Dimensional Analysis Units and Types Units are meters

Units and Types ] Units are meters, seconds, feet, tons, etc. ] Types of

Matching Units ] ] Conversion between units must be of the same type. Length

Conversion Factors ] A value is converted by applying the ratio of the conversion

Powers of Units ] It is useful to convert the dimensions of units into

Missing Units ] The energy in a compressed spring is given by U =

Dimensional Expressions ] The speed of waves in shallow water depends only on the

Base Quantities Acceleration g Height h Speed v ] dimensions: L/T 2 ] dimensions:

Checking a Result ] Terms do not match ] Terms match, this could be

Limitations ] ] Dimensional analysis only checks the units. Numeric factors have no units

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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 (L 2) • Volume (L 3) • 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? a) b)

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