Statics/Measurement and Units
In statics and mechanics, units can be expressed in terms of three basic dimensions: length, mass, and time. All other units are just a combination of the three basic units. For example, Force can be expressed in the unit of Newtons, a Newton is one kg*m/s^2. In other words, units exist for force, but force can also be described as a combination of the three basic units.
International System of Units (SI Units)
In the SI system of units, the three specified base units are the units of length, mass and time. The fourth unit, that of force, is then defined in terms of these.
- The unit of length is the meter, abbreviated m.
- The unit of mass is the kilogram, abbreviated kg.
- The unit of time is the second, abbreviated s.
- The unit of force, defined as one kilogram meter per second squared is the Newton (1 N = 1 kg-m/s^2). In other words, this is the force required to accelerate a mass of one kilogram at the rate of one meter per second squared.
When working with units that are large multiples of these units, or conversely small fractions, we often express them in terms of a system of multiples and submultiples with a prefix to describe the multiple or submultiple we are using. For example,
1000 meters = 1 kilometer (1000 m = 1 km).
A complete set of prefixes is:
1 000 000 000 000 000 = 10^15 peta (P)
1 000 000 000 000 = 10^12 tera (T)
1 000 000 000 = 10^9 giga (G)
1 000 000 = 10^6 mega (M)
1 000 = 10^3 kilo (k)
100 = 10^2 hecto (h)
10 = 10^1 deca (da)
0.1 = 10^-1 deci (d)
0.01 = 10^-2 centi (c)
0.001 = 10^-3 milli (m)
0.000 001 = 10^-6 micro (μ)
0.000 000 001 = 10^-9 nano (n)
0.000 000 000 001 = 10^-12 pico (p)
0.000 000 000 000 001 = 10^-15 femto (f)
0.000 000 000 000 000 001 = 10^-18 atto (a)
Other measurements used in both statics and mechanics are based on these four units.
For example, area is length times width and is given is square meters ().
Common SI units used in mechanics include the following:
Acceleration meter per second squared (m/s^2)
Angle radian (r)
Angular Acceleration radian per second squared (r/s^2)
Angular Velocity radian per second (r/s)
Area square meter (m^2)
Density kilogram per cubic meter (kg/m^3)
Energy Joule (J) or (N-m)
Force Newton (N) or (kg-m/s^2)
Frequency Herz (Hz) or (1/s)
Impulse Newton-second (kg-m/s)
Length meter (m)
Mass kilogram (kg)
Moment of a Force Newton-meter (N-m)
Power Watt (W) or (J/s)
Pressure Pascal (Pa) or (N/m^2)
Stress Pascal (Pa) or (N/m^2)
Time second (s)
Velocity meter per second (m/s)
Volume (solids) cubic meter (m^3)
Volume (liquids) litre (L) or (10-3 m^3)
Work Joule (J) or (N-m)
When talking about the volume of liquids, the commonly used unit is the cubic decimeter which is called the litre (L).
British and American Customary Units
While the International System of units is in common use throughout much of the world, engineers may still find British or American units in use. Hence, it is a good idea to have some familiarity with these units.
While the basic units in International System of units are length, mass, and time--with the unit of force defined in terms of these--in the British and American units, the base units are length, force and time, with mass being defined in terms of these.
- The unit of length is the foot (ft).
- The unit of force is the pound (lb).
- The unit of time is the second (s).
In fact, the unit of time is common to both systems of units.
The unit of mass in British and American units is defined as the mass that will be accelerated at a rate of 1 ft/s^2 when a force of 1 lb is applied. This unit is customarily called the slug. Mass is also occasionally described as lb(mass), A 1b(mass) is equal to the mass required to make one lb of weight when acted upon by the standard acceleration of gravity. The standard acceleration of gravity is about 32.2 ft/s^2, this means that one slug = 32.2 lb(mass).
Conversion from one System of Units to Another
While we can do all our calculations in one set of units or the other, as long as we are consistent, there are times we will want to convert from one system to the other.
- Unit of Length 1 ft = 0.3048 m
- Unit of Force 1 lb = 4.448 N
- Unit of Mass 1 slug = 1 lb-s^2/ft = 14.59 kg
As mentioned earlier, the second is the same in both systems of units and and so no conversion is required.
Common British and American Customary units and their SI equivalents are:
Acceleration ft/s^2 0.3048 m/s^2
in/s^2 0.0254 m/s^2
Area ft^2 0.0929 m^2
in^2 645.2 mm^2
Energy ft-lb 1.356 J
Force kip 4.448 kN
lb 4.448 N
oz 0.2780 N
Impulse lb-s 4.448 N-s
Length ft 0.3048 m
in 25.40 mm
mi 1.609 km
Mass oz mass 28.35 g
lb mass 0.4536 kg
slug 14.59 kg
ton 907.2 kg
Moment of a Force lb-ft 1.356 N-m
lb-in 0.1130 N-m
Moment of Inertia
Of an Area in^4 0.4162 x 10^6 mm^2
Of a mass lb-ft/s^2 1.356 kg-m^2
Momentum lb-s 4.448 kg-m/s
Power ft-lb/s 1.356 W
hp 745.7 W
Pressure lb/ft^2 47.88 Pa
lb/in^2 (psi) 6.895 kPa
Stress lb/ft^2 47.88 Pa
lb/in^2 (psi) 6.895 kPa
Velocity ft/s 0.3048 m/s
in/s 0.0254 m/s
mi/h or mph 0.4470 m/s or 1.609 km/hr
Volume (solids) ft^3 0.02832 m^3
in^3 16.39 cm^3
Volume (liquids) gal 3.785 l
qt 0.9464 l
Work ft-lb 1.356 J
Example
According to the official National Hockey League rulebook, "The official size of the (hockey) rink shall be two hundred feet (200') long and eighty-five feet (85') wide." What are the dimensions in SI units?
We need to convert the dimensions from feet to metres.
From the above table, 1 ft = 0.3048 m.
The length of the rink is 200 ft x 0.3048 m/ft = 60 m.
The width of the rink is 85 ft x 0.3048 m/ft = 26 m.
Dimension Analysis: [ft]*[m/ft]=[m]
Significant Digits
When we talk about measurements and calculations, we need to understand the degree of accuracy involved.
The accuracy of our calculations cannot be more precise than the accuracy of our measurements.
Suppose we are provided with a distance to an accuacy of one decimal place, say 9.8 m. We are told an object travels this distance in 0.81 seconds. It does not make sense to say the object is travelling at a velocity of 12.11111111 m/s, that is, to eight decimal places.
This is because neither the distance nor the time taken to travel this distance is specified to this degree of precision. In fact, they are both specified to an accuracy of only two significant digits.
For reasons we will discuss shortly, we can say the object is travelling at a velocity of 12.1 m/s.
For many calculations in statics, we work to at most three significant digits.
That is, we would only express our results with only three digits of accuracy, unless the left-most digit is a '1', in which case we can express our results to four digits. This is commonly known as slide-rule accuracy as that is the accuracy one would expect from a slide-rule.
1 - Both the principal SI units used in mechanics and the US Customary units and their SI equivalents are taken from Beer, Ferdinand P. and El Russell Johnston, Jr. "Vector Mechanics For Engineers, Statics" 3rd edition, McGraw Hill c 1977. It should be possible to find similar tables in other texts on this subject.