Dimensional Analysis. Units and Types  Units are meters, seconds, feet, tons, etc.  Types of units are length, mass, force, volume, etc.  The type.

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Presentation transcript:

Dimensional Analysis

Units and Types  Units are meters, seconds, feet, tons, etc.  Types of units are length, mass, force, volume, etc.  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 Units  It is useful to convert the dimensions of units into fundamental dimensions. Length (L)Length (L) Time (T)Time (T) Mass (M)Mass (M)  Units can be raised to a power, and so can the fundamental dimensions. Area (L 2 )Area (L 2 ) Volume (L 3 )Volume (L 3 ) Force (M L / T 2 )Force (M L / T 2 )

Missing Units  The energy in a compressed spring is given by U = ½ kx 2. U is the energy in kg m 2 /s 2, and x is the length in m. What are the correct units for k?  Use dimensional analysis:  Substitute units for dimensions: k has units of kg/s 2

Dimensional Expressions  The speed of waves in shallow water depends only on the acceleration of gravity g, with dimensions L/T 2, and on the water depth h. Which of the following formulas for the wave speed v could be correct? b) a)

Base Quantities Acceleration g  dimensions: L/T 2  length/time 2  example: m/s 2 Height h  dimensions: L  length  example cm Speed v  dimensions: L/T  length/time  example km/h

Checking a Result  Terms do not match  Terms match, this could be a valid formula.

Limitations  Dimensional analysis only checks the units.  Numeric factors have no units and can’t be tested. is also valid. is also valid. is not valid. is not valid. next