Math ReviewUnits Review

Science 111

CSUB

Jeff Lewis ©2008

Units and the Metric System

• This is material covered in Appendix A of the textbook.

• The (probably) more familiar units are the USCS: United States Customary System.

• The metric system is also known as the International System or SI system.

Why Why Why?

• Why use the metric system when you are more familiar with the U.S. system?

1. The metric system is the better system.2. The metric system is the standard

throughout the rest of the world.3. Anyone doing or learning science needs to

learn the metric system.

Standard (or Base) Units

• USCS

Length foot (ft)

Force pound (lb)

Time second (sec)

Note: the pound is a unit of force, not mass.

• SI

Length meter (m)

Mass kilogram (kg)

Time second (s)

Same time unit in both systems.

Better? Why?

• Why is the metric system a “better” system than the USCS?

• Not because the meter is better than the foot.

• What makes the metric system better are the metric prefixes used to construct larger and smaller size units.

Prefix Advantages

• The same prefixes are used with all units to make the new, larger or smaller, units.

• And the new units are always simple multiples of 10 (or 100 or 1000 etc.) larger or smaller.

Micro prefix

• micro- makes a new unit 1,000,000 times smaller.

Microgram, microsecond, micrometer, microliter, microvolt, microphone (whoops, ignore that last one).

1 microgram = .000001 grams = 1 µg

Milli prefix

• milli-makes a new unit 1/1000 times the size of the original unit.

Milligram, millisecond, millimeter, milliliter, millidegree, milliamp, millipede (oops).

1 millisecond = .001 seconds = 1 ms

Centi prefix

• centi- makes a new unit 1/100 (.01) the size when in front of anything.

Centimeter, centigram, centisecond, centidegree, centijoule.

1 centimeter = .01 meters = 1 cm

Kilo prefix

• kilo- make a new unit 1000 times larger.

Kilogram, kilometer, kilosecond, kilopound, kilowatt, kilovolt.

1 kilogram = 1000 grams = 1 kg

Mega prefix

• mega- makes a new unit 1,000,000 times larger.

Megameter, megasecond, megagram, megahertz, megaton, megaparsec.

1 megaton = 1,000,000 tons = 1 Mton

Standard Prefixes

• Whatever unit you want larger or smaller versions of, you use the same prefixes.

• Not like the USCS where every unit works differently (like inch-foot-yard-mile or ounce-pint-quart-gallon).

Learn the Metric System!

• Learn the prefixes we’ve talked about and how much each stands for.

• You’ll need to know the metric system for homework, labs, and exams.

MKS vs CGS

• In the mks version of the metric system, the meter, kilogram, and second are considered the “fundamental” units.

• In the cgs version, the “fundamental” units are centimeters, grams, and seconds.

• Does it matter which is fundamental? No!

We will use mks system

• The systems do differ when you talk about derived units (units that are combinations of the base units).

• mks: force – newtons, energy – joules• cgs: force – dynes, energy – ergs

• We’ll use mks, newtons and joules.

Conversion of Units

• A very important mathematical technique is being able to convert units.

• Meaning, being able to take a value expressed in one unit and figure out its equivalence expressed in a different unit.

Sample Conversion Problems

• For example, you might want to know– What is 5 kilometers in miles?– What is 100 meters in yards?– How many ounces are in one liter?– What is 90 kg in pounds?– How many minutes is 505 seconds?– How many meters is 345 centimeters?

Basic Method

• Look up the equivalence that relates the two units you are trying to convert between.

• Such as, 1 ft = 12 in or 1 km = 1000 m• This equivalence is then used to construct a

“conversion factor”, a fraction with one of the values on top and the other on the bottom.

Conversion Factors

( ) 1 ft 12 in ( )12 in

1 ft ( )1000 m 1 km

or or

What makes these conversion factors special is that each is equal to one (because numerator and denominator are the same!). Mathematically, this means we can multiply them anywhere, anytime, without changing the value.

Sample Problem with Solution

Problem: How many inches are in 15 feet?

That is, we are trying to convert 15 ft into the equivalent number of inches.

Solution: Create an equality, 15 ft = 15 ft

Then multiply by a conversion factor that will cancel the ‘ft’ and give ‘in’ instead.

15 ft = 15 ft = 180 in

Note how I chose the conversion factor with ft on the bottom so that the ft would cancel.

My original, trivial, 15 ft = 15 ft equality is still valid even though I multiplied on the right side only because I multiplied by a factor of one.

( )12 in 1 ft

New Problem with a Twist

Problem: What is an area of 100 square feet (10 ft by 10 ft or 100 standard floor tiles) in units of square inches?

Solution: Start with 100 ft2 = 100 ft2

(ft2 = ft x ft, an area unit).

The twist? To do this conversion we have to multiply by the conversion factor twice.

100 ft2 = 100 ft2 = 14,400 in2

The ft2 unit is really two factors of ft, so I had to convert both of them.

In converting volumes, there would be three length units to convert.

( )12 in 1 ft ( )12 in

1 ft

Problem with a Different Twist

Problem: How many seconds are there in one year?

That is, we are converting the duration of 1 yr into the equivalent number of seconds.

Solution: Start with 1 yr = 1 yr

The twist? I don’t know the equivalence factor between years and seconds (that is what we are trying to figure out).

Instead, I can do this with a chain of conversions.

1 yr = 1 yr

= 31536000 s = 31,536,000 s

= 3.15 x 107 s

I didn’t know the direct conversion from years to seconds but I knew the intermediate conversions.

Note that I figured out what to put on top and bottom based on how units will cancel.

( )365 day 1 yr ( )24 hr

1 day ( )60 min 1 hr ( ) 60 s

1 min

Unit Conversion Summary

• Converting units is a very common problem, especially in labs.

• I urge you to carefully follow the method I’ve outlined here.

• Students who have trouble usually don’t write the steps down and instead try just to do it in their head.

Practice Problems (do now!)

1. What is 5 kilometers in miles? (1 mi = 1.609 km)

2. What is 60 mi/hr in km/hr?3. What is 1 m/s in mi/hr? (1 km = 1000 m)4. What is 500,000 ft3 in m3? (1 m = 3.28 ft)5. What is 32.2 ft/s2 in cm/s2?

(100 cm = 3.28 ft)

Answers:

1. 3.11 mi

2. 96.54 km/hr

3. 2.24 mi/hr

4. 14,169 m3

5. 982 cm/s2

Scientific Notation

• Powers-of-ten notation:– 105 means 10 x 10 x 10 x 10 x 10 = 100,000– 1024 = 1 followed by 24 zeroes– 10-1 = 1/10 = 0.1– 10-4 = 1/10 x 1/10 x 1/10 x 1/10 = 1/10,000 =

0.0001– 3.21 x 103 = 3.21 x 1000 = 3,210– 3.21 x 10-5 = 0.0000321

Scientific Notation Advantages

• Scientific (or powers-of-ten) notation is a simple way to write out very large or very small numbers.

• While we won’t be doing much math this quarter, you will be expected to recognize and understand values written in scientific notation when you see them.

• And if you need to do a calculation…

Sci Not on your Calculator

• All scientific calculators come with a shortcut button for inputting numbers written in scientific notation.

• Look on your calculator for a button labeled “E”, “EE”, or “Exp”.– I’ll assume it’s called “EE” in the following.

• To enter the value 4.2 x 1015 into your calculator, you push “4 . 2 EE 1 5”

Sample Problem• Calculate 4.2 x 1015 / 2.1 x 10-5

• Solution: Push “4 . 2 EE 1 5 / 2 . 1 EE +/- 5”• Answer: 2 x 1020

• Notes: If your calculator says “2 20”, you need to realize that that means 2 x 1020.

• Without the EE button, you’d have to push “4 . 2 x 1 0 ^ 1 5 / ( 2 . 1 x 1 0 ^ +/- 5 )” and you would get the wrong answer without the parentheses.

Practice Problems6. Simplify (2 x 1010) x (3 x 1020)7. Simplify (25 x 1010) / (5 x 1012)8. Simplify (6.02 x 1023) (105)

18• Hint: 105 is “1 EE 5” or “1 0 ^ 5”, not “1 0 EE

5”

9. Simplify 4 π (1.496 x 108)2

10. Simplify 1.05 x 10-22

(10-14) (4.32 x 10-9)

Answers

6. 6 x 1030

7. .05 ( = 5 x 10-2)

8. 3.34 x 1027

9. 2.81 x 1017

10. 2.43

One way to do #10: “1 . 0 5 EE - 2 2 / 1 EE - 1 4 / 4 . 3 2 EE - 9 =”

Combined Units

• We learned about base units (kg, m, s) before, but some types of quantities have units that are combinations of these.

• Speed or velocity units: distance/time, units like mi/hr, m/s, or km/min.

• Acceleration units: distance/time/time, units like m/s2, mi/hr/sec, or ft/s2.

More Combined Units

• The metric unit of force, the newton (N), is a combined unit: N = kg m / s2

• The USCS units of mass, the slug, is a combined unit: slug = lb s2 / ft

• The metric unit of energy, the joule (J), is a combined unit: J = kg m2 / s2

– This can also be written as J = N m

Still More Combined Units

• The USCS unit of energy is the (lb ft), a combined unit without a special name.

• The metric unit of momentum is the (kg m/s), a combined unit without a special name.

• Area units are distance x distance, like m2.

• Volume units are distance cubed, like m3.

Temperature Units

• There are three commonly used temperature scales: Fahrenheit (°F), Celsius (°C), and Kelvin (K), these will be discussed in more detail in chapter 7.

• The conversion formulae:– C = (5/9) (F - 32)– F = (9/5) C + 32– C = K -273 K = C + 273

Practice Problems

11. Simplify (N m3) / (kg m/s)[N = kg m / s2]

12. Convert 78°F into °C[C = (5/9) (F - 32)]

13. Convert the answer in the previous problem into K

[K = C + 273]

Answers

11. (N m3) / (kg m/s) = (kg m4 / s2) / (kg m/s) = m3/s

12. C = (5/9) (78 - 32) = (5/9) (46) = 25.6

13. K = 25.6 + 273 = 298.6

There’s more?? Will this day never end?

Okay, maybe we should take a short break before reviewing basic algebra and doing review for the first exam.

Algebra Review• Algebra is the manipulation of an equation

to solve for an unknown.

• The basic rule is that the equality remains valid so long as you do the same thing to both sides of the equation.

• Example: 4 x = 12, solve for x.

• Solution: Divide both sides by 4 and you get x = 12/4 = 3

• Example: (3/x) = 15, solve for x.

• Solution: Multiply both sides by x, giving 15 x = 3, then divide both sides by 15, x = 3/15 = 0.2– Alternate solution 1. Think of the 15 as being

(15/1) and cross-multiply.– Alternate solution 2. Again think of 15 as

(15/1) and ‘flip’ both sides [giving (x/3) = (1/15), then multiply both sides by 3].

• Example: y4 = 1.6 x 108, solve for y.

• Solution: Take the “fourth-root” of both sides of the equation. This can be done by using either a “x√y” button on your calculator or by raising both sides of the equation by the (1/4) power.

y = (1.6 x 108)1/4 = 112.5

Practice Problems

14. (1/x) = 16, solve for x.

15. x3 = 216, solve for x.

16. 20 x2 = 4000, solve for x.

Yeah, I’m getting tired too. So that’s enough algebra. Now lets talk about the exam.