PHYS 20 LESSONS:

INTRO

Lesson 1: Intro to CH Physics

Measurement

A. EXPRESSING ERROR IN SCIENCE

Error is unavoidable in science.

There are 3 major sources of error:

Systematic Errors

Random Errors

Blunders

Systematic Errors

These are errors from identifiable causes.

They can be improved (reduced).

There are a variety of systematic errors possible ...

1. Instrument Error

There might be error in the measuring instrument itself.

e.g. Calibration error

0

1

2

This scale has a reading of 0.3 kg, even though there is nothing on it yet. As a result, all measurements will be 0.3 kg too high.

To reduce this error, you need to adjust this scale to a zero reading before you make a measurement.

2. Observation Error

The student / scientist may be observing the measurement

in a way that introduces error.

e.g.

Parallax Error (reading from an angle)

Consider measuring the length of an object with a thick ruler:

10 20 30

10 20 30

From one side, the reading may seem to be 14.0 mm

10 20 30

From the other side, the reading may seem to be 17.0 mm.

10 20 30

To reduce parallax error, try to look directly

above the ruler.

Other Forms of Systematic Error

3. Environmental Error

e.g. A strong wind affects the object's motion

4. Theoretical Error

e.g. You assume there is no friction, but it is significant.

5. Analysis Error

e.g. Rounding error - the more calculations you do withmeasurements, the greater the error

Random Errors

Errors that are unavoidable.

Some readings are too high, while others are too low.

e.g. Reaction times using a stopwatch

To improve this error, calculate the average.

Blunders

These are outright mistakes.

e.g.

t (s) d (cm)

0

1

2

3

4

5

3.7

4.6

5.8

1.7

7.5

8.7

This value clearly does not fit the pattern. It is a blunder.

When blunders are discovered, they are ignored (removed).

t (s) d (cm)

0

1

2

3

4

5

3.7

4.6

5.8

1.7

7.5

8.7

The blunder is ignored when you do the analysis.

ACTIVITY

Timing a tennis ball (with and without technology)

A1. Measurements

When you are taking a reading from a measurement device,

how many digits should you record?

The more digits, the more precise (i.e. the better) the device.

Rule for measurement

You can have only one uncertain digit in a measurement

Rule for measurement

You can have only one uncertain digit in a measurement

The last digit of a recorded measurement is assumed to be uncertain.

e.g.

4 3 7 . 4 kg

Uncertain digit (could be a 3 or 5)

e.g. Consider the following measurement:

What would the measurement be?

30 mm10 20

Measurement: 1 8 mm

The object is longer than 18 mm, but less than 19 mm.

Thus, the first two digits will definitely be 18

Since these are certain, we are still allowed more digits.

30 mm10 201819

Measurement: 1 8 . 3 mm

Since the length is about 3/10 between 18 and 19,

the next digit is likely a 3.

However, we are not certain about this digit.

30 mm10 201819

Measurement: 1 8 . 3 mm

This digit could also have been a 2 or a 4

It is impossible to know for sure with this ruler.

Thus, this is our one uncertain digit. We stop here.

30 mm10 201819

A2. Significant Digits

Significant digits are the digits that are the direct result

of reading the measuring instrument.

Not all digits in a measurement are significant.

It is important to know the conventions for significant digits,

since it is a key skill when doing arithmetic with measured quantities.

Rule for Measured Quantities

Locate the first nonzero digit from the left

(i.e. the leftmost digit)

This digit, and all digits to the right, are significant

Use this rule for any quantity that is the result of

measurement (i.e. a number with units)

e.g.

Consider the quantity 28 000 kg

How many significant digits are there?

2 8 0 0 0 kg

Find the first nonzero digit from the left

2 8 0 0 0 kg

This digit, and all digits to the right, are significant.

Thus, this quantity has 5 significant digits.

e.g.

Consider the quantity 0.00720 m

How many significant digits are there?

0 . 0 0 7 2 0 m

Find the first nonzero digit from the left

0 . 0 0 7 2 0 m

This digit, and all digits to the right, are significant.

Thus, this quantity has 3 significant digits.

Rule for Definitions and Counting Numbers

Not all quantities used in science are the result of

measurement. Any quantities based on counting or

definition are considered to be perfect numbers.

Thus, counting numbers and definitions have an infinite

number of significant digits. No uncertainty exists.

e.g.

Definition: 1 mm = 1 10-3 m Perfect numbers

Counting: 7 tennis balls Infinite significant

digits ()

A3. Scientific Notation

Definition:

A number is in scientific notation when it is a number

between 1 and 10, multiplied by a power of 10

e.g. 4.85 103 m/s

A number between Multiplied by a power

1 and 10 of 10

Why use scientific notation?

There are two reasons why this notation is very useful:

1. For very big or small numbers

2. A clear number of significant digits

1. For big or small numbers

When numbers get really large or really small, there are

many zeros in the number. This makes them difficult to read.

Scientific notation makes them much easier to handle.

e.g.

The radius of a hydrogen atom

is 0.000 000 000 0529 m

Much easier if we express it as

5.29 1011 m

2. Clear number of significant digits

What if your calculator showed the number 142 000 , but

the answer was supposed to have only 3 significant digits?

You can't simply remove the last 3 digits, since it would

become 142 (clearly not the same number).

But you can't leave the number as 142 000, since this

would have 6 significant digits.

What if your calculator showed the number 142 000 , but

the answer was supposed to have only 3 significant digits?

So, we put it into scientific notation:

142 000 (3 sig digs) = 1.42 105

This clearly has 3 significant digits

Method:

To change a number into scientific notation:

Move the decimal to the right of the first nonzero digit

Multiply the resulting number by the appropriate

power of 10

Each time you moved the decimal to the left,

increase the exponent by 1

Each time you moved the decimal to the right,

decrease the exponent by 1

Key things to remember:

1. Positive exponents for big numbers (greater than 1)

Negative exponents for small numbers (less than 1)

2. If you need to round:

When the next digit is 5 or greater, then round up

Otherwise, don't round up.

e.g Convert 84170 (3 sig digs) to scientific notation

8 4 1 7 0 . (3 sig digs)

The decimal is placed at the end of the number

8 4 1 7 0 (3 sig digs)

The decimal is moved 4 digits to the left

(placed after the 1st nonzero digit)

8 4 1 7 0 . (3 sig digs)

= 8 . 4 1 7 0 10 4

A number between Moved 4 digits

1 and 10 to the left

(big number = positive exponent)

8 4 1 7 0 (3 sig digs)

= 8 . 4 1 7 0 10 4

Only 3 sig digs

are allowed

8 4 1 7 0 (3 sig digs)

= 8 . 4 1 7 0 10 4

Since the next digit is 5 or greater,

you will round up

8 4 1 7 0 (3 sig digs)

= 8 . 4 1 7 0 10 4

= 8 . 4 2 10 4

e.g. Convert 0.000 000 056 1 (2 sig digs) to scientific notation

0 . 0 0 0 0 0 0 0 5 6 1 (2 sig digs)

The decimal is moved 8 digits to the right

(right after the 1st nonzero digit)

0 . 0 0 0 0 0 0 0 5 6 1 (2 sig digs)

= 5 . 6 1 10 8

A number between Moved 8 digits

1 and 10 to the right

(small number = negative exponent)

0 . 0 0 0 0 0 0 0 5 6 1 (2 sig digs)

= 5 . 6 1 10 8

Only 2 sig digs

are allowed

0 . 0 0 0 0 0 0 0 5 6 1 (2 sig digs)

= 5 . 6 1 10 8

Since the next digit is less than 5,

you do not round up

0 . 0 0 0 0 0 0 0 5 6 1 (2 sig digs)

= 5 . 6 1 10 8

= 5 . 6 10 8

Normal Sci Eng

Note: Scientific Notation using a TI-83

1. Change to Scientific Notation

Enter: Mode

Sci

4.73 E 62. To enter the number 4.73 106 :

Enter: 4.73 EE 6

2nd Comma