Natural Units Environment
The Natural Units environment is an environment designed for computations with units. This environment is set up by using the command with(Units[Natural]).
Various procedures are overloaded to handle units. For example, evaluates the sine of 3 degrees and returns .
Arithmetic operators are overloaded to extract units from their operands. For example, evaluates to and evaluates to , which can be converted to .
Note: Although easier to use, it is slower, in general, to perform computations in the Natural Units environment than to use the conversion routines at the toplevel.
If you do not want to interpret everything as a unit, you can either use the Units[Standard] package, where units must be specified with the Unit routine, or use the strict option to the Units[UseSystem] routine.
These computations can also be done in the Standard Units environment, or at the toplevel by using only conversion routines (see Default Units). Each example in this worksheet is also in the other worksheets to show how you can perform the computations in the other environments.
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Simple Examples


Add 4 feet to 3 inches.
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 (1.1) 
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 (1.2) 
How many meters are equivalent to 4 yards?
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 (1.3) 
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 (1.4) 
How many liters are equivalent to 5 UK gallons?
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 (1.5) 
How many liters are equivalent to 5 US gallons?
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 (1.6) 
How many US liquid gallons are equivalent to a UK gallon?
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 (1.7) 


Unit Names and Symbols


Unit symbols and various spellings of unit names are recognized by the package.
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 (2.1) 
This feature is expandable so that, for example, you can specify metr as an alternate spelling of meter.
Some units can be recognized with SI or IEC prefixes.
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 (2.2) 


Sample Questions with Solutions



How many miles do you travel in 35 minutes moving at 55 miles per hour?


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 (3.1.1) 
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 (3.1.2) 
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 (3.1.3) 


How many seconds are equivalent to 3 weeks?


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 (3.2.1) 


How many inches are equivalent to 5 feet 4 inches?


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 (3.3.1) 


Convert 50 km/h to cm/s.


The cm/s must be quoted. Otherwise, evaluates to .
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 (3.4.1) 
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 (3.4.2) 


How many seconds does it take an object, released from rest, to fall 20 meters?


From the equation , we derive:
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 (3.5.1) 
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 (3.5.2) 
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 (3.5.3) 


What is the rest energy of an electron?


Assume the mass of an electron is kilograms.
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 (3.6.1) 
Approximate the speed of light by meters per second.
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 (3.6.2) 
Using the formula , you can approximate the rest energy of the electron.
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 (3.6.3) 
A more precise answer can be found by converting the known unit the electron mass (em) to joules using energy conversions.
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 (3.6.4) 


Approximately what volume does 1,000,000,000 US dollars worth of gold occupy?


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 (3.7.1) 
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 (3.7.2) 
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 (3.7.3) 
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 (3.7.4) 
What is the length of one side of a cube with this volume?
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 (3.7.5) 


An Su27 Flanker can travel at mach 1.1 at sea level. How fast is this in miles per hour, miles per second, and meters per second?


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 (3.8.1) 
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 (3.8.2) 
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 (3.8.3) 


Approximately how many meters are there in 3.5 miles?


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 (3.9.1) 
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 (3.9.2) 


Given 50 US gallons of water, how many 750 mL bottles could you fill?


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 (3.10.1) 
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 (3.10.2) 


Given nylon with a linear mass density of 20 deniers, what length of thread is used in an object weighing 12 grams?


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 (3.11.1) 


What is the volume in cubic inches of a 2 liter engine.


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 (3.12.1) 
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 (3.12.2) 


For a given phenomena with a frequency of 1.420,405,761 GHz, find the:


1. period,
2. number of cycles per year, and
3. number of cycles since the beginning of the earth.
First you must convert the frequency from GHz to Hz (cycles per second).
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 (3.13.1) 
The period is:
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 (3.13.2) 
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 (3.13.3) 
The number of cycles per year is:
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 (3.13.4) 
The number of cycles since the beginning of the earth is:
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 (3.13.5) 


Given inductance and capacitance, find the resistance in microohms.


The following formula relates the resistance to the inductance and capacitance.
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 (3.14.1) 
Use an inductance of 124 nanohenries and a capacitance of 3.52 microfarads.
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 (3.14.2) 
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 (3.14.3) 
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 (3.14.4) 


Given a molar energy, find the mass energy in Btu's per pound.


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 (3.15.1) 
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 (3.15.2) 
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 (3.15.3) 
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 (3.15.4) 


Given a distance function, find the speed by differentiation, speed at 2.5 seconds by evaluation, and distance traveled between 1 and 2.5 seconds by integration and by subtraction of the distance function values.


Note: t is a unit of mass, the tonne. Therefore, t1 is used to represent the time variable.
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 (3.16.1) 
To find the speed function, differentiate the distance function.
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 (3.16.2) 
To find the speed at 2.5 seconds, evaluate the speed function at t1=2.5.
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 (3.16.3) 
By using a definite integral, you can determine the distance traveled from the speed function.
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 (3.16.4) 
The distance traveled can also be calculated directly from the distance function.
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 (3.16.5) 
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 (3.16.6) 


Find the minimum and maximum length of 1.2 yards, 1 meter, 3.2 feet, and 0.6 fathoms.


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 (3.17.1) 
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 (3.17.2) 


Given a torque of 3 newton meters, how much energy is required to move a lever through 10 degrees?


The energy required is the product of the torque and the angle in radians.
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 (3.18.1) 
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 (3.18.2) 
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 (3.18.3) 


The HyperX can travel at speeds up to 7200 miles per hour. How long would it take to circle the earth at maximum speed (assuming it could carry sufficient fuel)? How far does it travel in a 10 second flight at maximum speed?


To find the time to circle the earth, divide the distance by the speed.
The meter was originally defined as 1/10,000,000 th the distance from the North Pole to the Equator on the meridian passing through Paris. Therefore, 40,000 kilometers is a good approximation of the circumference of the earth.
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 (3.19.1) 
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 (3.19.2) 
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 (3.19.3) 
To find the distance traveled, multiply the speed by the time.
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 (3.19.4) 
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 (3.19.5) 
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 (3.19.6) 


Given 1032 UK gallons of oil, how many cylindrical cans with a height of 1.2 feet and diameter of 0.9 feet could you fill?


The volume of a cylinder is .
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 (3.20.1) 
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 (3.20.2) 
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 (3.20.3) 


Given an power gain from 332 microwatts to 23 milliwatts, what is the gain in decibels? What would the decibel gain be if the increase were a voltage increase?


A gain is a quotient of the final value divided by the initial value.
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 (3.21.1) 
To determine the decibel gain, first take the ln of the gain.
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 (3.21.2) 
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 (3.21.3) 
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 (3.21.4) 
Power is proportional to the square of the voltage. Therefore, the decibel increase corresponding to the voltage gain should be a factor of 2 times that of the power gain.
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 (3.21.5) 
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 (3.21.6) 
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 (3.21.7) 
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 (3.21.8) 
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