PE = potential energy [J]
H = height [ft]
r = density [g>cm3]
7.8 CONVERSION INVOLVING EQUATIONS
Engineering problems are rarely as simple as converting from one set of units to another. Normally, an equation is involved in the problem solution. To minimize the likelihood of mistakes, we adopt the following procedure for all problems. While this procedure may seem to overanalyze simple problems, it is relatively foolproof and will become more and more useful as the material progresses in difficulty.
Equation Procedure
1. Given a problem, first convert all parameters into base SI units, combinations of these units, or accepted non-SI units. Use the five-step conversion procedure pre-viously described.
2. Perform all necessary calculations, as follows:
(a) Determine the appropriate equation.
(b) Insert the known quantities and units. Be sure to carry the units through until the end!
(c) Calculate the desired quantity.
This gives the answer in SI units.
3. Convert the final answer to the required units and express the answer in “reason-able” terms.
LEARN TO: Solve an equation for a desired quantity using the 3-step procedure
Recognize the importance of converting to base SI units in equation solutions
EXAMPLE 7-16 On a trip from Alphaville to Betaville, you can take two main routes. Route 1, which goes through Gammatown, is 50 kilometers [km] long; however, you can only drive an average speed of 36 miles per hour [mph]. Route 2 travels along the freeway, at an average speed of 50 mph, but it is 65 km long. How long does it take to complete each route? State the time for each route in minutes [min].
Step One: Convert to Base SI Units
Method Route 1 Route 2
(1) Term to be converted 36 mph 50 mph
(2) Conversion formula
(3) Make a fraction (equal to one)
(4) Multiply 36 mi
h ` 1 km
0 .621 mi 50 mi h ` 1 km
0 .621 mi
(5) Cancel, calculate 58 km/h 81 km/h
Step Two: Calculate
Method Route 1 Route 2
(1) Determine appropriate equation
Distance = (velocity) (time) which can be rewritten as . . . Time = distance/velocity (2) Insert known quantities Time= 50 km
58kmh Time= 65 km 81kmh
(3) Calculate Time = 0.86 h Time= 0.8 h
Step Three: Convert from Base SI Units to Desired Units
Method Route 1 Route 2
(1) Term to be converted 0.86 h 0.8 h
(2) Conversion formula
(3) Make a fraction (equal to one)
(4) Multiply 0 .86 h ` 60 min
1 h 0 .8 h ` 60 min 1 h (5) Cancel, calculate, be reasonable 52 min 48 min NOTE
km km h
= km h km = h
EXAMPLE 7-17 You are designing a bottle to store juice for a large food manufacturing plant. The bottle is cylindrical in shape, with a 3 inch diameter and a height of 0.45 meters. What is the volume of the bottle in units of cubic centimeters?
The equation for the volume of a cylinder is: Vcylinder = pr2H.
Note that several common geometric formulas are provided for you in the end pages of this text.
To solve this problem so the end result is in cubic centimeters, we must convert both the radius and height into units of centimeters before plugging the values into the equation.
COMPREHENSION
CHECK 7-17 A basketball has a diameter of approximately 27 centimeters. Find the volume of the basketball in units of gallons.
COMPREHENSION
CHECK 7-16 Eclipses, both solar and lunar, follow a cycle of just over 18 years, specifically 6585.32 days. This is called the Saros Cycle. One Saros Cycle after any given eclipse an almost identical eclipse will occur due to fact that the Earth, the Moon, and the Sun are in essentially the same positions relative to each other.
The Sun, and the entire solar system, is moving relative to the Cosmic Microwave Background Radiation (the largest detectable frame of reference) at roughly 370 kilometers per second. How far does our solar system travel through the uni-verse in one Saros Cycle? Express your answer in the following units:
(a) meters, with an appropriately chosen prefix;
(b) light-years: one light year = 9.46 * 1015 meters.
Step One: Convert to Base SI Units
Method Diameter Height
(1) Term to be converted 3 inches 0.45 meters
(2) Conversion formula
(3) Make a fraction (equal to one) 3 in 2 2.54 cm
1 in 0.45 m 2 100 cm 1 m (4) Multiply
(5) Cancel, calculate 7.62 cm 45 cm
Step Two: Calculate
(1) Determine appropriate equation Vcylinder = pr2H
D = 2r so... r = 1/2 D (2) Insert known quantities r = 1/2 (7.62 cm) = 3.81 cm
V = p (3.81 cm)2 (45 cm) (3) Calculate, be reasonable V = 2,052 cm3
In-Class Activities
ICA 7-1
Express the following values using scientific notation, engineering notation, and using the correct SI prefix.
Standard Scientific Engineering With Prefix
(a) 389,589,000 J
(b) 0.0000000008 Pa
ICA 7-2
Complete the following table:
Meters Centimeters Millimeters Micrometers Nanometers Abbreviation [m]
Example 9E-08 9E-06 9E-05 0.09 90
(a) 50
(b) 5
ICA 7-3
Complete the following table:
Inches Feet Yards Meters Miles
Abbreviation [in]
(a) 90
(b) 2
ICA 7-4
Complete the following table:
Cubic Inch Fluid Ounces Gallon Liter Cubic Foot
Abbreviation [in 3 ]
Example 716 400 3.12 11.8 0.414
(a) 3
(b) 5
ICA 7-5
Complete the following table:
Miles per Hour
Kilometers per Hour
Yards per Minute
Feet per Second
Abbreviation [mph] or [mi / h]
(a) 100
(b) 55
ICA 7-6
Complete the following table:
ICA 7-7
A category F5 tornado can have wind speeds of 300 miles per hour [mph]. What is this velocity in units of meters per second?
ICA 7-8
A new hybrid automobile with regenerative braking has a fuel economy of 55 miles per gallon [mpg] in city driving. What is this fuel economy expressed in units of feet per milliliter?
ICA 7-9
The AbioCor TM artificial heart pumps at a rate of 10 liters per minute. Express this rate in units of gallons per second.
ICA 7-10
If a pump moves water at 2 cubic feet per hour, what is this rate in units of cubic centimeters per second?
ICA 7-11
In China, one “bu” is 1.66 meters. The average height of a human is 5 feet, 7 inches. Convert this height to units of bu.
ICA 7-12
In China, one “cun” is 3.5 centimeters. A “cubit” is defined as 18 inches. Convert 50 cubits to units of cun.
ICA 7-13
In China, one “fen” is defined as 3.3 millimeters. Ten nanometers is the thickness of a cell mem-brane. Convert 10 nanometers to units of fen.
Gallons per Minute
Cubic Feet per Hour
Liters per Second
Fluid Ounces per Day Abbreviation [gpm] or [gal/min]
(a) 15
(b) 20