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 kilometers per hour has dimensions of a velocity.

Matching Units Conversion between units must be of the same type. Length conversion: 1 in = 2.54 cm Time conversion: 1 hr = 3.6 x 103 s No conversion between different types of units. 1 in is not equivalent to some seconds

Conversion Factors A value is converted by applying the ratio of the conversion factors. How many inches in 50. cm? 50. cm (1 in / 2.54 cm) = (50. / 2.54) in = 20. in Many conversion factors use scientific notation. How many seconds in a year? 1 yr (365 d/yr) (24 hr/d) (3.6 x 103 s/hr) = 31500 x 103 s = 3.15 x 107 s

Powers of Units It is useful to convert the dimensions of units into fundamental dimensions. Length (L) Time (T) Mass (M) Units can be raised to a power, and so can the fundamental dimensions. Area (L2) Volume (L3) Force (M L / T2)

Missing Units Use dimensional analysis: The energy in a compressed spring is given by U = ½ kx2. U is the energy in kg m2/s2, 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/s2

Dimensional Expressions The speed of waves in shallow water depends only on the acceleration of gravity g, with dimensions L/T2, and on the water depth h. Which of the following formulas for the wave speed v could be correct? a) b)

Base Quantities Acceleration g dimensions: L/T2 length/time2 example: m/s2 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 not valid.