Measurement in Physics

AP Physics C

SI units for PhysicsThe SI stands for "System International”.  There are 3

fundamental SI quantities we will be using this semester. They basically breakdown like this:

SI Quantity SI Unit

Length Meter

Mass Kilogram

Time Second

Of course there are many other units to consider. Many times, however, we express these units with prefixes attached to the front. This will, of course, make the number either larger or smaller. The nice thing about the prefix is that you can write a couple of numbers down and have the unit signify something larger.

Example: 1 Kilometer – The unit itself denotes that the number is actually larger than "1" considering  fundamental units. The fundamental unit would be 1000 meters

Most commonly used prefixes in PhysicsPrefix Factor Symbol

Mega- ( mostly used for radio station frequencies) x 106 M

Kilo- ( used for just about anything, Europe uses the Kilometer instead of the mile on its roads)

x 103 K

Centi- ( Used significantly to express small distances in optics. This is the unit MOST people in AP forget to convert)

x 10-2 c

Milli- ( Used sometimes to express small distances)

x 10-3 m

Micro- ( Used mostly in electronics to express the value of a charge or capacitor)

x 10-6 m

Nano ( Used to express the distance between wave crests when dealing with light and the electromagnetic spectrum)

x 10-9 n

Tip: Use your constant sheet when you forget a prefix value

The importance of the metric system prefixes

ExampleIf a capacitor is labeled 2.5mF(microFarads),

how would it be labeled in just Farads?The FARAD is the fundamental unit used when discussed capacitors!

2.5 x 10-6 F Notice that we just add the factor on the end and use the root unit.

The radio station XL106.7 transmits at a frequency of 106.7 x 106 Hertz. How would it be written in MHz (MegaHertz)?A HERTZ is the fundamental unit used when discussed radio frequency!

106.7 MHz Notice we simply drop the factor and add the prefix.

SI: Derived Units

Physical Quantity Unit Name Symbol

area square meter m2

volume cubic meter m3

speed meter persecond m/s

accelerationmeter per

second squared m/s2

weight, force newton N (kg·m/s²)

energy, work joule J (N·m)

units that come from multiplying or dividing fundamental units

Dimensional Analysis

Suppose we want to convert 65 mph to ft/s or m/s.

Dimensional Analysis is simply a technique you can use to convert from one unit to another. The main thing you have to remember is that the GIVEN UNIT MUST CANCEL OUT.

hour

miles65

sec60

min1

min60

1

hour

mile

ft

1

5280

160601

52801165

s

ft95

s

ft95

ft

meter

281.3

1

281.31

195 sm /29

Trigonometric FunctionsMany concepts in physics

act at angles or make right triangles. Let’s review common functions.

)(tan

Theoremn Pythagorea

1-

222

adj

opp

bac

ExampleA person attempts to measure the height of

a building by walking out a distance of 46.0 m from its base and shining a flashlight beam toward its top. He finds that when the beam is elevated at an angle of 39 degrees with respect to the horizontal ,as shown, the beam just strikes the top of the building.  a) Find the height of the building and b) the distance the flashlight beam has to travel before it strikes the top of the building.

What do I know? What do I want?

Course of action

•The angle•The adjacent side

The opposite side

USE TANGENT!

ExampleA truck driver moves up a straight mountain highway, as shown above. Elevation

markers at the beginning and ending points of the trip show that he has risen vertically 0.530 km, and the mileage indicator on the truck shows that he has traveled a total distance of 3.00 km during the ascent. Find the angle of incline of the hill.

What do I know? What do I want?

Course of action

•The hypotenuse•The opposite side

The Angle USE INVERSE SINE!

Measurement & Precision

The precision of a measurement depends on the instrument used to measure it.

For example, how long is this block?

Measurement & Precision

Imagine you have a piece of string that is exactly 1 foot long.

Now imagine you were to use that string to measure the length of your pencil. How precise could you be about the length of the pencil?

Since the pencil is less than 1 foot, we must be dealing with a fraction of a foot. But what fraction can we reliably estimate as the length of the pencil?

Measurement & Precision

Suppose the pencil is slightly over half the 1 foot string. You guess, “Well it must be about 7 inches, so I’ll say 7/12 of a foot.”

Here’s the problem: If you convert 7/12 to a decimal, you get 0.583.

Can you reliably say, without a doubt, that the pencil is 0.583 and not 0.584 or 0.582?

You can’t. The string didn’t allow you to distinguish between those lengths… you didn’t have enough precision.

So, what can you estimate, reliably?

Measurement & Precision

Basically, you have one degree of freedom… one decimal place of freedom.

So, the only fractions you can use are tenths! You can only reliably estimate that the pencil is 0.6 ft

long. It’s definitely more than 0.5 ft long and definitely less than 0.7 ft long.

Thus, precision determines the number of significant figures we use to report measurements.

In order to increase the precision of their measurements, physicists develop more-advanced instruments.

How big is the beetle?

Copyright © 1997-2005 by Fred Senese

Measure between the head and the tail!

Between 1.5 and 1.6 in

Measured length: 1.54 in

The 1 and 5 are known with certainty

The last digit (4) is estimated between the two nearest fine division marks.

Significant Figures

Indicate precision of a measured value 1100 vs. 1100.0 Which is more precise? How can you tell? How precise is each number? Determining significant figures can be tricky. There are some very basic rules you need to

know. Most importantly, you need to practice!

Counting Significant Figures

The Digits Digits That Count Example # of Sig Figs

Non-zero digits ALL          4.337 4

Leading zeros(zeros at the BEGINNING) NONE          0.00065 2

Captive zeros(zeros BETWEEN non-zero digits) ALL          1.000023 7

Trailing zeros (zeros at the END)

ONLY IF they follow asignificant figure AND

there is a decimalpoint in the number

89.00 but

8900

4 2

Leading, Captive AND Trailing Zeros

Combine therules above

0.003020 but

3020

4

3

Scientific Notation ALL         7.78 x 103 3

Calculating With Sig Figs

Type of Problem Example

MULTIPLICATION OR DIVISION:

Find the number that has the fewest sig figs. That's how many sig figs should be in your answer.

3.35 x 4.669 mL = 15.571115 mLrounded to 15.6 mL

3.35 has only 3 significant figures, so that's how many should be in the answer.  Round it off to 15.6 mL

ADDITION OR SUBTRACTION:

Find the number that has the fewest digits to the right of the decimal point. The answer must contain no more digits to the RIGHT of the decimal point than the number in the problem.

64.25 cm + 5.333 cm = 69.583 cm rounded to 69.58 cm

64.25 has only two digits to the right of the decimal, so that's how many should be to the right of the decimal in the answer. Drop the last digit so the answer is 69.58 cm.

Scientific Notation

Number expressed as: Product of a number between 1 and 10 AND a power of 10

5.63 x 104, meaning 5.63 x 10 x 10 x 10 x 10 or 5.63 x 10,000

ALWAYS has only ONE nonzero digit to the left of the decimal point

ONLY significant numbers are used in the first number First number can be positive or negative Power of 10 can be positive or negative

When to Use Scientific Notation

Astronomically Large Numbers mass of planets, distance between stars

Infinitesimally Small Numbers size of atoms, protons, electrons

A number with “ambiguous” zeros 59,000

HOW PRECISE IS IT?

Order of Magnitude

Approximation based on a number of assumptions may need to modify assumptions if more precise

results are needed Order of magnitude is the power of 10 that

applies

Order of Magnitude – Process Estimate a number and express it in scientific notation

The multiplier of the power of 10 needs to be between 1 and 10

Divide the number by the power of 10 Compare the remaining value to 3.162 ( )

If the remainder is less than 3.162, the order of magnitude is the power of 10 in the scientific notation

If the remainder is greater than 3.162, the order of magnitude is one more than the power of 10 in the scientific notation

Easier – Find the logarithm (base 10) of the number, round it to the nearest whole number, and use that as the power of 10

10

Using Order of Magnitude

Estimating too high for one number is often canceled by estimating too low for another number The resulting order of magnitude is generally

reliable within about a factor of 10 Working the problem allows you to drop

digits, make reasonable approximations and simplify approximations

With practice, your results will become better and better

Uncertainty in Measurements There is uncertainty in every measurement –

this uncertainty carries over through the calculations May be due to the apparatus, the experimenter,

and/or the number of measurements made Need a technique to account for this uncertainty

We will use rules for significant figures to approximate the uncertainty in results of calculations

Percent Error

Experimental what you do in class (lab) Theoretical what you look up in a book

(internet)

experimental - theoretical% Error

theoretical

Percent Difference

x1 and x2 are two experimental values

Percent difference is comparing two experimental values, whereas percent error compares one experimental value with the actual/accepted value.

100221

21 *)(

)(%

xx

xxDifference