental dimensions.•Length (Length (LL))•Time (Time (TT))•Mass (Mass (MM))Units can be raised to a power, and so can the Units can be raised to a power, and so can the fundamental dimensions.fundamental dimensions.•Area (Area (LL22))•Volume (Volume (LL33))•Force (Force (MM LL / / TT22))Missing UnitsMissing UnitsThe energy in a compressed The energy in a compressed spring is given by spring is given by UU = ½ = ½ kxkx22. . UU is the energy in kg m is the energy in kg m22/s/s22, and , and xx is the length in m. What are is the length in m. What are the correct units for the correct units for kk??Use dimensional analysis:Use dimensional analysis:Substitute units for Substitute units for dimensions:dimensions:•kk has units of kg/s has units of kg/s2222222222/21TMLTMLkkLTMLkxUDimensional ExpressionsDimensional ExpressionsThe speed of waves in shallow water depends only on the The speed of waves in shallow water depends only on the acceleration of gravity acceleration of gravity gg, with dimensions , with dimensions LL//TT22, and on the , and on the water depth water depth hh. Which of the following formulas for the wave . Which of the following formulas for the wave speed speed vv could be correct? could be correct?ghv b)a)221ghv Base QuantitiesBase QuantitiesAcceleration Acceleration ggdimensions: dimensions: LL//TT22length/timelength/time22example: m/sexample: m/s22 Height Height hhdimensions: dimensions: LLlengthlengthexample cmexample cmSpeed Speed vvdimensions: dimensions: LL//TTlength/timelength/timeexample km/hexample km/hChecking a ResultChecking a ResultTerms do not matchTerms do not matchTerms match, this could be a Terms match, this could be a valid formula.valid formula.221ghv 22LTLTL23TLTLghv LTLTL2TLTLLimitationsLimitationsDimensional analysis only checks the units.Dimensional analysis only checks the units.Numeric factors have no units and can’t be tested.Numeric factors have no units and can’t be tested. is also valid.is also valid. is not valid.is not valid.3ghv next4
View Full Document
Unlocking...