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) = 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 (M L / T2)

6. 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

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. 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

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

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