Dimensional Analysis: Understanding Units, Types, and Dimensions, Slides of Physics

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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.

Powers of Units ^ It is useful to convert the dimensions of units intofundamental dimensions.•^ Length (

L )

-^ Time (

T )

-^ Mass (

M )

^ Units can be raised to a power, and so can thefundamental dimensions.•^ Area (

2 L )

-^ Volume (

3 L )

-^ Force (

M^ L^ /^ T

2 )

Missing Units ^ The energy in a compressedspring is given by

U^ = ½^ kx

U^ is the energy in kg m

22 /s, and

x^ is the length in m. What arethe correct units for

k?

^ Use dimensional analysis: ^ Substitute units fordimensions:•^ k

has units of kg/s

2 2 (^2222)

12 2 2 2 /

ML T

kxU MLkLT MLk T



  

Dimensional Expressions ^ The speed of waves in shallow water depends only on theacceleration of gravity

g , with dimensions

(^2) L / T , and on the

water depth

h. Which of the following formulas for the wave speed^ v

could be correct?^ ghv 

a)^ b)

1 2 gh^2

v^ 

Base Quantities Acceleration

g ^ dimensions:

2 L / T

^ length/time

2 ^ example: m/s

Height 2

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 avalid formula.

1 2 gh^2

v^ 

L^2 L^2 T

L^  T

3 L^2 T

L^  T

ghv  L^ L T L^ T ^2 L^ T L^  T

Limitations ^ Dimensional analysis only checks the units. ^ Numeric factors have no units and can’t be tested.

is also valid.is not valid.

ghv  3

next

4 

^ ghv