7.4
LEARN TO: Follow the 5-step conversion procedure to convert units when multiple steps are required
EXAMPLE 7-5 Convert the length 40 yards [yd] into units of millimeters [mm].
Method Steps
(1) Term to be converted 40 yd
(2) Conversion formula 1 yd = 3 ft 1 ft = 12 in 1 in = 2.54 cm 1 cm = 10 mm (3) Make fractions (equal to one) 3 ft
1 yd 12 in
1 ft 2 .54 cm
1 in 10 mm 1 cm
(4) Multiply 40 yd ` 3 ft
1 yd`12 in
1 ft `2 .54 cm 1 in `10 mm
1 cm
(5) Cancel, calculate, be reasonable 37,000 mm
EXAMPLE 7-6 Convert 55 miles per hour [mph] to units of meters per second [m/s].
Note that we have two units to convert here, miles to meters, and hours to seconds.
Convert the volume of 40 gallons [gal] into units of cubic feet [ft 3 ].
By examining the “Volume” box in the conversion table, we see that the following facts are available for use:
1 L = 0.264 gal and 1 L = 0.0353 ft3
By the transitive property, if a = b and a = c, then b = c. Therefore, we can directly write 0.264 gal = 0.0353 ft3
VOLUME 1 L = 0.264 gal 1 L = 0.0353 ft 3 1 L = 33.8 fl oz 1 mL = 1 cm 3
Method Steps
(1) Term to be converted 55 mph
(2) Conversion formula 1 km = 0.621 mi 1 km = 1,000 m 1 h = 60 min 1 min = 60 s (3) Make fractions (equal to one)
(4) Multiply 55 mi
h ` 1 km
0 .621 mi`1,000 m 1 km ` 1 h
60 min `1 min 60 s (5) Cancel, calculate, be reasonable 24.6 m/s
Method Steps
(1) Term to be converted 40 gal
(2) Conversion formula 0.264 gal = 0.0353 ft3
(3) Make a fraction (equal to one) 0.0353 ft3 0.264 gal
(4) Multiply 40 gal ` 0.0353 ft3
0 .264 gal (5) Cancel, calculate, be reasonable 5.3 ft3 EXAMPLE 7-7
NOTE
When raising a quantity to a power, be sure to apply the power to both the value and the units.
One frequently needs to convert a value that has some unit or units raised to a power, for example, converting a volume given in cubic feet to cubic meters. It is criti-cal in this case that the power involved be applied to the entire conversion factor, both the numerical values and the units.
EXAMPLE 7-8 Convert 35 cubic inches [in 3 ] to cubic centimeters [cm 3 or cc].
Method Steps
(1) Term to be converted 35 in 3
(2) Conversion formula 1 in = 2.54 cm
(3) Make fractions (equal to one)
12.54 cm2
3
11 in23
(4) Multiply
35 in3`(2 .54)3 cm3
1 in3
(5) Cancel, calculate, be reasonable 574 cm 3
Note that in some cases, a unit that is raised to a power is being converted to another unit that has been defined to have the same dimension as the one raised to a power. This is difficult to say in words, but a couple of examples should clarify it.
If one is converting square meters [m 2 ] to acres, the conversion factor is not squared, since the conversion provided is already in terms of length squared: 1 acre = 4,047 m 2 .
If one is converting cubic feet [ft 3 ] to liters [L], the conversion factor is not cubed, since the conversion provided is already in terms of length cubed: 1 L = 0.0353 ft 3 .
This picture shows 5-gallon water bottles made from polycarbonate. Millions of these bottles are made each year around the world to transport clean water to remote locations.
The use of polycarbonate to contain products for consumption has raised safety concerns because bisphenol A is leached from the plastic into the stored liquid. In July 2012, the US Food and Drug Administration banned the use of BPA in bottles and cups used by infants and small children.
Photo courtesy of E. Stephan
COMPREHENSION
CHECK 7–5 In January 2008, Scientific American reported that physicists Peter Sutter and Eli Sutter of Brookhaven National Laboratories made a pipette to measure droplets in units of a zeptoliter. Previously, the smallest unit of measure in a pipette was an attoliter. Convert the measurement of 5 zeptoliters into units of picoliters.
RULES OF THUMB
1 quart ≈ 1 liter 1 cubic foot ≈ 7.5 gallons
1 cubic meter ≈ 250 gallons 1 cubic meter ≈ 5, 55-gallon drums 1 cup ≈ 250 milliliters 1 golf ball ≈ 1 cubic inch
COMPREHENSION
CHECK 7-6 Officially, a hurricane is a tropical storm with sustained winds of at least 74 miles per hour. Convert this speed into units of kilometers per minute.
COMPREHENSION
CHECK 7-7 Many toilets in commercial establishments have a value printed on them stating the amount of water consumed per flush. For example, a label of 2 Lpf indicates the consumption of 2 liters per flush. If a toilet is rated at 3 Lpf, how many flushes are required to consume 20 gallons of water?
LESSONS OF THE MARS CLIMATE ORBITER
Some of you may have heard that the loss of the Mars Climate Orbiter (MCO) spacecraft in 1999 was due to a unit conversion error. The complete story is rather more complicated and illustrates a valuable lesson in engineering design. Most engineering failures are not due to a single mis-take, since built-in redundancies and anticipation of failure modes make this unlikely. Three primary factors (plus bad luck) conspired to send the MCO off course.
First, the spacecraft was asymmetrical, with the body of the spacecraft on one side and a large solar panel on the other. You might think shape is not an issue in the vacuum of space,
but in fact it is, and the NASA engineers were
(continued )
well aware of the potential problems. The panel acted like a sail, causing the craft to slowly change its orientation and requiring the MCO to make occasional small corrections by firing thrusters onboard the craft. This was a perfectly manageable “problem.”
Second, the software on the spacecraft expected thruster data in SI units, requiring the force expressed in newtons. On the Earth, a separate system calculated and sent instructions to the MCO concerning when and how long to fire its thrusters. The Earth-based system relied on software from an earlier Mars mission, and the thruster equations had to be modified to correct for the thrusters used on the new spacecraft. The original software had been written correctly, with the conversion factor from pound-force to new-tons included. However, this conversion was neither documented nor obvious from the code, being buried in the equations. When the equations were rewritten, the program-mers were unaware of the conversion factor and it was left out of the new code. This sent incorrect thruster-firing data to the MCO, specifically being too small by a factor of 4.45. This problem alone was manageable by comparing the calculated trajectory with tracking data.
Finally, after the third trajectory correction, the MCO entered “safe mode” while adjusting the solar panel, indicating a fault on the craft. At about the same time, the preliminary indications that the spacecraft trajectory was flawed began to come in.
Unfortunately, the engineers spent the next several weeks trying to determine what caused the craft to enter safe mode, falsely assuming the preliminary trajectory data was in error and waiting for longer-term tracking to give a better estimate. In the end, the spacecraft arrived at Mars about 100 kilometers off course.
Here is where the bad luck comes in. Other configurations of the craft or trajectory might have caused the 100 kilometer error to be away from Mars or parallel to the sur-face, in which case the trajectory could have been corrected later. Unfortunately, the tra-jectory was 100 kilometers lower than expected, and the MCO was probably destroyed by heating and stresses as it plunged through the Martian atmosphere. Cost: well over
$100 million.