Objectives- Express lengths, mass, time in SI units. - Convert distances between different units. - Describe time intervals in hours, minutes, and seconds. - Convert time in mixed units to time in seconds. - Describe the mass of objects in grams and kilograms.

Unit I Units and Measurement

It All Starts with a Ruler!!!

I. Two Systems of Units

1. Metric system and International System of Units

meter

kilogramsecondKelvin

2. English system inches, feet, yards, and miles.

pound Fahrenheit

meter (m): unit of length

kilogram (kg): unit of mass

second (s): unit of time

3. SI units

 meter, (SI unit symbol: m), is the fundamental unit of length in the International System of Units (SI).

Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole (at sea level).

Since 1983, it has been defined as "the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second."

National Prototype Metre Bar ( alloy of ninety percentplatinum and ten percent iridium) in  International Bureau of Weights and Measures (BIPM: Bureau International des Poids et Mesures) to be located in Sèvres, France.

 kilogramme ( kg), is the base unit of mass in the International System of Units (SI)

Is defined as being equal to the mass of the International Prototype of the Kilogram (platinum–iridium alloy) in International Bureau of Weights and Measures in Sèvres, France

Second (sec or s) The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

UNITS (Systéme Internationale)

4. Examples

The units for length, mass, and time (as well as a few others), are regarded as base SI units.

These units are used in combination to define additional units for other important physical

quantities such as force and energy.

II THE CONVERSION OF UNITSA) relation between different units

1 ft = 0.3048 m

1 mi = 1.609 km

1 liter = 10-3 m3

Example 1.1

Grandma traveled 27 minutes at 44 m/s.How many miles did Grandma travel?

44.3 miles

mile/min=1.64mile/min

27𝑚𝑖𝑛×1.64𝑚𝑖𝑙𝑚𝑖𝑛

=44.3𝑚𝑖𝑙𝑒𝑠

B) How to convert

Example 1 The World’s Highest Waterfall

The highest waterfall in the world is Angel Falls in Venezuela,with a total drop of 979.0 m. Express this drop in feet.

Since 3.281 feet = 1 meter, it follows that

(3.281 feet)/(1 meter) = 1 and (1 meter) / (3.281 feet)=1

For meter feet:

feet 3212meter 1

feet 281.3meters 0.979 Length

Convert 100km to miles

• A football field is 100 yards long.• What is this distance expressed in meters?

C) unit convert chart

D) Summary

Reasoning Strategy: Converting Between Units

1. In all calculations, write down the units explicitly.

2. Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically.

3. Use the conversion factors in reference tables. Be guided by the fact that multiplying or dividing an equation by a factor of 1 does not alter the equation.

time• Two ways to think about time:

– What time is it? • 3 P.M. Eastern Time on April 21, 2004,

– How much time has passed?• 3 hr: 44 min: 25 sec.

• A quantity of time is often called a time interval.

Converting Mixed Units

1. You are asked for time in seconds.2. You are given a time interval in mixed

units.1 hour = 3,600 sec 1 minute = 60 sec

3. Do the conversion:1 hour = 3,600 sec26 minutes = 26 × 60 = 1,560 sec

4. Add all the seconds:t = 3,600 + 1,560 + 31.25 = 5,191.25 sec

Time Units

E) Practice

Example 2 Interstate Speed Limit

Express the speed limit of 65 miles/hour in terms of meters/second.

Use 5280 feet = 1 mile and 3600 seconds = 1 hour and 3.281 feet = 1 meter.

s

m29

s 3600

hour 1

mile

1609m

hour

miles 6511

hour

miles 65 Speed

second

meters29 Speed

More practice

1. Convert 789 cm2 to m2

2. Convert 75.00 km/h to m/s

75.00 km x 1000 m x 1 h___ = 20.83m/s

h 1 km 3600 s

1m=100cm, 1m2=100cm *100cm=10000cm2

=0.0789

III Limits of Measurement

A). Accuracy and Precision

• Accuracy - a measure of how close a measurement is to the true value of the quantity being measured.

Example: Accuracy• Who is more accurate when

measuring a book that has a true length of 17.0cm?

Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm

Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm

• Precision – a measure of how close a series of measurements are to one another. A measure of how exact a measurement is.

Example: Precision

Who is more precise when measuring the same 17.0cm book?

Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm

Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm

Example: Evaluate whether the following are precise, accurate or both.

Accurate

Not Precise

Not Accurate

Precise

Accurate

Precise

--Why significant figures is important?--What does significant figures in a number consist of?--How to record measurement with proper significant figures? (digram)--List the rules in counting significant figures (zero rules)--Rules in calculation:

Multiplication ruleDivision ruleAddition & subtraction ruleRule in calculate average

B) Significant Figures

• The significant figures in a measurement include all of the digits that are known, plus one last digit that is estimated.

Centimeters and Millimeters40.16 cm

The length of this miniature piezo electric motor is:

8.0 mm

B.1) Finding the Number of Sig Figs:• All non-zero digits are significant.• Zeros between two non-zero digits are

significant.• Leading zeros are not significant.• Trailing zeros in a number containing a decimal

point are significant.• trailing zeros in a number not containing a

decimal point can be ambiguous. (scientific notation is the solution)

• One convention about trailing zero

A bar placed over ( or under) the last significant figure; any trailing zeros following this are insignificantExample:500 has 1 s.f. 5500 has 2 s.f. 500. has 3 s.f.

Scientific notationWrite number in form:

Standard decimal notation Scientific notation

2 2×100

300 3×102

4,321.768 4.321768×103

−53,000 −5.300×104

6,720,00,000 6.72000×109

0.2 2×10−1

0.000 000 007 51 7.51×10−9

How many sig figs?

1 10302.00

100.00 970

0.001 0.00250

10302 1.0302x104

7

2

3

5

2

5

1

5

B.2)Sig Figs in Addition/Subtraction

Express the result with the same number of decimal places as the number in the operation with the least decimal places.

Ex: 2.33 cm + 3.0 cm

5.3 cm (Result is rounded to one decimal place)

B.3) Sig Figs in Multiplication/Division

• Express the answer with the same sig figs as the factor with the least sig figs.

• Ex: 3.22 cm

x 2.0 cm

6.4 cm2

(Result is rounded to two sig figs)

2330 cm+ 3.0 cm 2330 cm

2330𝑚3.0𝑠

=780𝑚/ 𝑠

More example

B.4) Constant and Counting Numbers

• Constant number have infinite sig. figs.

• Counting numbers have infinite sig figs.

• Ex: 3 apples • Eg. π=3.1415926……

C) practice1.Calculate Volume of sphere with ,55.0 mr

33

33

70.06969.0

)55.0(3

4

3

4

mm

rV

2. Perimeter of the big circle

mm

mrP

5.34557.3

)55.0(22

2. Try the following

7.895 + 3.4=

(8.71 x 0.0301)/0.056 = =

A= =

13m

4.91m2

IV Dimension Analysis – some simple rules

1.In : The product unit is the product of the individual unit of each of those variables. (Ditto for ratios.)

2. : Different terms can only added together in a sum if each term in the sum has the same unit type. (Ditto for subtraction.)

Example 1

- impossible: 40m + 20m/s or 12.5 s - 20m2

- Can Do: 50.0m + 20.55m=70.6mand 40m/s +11m/s =51m/s

- Can Do, but need to convert into same unit:

40m + 11cm = 40m + 11cm = 40.11m

Example 2

The above expression yields:

1.5 m 3.0 kg ?

a)4.5 m kgb)4.5 g kmc)A or Bd)Impossible to evaluate (dimensionally invalid)

IV Scalars and Vectors

1. DefinitionA scalar quantity is one that can be described

by a single number:

temperature, speed, mass

A vector quantity deals inherently with both magnitude and direction:

velocity, force, displacement

2. Graph a Vector

By convention, the length of a vectorarrow is proportional to the magnitudeof the vector.

8 lb4 lb

Arrows are used to represent vectors. Thedirection of the arrow gives the direction ofthe vector.

3. Vector Addition and Subtraction

Often it is necessary to add one vector to another.

A)

example 1

5 m 3 m

8 m

B) Example 2

Example 2 continue

2.00 m

6.00 m

Example 2 continue

2.00 m

6.00 m

222 m 00.6m 00.2 R

R

m32.6m 00.6m 00.2 22 R

C)

When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.

Example 3

A

B

BA

A

B

BA

Given

A B