The second step in dimensional analysis is therefore to set up a mathematical problem that uses one or more conversion factors to get to the units you are interested in. If you multiply by a conversion factor in the form of a ratio, you are really only multiplying by 1, since the two parts of the ratio equal each other: In the egg problem, the statement that "1 dozen eggs = 12 eggs" is a conversion factor.Ģ. The first step in dimensional analysis is therefore identifying the conversion factor(s) you will need to make your conversion. A conversion factor is a statement of the equal relationship between two units. Dimensional analysis uses three fundamental facts to make these conversions, which lead to the steps in the conversion process:ġ. A notable exception is the conversions among temperature units (see our Temperature module for details). Generally, unit conversions are most easily solved using a process called dimensional analysis, also known as the factor-label method. b.was destroyed in the Martian atmosphere.a.returned to Earth several months early.Whether you realize it or not, when you do this problem in your head, you're figuring it out like this:ĭue to the lack of a common unit for calculations, the Mars Climate Orbiter Even such a simple problem as figuring out that two dozen eggs equals 24 eggs is, at its heart, a unit conversion problem. While most people are not controlling NASA space probes, unit conversion is something that happens every day, in all walks of life. Ideally, people should be comfortable with a variety of ways of converting units in order to allow for collaboration among individuals from a variety of backgrounds. As a result, unit consistency may not be possible within or between teams around the world. Most people, however, are most comfortable working in whatever units they grew up using. The Orbiter loss illustrates the need for consistent use of units. While each individual adjustment mistake was very small, this mistake grew larger and larger over multiple adjustments, resulting in the craft's premature demise in the Martian atmosphere. Since 1 pound = 4.45 Newtons, 4.45 times too much thrust was applied each time the thrusters were used. Unfortunately, the NASA engineers measured this force in pounds (a non-metric unit), while the JPL team worked in Newtons (a metric unit), and the software that calculated how long the thrusters should be fired did not make the proper conversion. The designers had planned for this, and jet thrusters were turned on by the flight controllers to apply a force, making numerous small corrections to readjust its course. Throughout the journey from Earth, solar winds pushed against the solar panels of the probe, throwing the spacecraft off course by a small amount. What caused this disaster? The problem arose in part from a simple, seemingly innocent, mistake. Figure 1: An artists rendition of the Mars Climate Orbiter. The consequences were catastrophic: when the scientists and engineers commanding the probe lost communication, they could only assume that the spacecraft was incinerated by the friction from an atmospheric entry that it was never supposed to make. Now the probe would actually enter the planet's thin atmosphere, something for which it was never designed. But new calculations based on the current flight trajectory showed the Orbiter skimming within 60 km of the Martian surface. They had planned for the probe to reach an orbit approximately 180 km off the surface of Mars – well beyond the planet's thin atmosphere. As the Orbiter reached its final destination, the flight controllers began to realize that something was wrong. It had been in flight for over nine months, covering more than 415 million miles of empty space on its way to Mars. The probe was one of several planned for Mars exploration, and would stay in orbit around the planet as the first extraterrestrial weather satellite. On September 23, 1999, NASA's $125 million Mars Climate Orbiter approached the red planet under guidance from a team of flight controllers at the Jet Propulsion Laboratory.
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