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Coaster Dynamics
Physics Primer
Chapter 2
Measurement and Systems of Units
Standard Units
In the application of science, precise measurement of physical quantities and phenomenon are required to accurately describe the systems being studied.
When developing a measurement system, certain key quantities must be chosen as fundamental quantities, from which other derived quantities are defined. This context is loosely analogous to state variables and derived variables in a state-determined system (as discussed in Chapter 1).
The choice of fundamental quantities and units in a measurement system is somewhat arbitrary -- as is the case with the selection of state variables. However, the choice of fundamental quantities will affect their usage -- which can make the chosen system either advantageous or inconvenient.
Measurement Systems
There are 3 measurement systems frequently employed today in common usage and in science: the Metric System, the Gausian System, and the British Engineering System.
In the Metric (mks) System, the fundamental quantities are length, mass, and time, with fundamental units of meters, kilograms, and seconds.
In the Gausian (cgs) System, the fundamental quantities are also length, mass, and time, with fundamental units of centimeters, grams, and seconds.
In the British (fps) Engineering System, the fundamental quantities are length, force, and time, with fundamental units of feet, pound-force, and seconds. There is also an alternate basis for the British (fss) Engineering System which uses a fundamental quantity of mass, with units of slugs. The use of this system is typically limited to few applications, such as fluid mechanics.
The measurement systems are summarized in Table 2-A.
Metric (mks) Gausian (cgs) British (fps) British (fss) Mass kilogram gram pound-mass slug Length meter centimeter foot foot Time second second second second Force newton dyne pound-force pound-force Temp ° kelvin ° kelvin ° rankine ° rankine Table 2-A. Common Systems of Units.
The one unit shared in all the systems is that for time -- the second. Originally, the second was defined in terms of a precise fraction a solar year. However, that definition has been replaced with a more accurate one based on the measurable oscillation frequency of the cesium atom. Using this basis, the definitions and procedures for constructing atomic clocks have been standardized within the scientific world, and are precise to within one second in 5000 years.
The definition of a meter was originally based on the length of the standard meter -- a bar made of platinum-iridium alloy that is stored in Paris, France. Because a physical bar is subject to environmental changes, it is no longer used as the standard. Instead, the meter is now defined in terms of the precise number of wavelengths of light emitted from a particular isotope of krypton (and no... Superman has nothing to do with this definition). Similarly, rather than relying on the length of a physical instrument, the foot is now defined in terms of its fraction of one meter.
The definition of a kilogram of mass remains tied to the standard kilogram -- a cylinder made of platinum-iridium alloy maintained by the International Bureau of Weights and Measures in Paris, France, that is defined to have a mass of exactly 1.0 kilogram.
For the Metric and Gausian Systems, force is a derived unit, defined by means Newton's Second Law of Motion. That is, a force of one newton (named after Isaac Newton) will accelerate a mass of one kilogram at
one m/s². Similarly, in the Gausian System, a force of one dyne will accelerate a mass of one gram at one cm/s².For the British Engineering System, force is a fundamental unit, defined to be the force that gravity exerts on a one-pound mass on earth at sea level. Since earth's gravity is used as part of the definition of a pound-force, to be precisely correct, it must be specified at sea level. Analogous to the Metric System, a pound-force will accelerate a pound-mass at the "unit acceleration" of 32.174 ft/s². However, in the British System, mass is a derived quantity, not a fundamental quantity.
One of the unfortunate complications of the British System is the use of the term "pound" for both a force and a mass. Thus, they should be distinguished by specifying either "pound-force" or "pound-mass", or lbf or lbm.
Another complication resulting from this choice of units is that in some cases,
a special conversion factor (pronounced "g-of-c") is needed when relating masses and forces:gc = 32.174 (lbm · ft)/(lbf · s²).
Conversion of Units
Some useful unit conversion factors are summarized in Table 2-B.
Metric (mks) Gausian (cgs) British (fps) British (fss) Mass 1.00 kg 1000 g 2.20 lbm 0.0685 slug Length 1.00 m 100 cm 3.28 ft 3.28 ft Force 1.00 newton 10,000 dynes 0.225 lbf 0.225 lbf Table 2-B. Common Unit Equivalence Relationships.
To convert from one system of units to another, a useful technique is to form the ratio of equivalent quantities of different units as a number equal to one. For example
(1.00 m)/(3.28 ft) = 1.
Then, multiply the quantity to be converted by the appropriate ratios. For instance, to convert an acceleration of 9.81 m/s² to ft/s²:
9.81 m/s² = ( 9.81 m/s² )( 3.28 ft/m ) = 32.2 ft/s².
Reference Frames
Physical quantities measured by observers who are in different locations, or who are moving with respect to one another, will have different values. However, the quantities themselves are not different -- only the observed numerical measurements are different.
For example, if you were standing at the side of a road and you measured the speed of a car passing by with your brother inside, the car might be going at 30 miles per hour. If your sister were then in a car following your brother at the same speed, you would measure a speed of 30 miles per hour for both your brother and sister. However, from your sister's viewpoint, in her reference frame of measurement, your brother's car would appear to be motionless.
Every observer uses their own reference frame of measurement. Nevertheless, experiment shows that, despite the different numerical measurements, the relationships between physical quantities remain unchanged for all reference frames. In other words, the laws of physics remain the same for all reference frames. This is called the Equivalence of Reference Frames, and is a fundamental principle in physics.
To simplify our calculations, all measurements in
Coaster Dynamics use the Metric (mks) System, and a fixed Global Coordinate System with its origin directly below the roller coaster track's begin/end point, with the upward direction (pointing away from the earth's center) along the positive Y-dimension. This is shown in Figure 2-A.
Figure 2-A. Global Coordinate System Used in
Coaster Dynamics.G-Force
As discussed above, the quantities used to measure force differ depending on the system of units being employed. Furthermore, the unit of force defined for the Metric System, which is used throughout
Coaster Dynamics, is the newton. The newton is a unit most people cannot relate to easily -- even in countries that primarily use the Metric System.To simplify our lives... we can measure force in terms of a "G" -- which is a dimensionless unit.
A "G" is the force on an object divided by the force exerted on that object by gravity on earth at sea level. In other words, if you are standing on earth at sea level, and there are no other forces acting on you, the force acting on you has magnitude 1.0 G. If the force on you were 2.0 G's, you would seem to weigh twice your normal weight.
Actually, when measured using a Global Coordinate System attached to earth with the positive direction pointing upward, away from the center of the earth, the force exerted on you by gravity while standing on the ground would be -1.0 G (and according to Newton's Third Law of Motion, the force exerted on you by the ground is +1.0 G).
To be precisely correct, you can impress your family and friends by saying that "a force of magnitude 1.0 G is defined as a force that causes a unit of acceleration when applied to a unit of mass" -- this is fundamentally correct regardless of the system of measurement being used.
For roller coaster enthusiasts, G's are an easy way to measure force in a loop. These forces are what press you into your seat when going through a tight loop. High G forces are fun for thrill seekers... but care must be taken to limit the forces to a safe level. As a point of reference, a typical "Top Gun" jet fighter pilot may begin to loss consciousness if subjected to a sustained force of 5 G's, or more.
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