<Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field>

<Text-field layout="Heading 1" style="Heading 1">Overview of the Units Package</Text-field>

The Natural option tells Maple that the unit variables will be expressed as ordinary variables without special notation. Consequently, one should take care not to create a new user variable called km, or miles, etc. There is an option to work in a "protected" mode where such ambiguity is prevented.

<Text-field layout="Heading 2" style="Heading 2">Some simple examples</Text-field>

Consider the following definition for a length. Notice that Maple casts the answer in meters [m]. SI is the default unit system used by the Units package.

myDist := 4*ft + 3*inches;

You can easily express the above in standard US (FPS - Foot Pound Second) system using the convert command

convert( myDist, system, FPS); evalf( % );

The Units package knows that a NiMqKCUja2dHIiIiJSJtR0YlKiQlInNHIiIjISIi is a Newton

myForce := 5*g * 3 * m/s^2;

Maple is smart enough to understand when units conflict -- for example, adding a length (ft) to a mass (kg) does not make sense, and Maple returns an error if the user tries to do so.

<Text-field layout="Heading 2" style="Heading 2">Which units systems and dimensions does the Units package know?</Text-field>

The 8 standard unit systems recognized by the Units package.

GetSystems();

The Units package also knows the dimensions of all standard quantities, for example area, frequency, torque, etc.

GetDimension( area ); GetDimension( frequency ); GetDimension( torque );

You can ask Maple to list the base quantities the Units package knows.

GetDimensions();

What units does the Units package recognize? Specify the dimension, in this case "force", and you can find out using the GetUnits function.

GetUnits( dimension=force );

Here are all 569 units known by the package

GetUnits();

<Text-field layout="Heading 2" style="Heading 2">Adding your own units</Text-field>

You can extend the set of units recognized by the Units package. In this example, we use the Units package to encapsulate the following information about army groups:

1 platoon = 24 soldiers 1 company = 3 platoons 1 battalion = 3 companies 1 brigade = 3 battalions 1 division = 3 brigades 1 corps = 3 divisions

We reinitialize the Units environment.

restart: with(Units):

Define the army groups as units.

AddBaseUnit(soldier, context = military, dimension = humanlife, spellings = soldiers); AddUnit(platoon, context = military, conversion = 24 * soldier, spellings = platoons); AddUnit(company, context = military, conversion = 3 * platoon, spellings = companies); AddUnit(battalion, context = military, conversion = 3 * company, spellings = battalions); AddUnit(brigade, context = military, conversion = 3 * battalion, spellings = brigades); AddUnit(corps, context = military, conversion = 3 * brigade, spellings = corps);

Now we add the "military" system to the 8 systems already recognized by the package.

AddSystem(military, check, soldier); with(Units[Natural]): UseSystem( military ):

How many soldiers are in 5 battalions?

convert(5*battalion, units, soldier);

After a fierce battle, your fighting forces are down to 2 corps and a half-strength company. How many soldiers do you have left?

2*corps + (1/2)*company;

How many platoons is that?

evalf[4](convert(%, units, platoon));

<Text-field layout="Heading 1" style="Heading 1">Example Problems Involving Units</Text-field>

<Text-field layout="Heading 2" style="Heading 2">Reinitialize</Text-field>

restart; with(Units[Natural]): with (ScientificConstants):

<Text-field layout="Heading 2" style="Heading 2">How much room does 1 billion dollars worth of gold take up?</Text-field>

First, define the dollar value of the gold, cost per troy ounce (assume $268), and the density.

val := 10^9 * USD; costPerOz := 268.00 * USD/oz[troy]; density := 19.3 * g/cm^3;

The resulting mass and volume are:

mass := val/costPerOz; volume := mass/density;

What is the length of one side of a cube having this volume?

length_side := root[3](volume);

<Text-field layout="Heading 2" style="Heading 2">What is the sin of 45 degrees?</Text-field>

Users might initially try the following (which is incorrect):

evalf(sin(45));

The correct but tedious way without the Units package.

evalf(sin((Pi/180)*45));

With Maple's Units package, you can simply state 45*degrees, and functions will understand how to manage the information. This one example will likely save many a high school student at least 20 minutes of grief.

evalf(sin(45*degrees));

<Text-field layout="Heading 2" style="Heading 2">How many seconds in 3 weeks?</Text-field>

t := 3 * week;

convert( t, units, s);

<Text-field layout="Heading 2" style="Heading 2">How far do you travel in 35 minutes going 55 miles per hour? Express answer in meters and miles.</Text-field>

d := 55*mph * 35*minutes;

convert( d, units, mi );

evalf( d );

<Text-field layout="Heading 2" style="Heading 2">Given a molar energy, find the mass energy in Btu's per pound.</Text-field>

molar_energy := 523.432*Btu/mol(carbon);

carbon_mass := 12*kg/mol(carbon);

mass_energy := molar_energy / carbon_mass;

convert(%, units, 'Btu/lb');

<Text-field layout="Heading 2" style="Heading 2">Given a torque of 3 Newton meters, how much energy is required to move a lever through 10 degrees?</Text-field>

energy := torque * angle;

eval(energy, [torque = 3*N*m(radius), angle = 10*deg]);

evalf(%);

<Text-field layout="Heading 2" style="Heading 2">What is the force of gravity on a space vehicle</Text-field>

What is the gravitational force on a 500lb satellite orbiting 100 miles above ground.

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bml5bUphbWBXYFtIYUdxZjx3YnFGZT9JbUpJYFNIX1VIW3d3ZV94aE5oZ3lZaVpeYGhhblZGbUNmXV1mYFNIW3dfXFxjeGlkQXZHcWlLPlt3V2BjeHFdWHZJcWdHTmJZSXJkaWdcXG9pXXFeWl54XWZxWGFqa3hebj9rUHFiV0B0OnFaVEg+PVc6XmpAPjs/dDpKQlpHPjw+QjtGbjtDRzs8OzpBTjpqO1o8Ozw7XFw6QkpCQlo6UkpLTWJTOmpKQTo+O15KSlZhbXlJSlRQcEI9THdVc09IOztCOk1US1dES1dnSjtlWjE6

We have the following definitions:

m1 - mass of satellite m2 - mass of earth G - gravitational constant height - height of the satellite

radius - radius of earth (given as: 25000*5280 feet is the circumference of earth)

m1 := 100*lb; m2 := 6.0*10^24*kg; G_const := Constant (G, units); height := 100*mile; radius := 25000*5280*ft/(2*Pi);

Distance from the center of mass of the satellite to the center of mass of earth.

dist := radius+height;

Force := simplify (G_const*m1*m2/dist^2);

<Text-field layout="Heading 2" style="Heading 2">What is the energy of an electron?</Text-field>

mass := 9.11e-31 * kg; c := 3.00e8 * m/s; E := mass * c^2;

<Text-field layout="Heading 2" style="Heading 2">For a given phenomena with a frequency of 1.42... GHz, what are:</Text-field>

1. the period,

2. the number of cycles per year, and

3. the number of cycles since the beginning of the earth

frequency := 1.420405761 * GHz;

The period in seconds and nano seconds.

1/frequency; convert(%, units, nanoseconds);

Cycles per year:

convert(frequency, units, '1/yr');

<Text-field layout="Heading 2" style="Heading 2">Given inductance and capacitance, find the resistance in microohms.</Text-field>

resistance := sqrt(inductance / capacitance);

eval(resistance, [inductance = 124.*nH, capacitance = 3.52 * uF]): evalf(%);

convert(%, units, uOmega);

<Text-field layout="Heading 2" style="Heading 2">Given an power gain from 332 microwatts to 23 milliwatts, what is the gain in decibels? What would the gain have been had the increase had been a voltage increase?</Text-field>

Current gain.

gain := 23*mW / (332*uW(base));

convert(ln (gain), units, dB): evalf (%);

Voltage gain.

voltage_gain := 23*mV / (332*uV(base));

convert(ln(voltage_gain), units, dB): evalf (%);