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Dimensional Analysis

Learn about dimensional analysis, unit types, conversion factors, powers of units, and missing units in expressions. Practice converting units with examples. Understand the limitations of dimensional analysis.

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Dimensional Analysis

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  1. Dimensional Analysis

  2. 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.

  3. 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

  4. 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

  5. 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 (ML / T2)

  6. 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 Missing Units

  7. 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)

  8. Acceleration g dimensions: L/T2 length/time2 example: m/s2 Base Quantities Height h • dimensions: L • length • example cm Speed v • dimensions: L/T • length/time • example km/h

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

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

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