Unit 1 - Measurements

Unit 1 covers how to make measurements and how to do math with those measurements. These pages exist to explain more specific concepts.

Significant Figures - The figures that have useful information

Scientific Notation - A way to write extreme numbers legibly

Uncertainty - How far off the measurement can be

Accuracy - Closeness to the accepted value

Precision - Closeness of points of data to each other

Zero-Offset Error - An error that occurs when the line of best fit should pass through zero, but does not.

## 1 - 1: Orders of Magnitude and Scientific Notation

An order of magnitude can be expressed as 10 to an exponent. The more digits that exist to the left of the decimal place, the higher the order of magnitude.

Scientific notation is a way to express very large or very small numbers without too many digits. It is written as a number between 1 and 10 multiplied by 10 to some power.

## 1 - 2: Measurement and Significant Figures

There are 7 fundamental units of the SI system. Memorize this!!!

**Ampere** - for current of an electrical circuit

**Candela** - light intensity

**Kelvin** - measurement for temperature. Take celsius and subtract it by 273

**Kilogram** - weight measurement. If you do not know this unit, then good luck on your exam

**Meter** - distance measurement.

**Mole** - unit for any number. Mole means 6.02x1023 of that number. e.g. 1 mole of donuts is 6.02x1023 of donuts.

**Seconds** - 3, 2, 1, blast off!

When you take data and analyze it, it can be precise or accurate. They are not the same.

**Precision** ("Focus") - refers to the closeness of the data points to each other. A very precise data collection would have all of the data focus very closely to one spot.

**Accuracy** ("Strike deadly") - refers to the closeness of the data points to accepted value. A very accurate data collection would have all the data striking the close to the value you want, which in this case, the accepted value.

You want to focus and strike deadly by having all the data points onto the right target:

http://cdn.antarcticglaciers.org/wp-content/uploads/2013/11/precision\_accuracy.png

**Welcome to the league of sigfigs!**

The four rules for entering this league are:

1. Zeros at the beginning are not significant: 0.0143 = 3 sigfigs.
2. Zeros within a number are significant: 10101 = 5 sigfigs
3. Zeros at the end of a decimal number are significant: 1234.9000 = 8 sigfigs
4. Zeros at the end of a whole number might be significant. if there is a decimal at the end, then yes. otherwise no. 100 = 1 sigfig. 100. = 3 sigfig.

Well then, let's get the game started.

* When you multiply or divide, leave as many sigfigs as the number in the equation with the least sigfigs. Do not count constants. e.g. 12 x 101 = 1212 -> 1.2 x 103
* When adding or subtracting, leave as many decimal places as the number in the equation with the least amount of decimals

## 1 - 3: Uncertainty in Measurements

Uncertainty refers to how far off a measurement can be by using the error bars. We have uncertainties due to the fear of someone going herpa derpa in the experiment.

To determine uncertainty when measuring:

**Digital Devices** (Those with number display): use the smallest decimal on display

**Analog** (Those with lines to show increments. E.g. clocks, beakers): Use half of the smallest increment.

To determine uncertainty when calculating:

When calculating the uncertainty for repeated trials: (largest value - smallest value)/2 = uncertainty

When multiplying or dividing: add the relative (percentage) uncertainties together. E.g. (2 ± 0.5) x (4 ± 2) = (2 ± 25%) x (4 ± 50%) = 8 ± 75% = 8 ± 6

When adding or subtracting: add the absolute uncertainties together: E.g. (2 ± 0.5) + (4 ± 2) = 6 ± 2.5

When raising a value to an exponent, then multiply the relative uncertainty by the exponent. E.g. (2 ± 0.5)2 = (4 ± 50%)

When multiplying/dividing a measurement by a constant, then multiply/divide that uncertainty too.

## 1 - 4: Uncertainty and Graphical Analysis

Graphical analysis means "HEY LOOK! LET'S MAKE A LINE OUT OF THE DATA!" by drawing two lines:

First line that has the max slope (steepest) line possible while still fitting through all the error bars.

Second line that has the least slope (gradual) line possible while still fitting through all the error bars.

Draw a third line that has the middle slope between the max and the least. This is the line you want. If you have a number for the slopes, add the max and min slopes together and divide by 2.

## 1 - 5: Analyzing Non-Linear Relationships

Well some relationships don't work well because the difficulty of keeping a relationship exponentially increases. Find a better partner is the usual solution, and forget the break-up! But physics isn't like that. Physics says "Make them straight" and be forceful onto this relationship.

Suppose that there is a quadratic relationship, with y = 2x2. when you plot it on the graph, x will have 1, 2, 3, 4... while y will have 2, 8, 18, 32... So it appears to be a curve exponentially increasing up. Physics is like NAW BRUH THIS RELATIONSHIP IS A FAILURE. LEMME JUST MAKE IT BETTER.

So, physics decided to plot y against x2 instead of y against x like before. x will have 1, 4, 9, 16... while y will have 2, 8, 18, 32. Since both variables increase exponentially, they produce a linear graph.

Because physics love to interrupt relationships, some relationships likes to stay hidden, and only leave footprints behind by giving a table of values. Physics puts on his detective caps and plots the points. Suppose the values are:

X Y

1 100

2 50

3 25

Physics knows the following:

1. If the y values are increasing faster and faster (e.g. y -> 1, 4, 9, 16), then the exponent value on x is greater than 1
2. If the y values are decreasing slower and slower (e.g. y -> 20, 10, 5) , then the exponent value on x is less than 0
3. If the y values are increasing slower and slower, then the exponent value is between 0 and 1

In this case, physics has determined that it is case 2.

The only way to straighten it up now is to use logger pro. On the exam, this won't happen, so forget about that.