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Units and Dimensions

In physics we measure quantities such the length of a room or the mass of an electron. The measurement results in a physical quantity consisting of a pure number and a unit. Physicists also discuss dimensions of physical quantities. The System International (SI) is based on 4 fundamental dimensions, length, L, time, T, mass, M, and charge Q. All quantities in this course are combinations of the first three "dimensions." Each of these dimensions must be measured in some type of unit.

The base units that we will use are the SI units, sometimes called mks. Other systems of units are the cgs and US Engineering. They are defined in the text.

Quantity

Dimension

SI Unit

cgs

US Engineering (1)

US Engineering (2)

Length

L

meter, m

centimeter, cm

foot. ft

meter

Mass

M

kilogram, kg

gram, g

slug

kg-mass, kgm

Time

T

second, s

second, s

second, s

second, s

The base units and dimensions are used to find units of derived quantities. Some of the unit combinations have their own names.

Quantity

  

SI Unit

cgs

US Engineering (1)

US Engineering (2)

Area

L2

m2

cm2

ft2

m2

Speed

L/T

m/s

cm/s

ft/s

m/s

Acceleration

L/T2

m/s2

cm/s2

ft/s2

m/s2

Force

M L/T2

Newton, N

1 N = 1 kg m/s2

dyne

1dyne = 1 g cm/s2

pound, lb

1 lb = 1 slug ft/s2

kilogram-force, kgf

1 kgf = 9.80665 kgm m/s2

Named Prefixes

Your text lists the named prefixes for powers of 10. Most of the prefixes represent powers of 10 spaced by 103 eg. milli = 10-3, micro = 10-6, etc. Calculators often call this Engineering notation. The following will be the common prefixes in this course.

micro 

(mu)

10-6

milli

m

10-3

centi

c

10-2

kilo

k

103

mega

M

106

Determining Units in a Formula:

Eg. The speed, v, of a water wave in shallow water of depth h is v = A1/2 h1/2 where A is a constant. Find the dimensions and units of A.

Rewrite the equation as A = v2/ h, then the dimension of A, [A] = [v2] [h] = [L/T]2 (1/[L]) = [L/T2], and so the SI units are m/s2.

In the following examples, determine the dimensions of the quantity and its SI units.

1. For a car moving through the air, the acceleration, a, is given by a = B v + C v2 where v is speed and B and C are constants. Determine the units of B and C.

2. The rate of flow of a liquid out of a tube of radius r and length x is given by

V/ t = P  r4 /(D x)

where V is the volume and P is pressure in SI units of kg/(s2 m). Find the units of the viscosity, D.

3. When a mass is attached to a spring, the acceleration is a = k x/m where a is acceleration, x is a length, and k is a spring constant. Find the units of k.

Converting Units

Use the units as a guide to doing conversions. For example,

Convert 25.0 pounds/square inch into newtons/square meter.

1. Find some conversion factors, either in the text(usually in an Appendix or flyleaf), or at Rowlett's compendium of units at http://www.unc.edu/~rowlett/units/index.html.

a. 1 pound = 4.45 newtons

b. 1 inch = 2.54 cm

c. 100 cm = 1 m

2. Set up the units, remembering to square where needed

3. Put in the numbers and calculate

4. Round the answer to make the result have a reasonable number of significant figures. Here I had 3 sig. fig. to start (25.0), and I assume that the conversion factors are exact, so I use 3 sig fig in the answer, 25.0 pounds/square inch = 172000 newtons/square meter = 172 kN/m2, where I use the k = kilo to make the number easier to read.

Your Conversion:

Rainfall is sometimes measured in "inch-acres", that is the volume of water that corresponds to a one inch depth on an acre of land. The density of water is 1 gram/cubic centimeter. Convert this density to (metric tons)/(inch-acre). Some conversion factors are: 1 metric ton (tonne) = 1000 kg, 1 inch = 2.54 cm, and 1 acre = 4840 square yards. You may work in groups at your table, but each person should turn in a sheet with the solution neatly presented.