Math: The language of Physics

Measurement: a comparison between an unknown quantity

and a standard.

Scientific Measurements

Using the Metric System (SI) for continuity:

Being a decimal system, we use prefixes to change between powers of 10.

Ie: 1/1000 of a gram = one milligram

Quantity Unit Symbol

Length meter mMass kilogram kgTime Second sTemperature Kelvin KElec. Current ampere A

Scientific Notation

We use Scientific notation for expressing numbers that are VERY LARGE or very small.

We write the numerical part of a quantity as a number between 1 and 10 multiplied by a whole-number power of 10.

► The average distance from the sun to Mars is 227,800,000,000m

This is written as 2.278 x 1011m

Moving the decimal to the LEFT a # of places is POSITIVE.

► The average mass of an electron is about

0.000,000,000,000,000,000,000,000,000,000,911 kg

This is written as 9.11 x 10-31 kg

Moving the decimal to the RIGHT a # of places is NEGATIVE.

Your turn to practice

1. Express the following quantities in scientific notation

a. 5800 m c. 302,000,000 m

b. 450,000 m d. 86,000,000,000 m

2. Express the following quantities in scientific notation

a. 0.000,508 kg c. 0.0003600 kg

b. 0.000,000,45 kg d. 0.004 kg

Converting Units with D. A.

►You can easily convert from a given unit to a needed unit using a series of Conversion Factors (fractions that equal one like

4 quarters / $1 or 5280 ft / 1 mile

►It is IMPERATIVE that you show your UNITS while you SHOW YOUR WORK.

What is the equivalent in kg of 465 g?

Recall that 1 kg = 1000 g

456g/1 ( 1kg / 1000g) = 456g*kg / 1000g = .456kg

Your turn to practice

Convert each of the following length measurements as directed.

a. 1.1 cm to meters c. 2.1 km to meters

b. 76.2 pm to mm d. 2.278 x 1011m to km

Convert each of the following mass measurements to kilograms.

a. 147 g c. 7.23 µg

b. 11 Mg d. 478 mg

Combinations with Scientific Notation

Adding and subtracting:

(4 x 108m) + (3 x 108m)

= (4 + 3) x 108m = 7 x 108m

(4.1x10-6kg) – (3.0x10-7kg)

= 4.1x10-6kg – 0.30x10-6kg

= (4.1-0.30)x10-6kg

=3.8x10-6kg

Multiplying and Dividing with Scientific Notation

First Multiplying:

(4x103 kg) (5x1011m)

=(4 x 5) x 103+11 kg*m

=20x1014kg*m

=2x1015 kg*m

Now Dividing

8x106 m3 / 2x10-2 m2

=8/2 x 106-(-2) m3-2

=4 x 108 m

See? Easy!

Measurement Uncertainties►Scientific results need to be reproducible.

►All measurements have a degree of uncertainty.

►Precision- the degree of exactness of a measurement. (to within 1/2 of the smallest measurement increment)

►Accuracy-how close results compare to a standard. Be sure to calibrate (zero) your instrument before using it.

Significant Digits

►When making a measurement, record your quantity by estimating 1 position beyond that which you can measure with that tool. In a meter stick, a pencil’s length might be recorded as 19.6cm. If the pencil’s end is somewhere between 0.6 and 0.7cm, then you estimate how far between and record the measurement as 19.62cm. All nonzero digits are significant. This has 4 significant digits.

What about Zeros?

►All final zeros after the decimal are significant.

►Zeros between 2 significant digits are always significant.

►Zeros used only as place holders are NOT significant.

All of the following have 3 significant digits:

245m 18.0g 308km 0.00623g

Your turn to Practice

State the number of significant digits in each of the following measurements:

a. 2804 m e. 0.003,068 m

b. 2.84 km f. 4.6 x 105 m

c. 0.007060 m g. 4.06 x 10-5 m

d. 75.00 m h. 1.20 x 10-4 m

Math functions with sig. figs

►In recording results of experiments, the answer can never be more precise than any individual measurement involved in calculating that answer.

►For adding and subtracting, first perform the function and then round to the appropriate decimal having the least precise value.

►EX: 24.686 m + 2.343 m + 3.21 m = 30.239 m rounded off correctly to 30.24 m

Multiplying & Dividing

►Perform the calculation, note the factor with the least significant digits, then round this answer to the same number of significant digits.

►EX: 3.22 cm x 2.1 cm = 6.762 cm2; 6.8 cm2

►EX: 36.5m / 3.414s = 10.691m/s; 10.7m/s

Your turn to Practice

► Solve the following problems:

1. Add 1.6 km + 1.62 m + 1200 cm

2. 10.8 g – 8.264 g

3. 3.2145 km x 4.23 km

4. 18.21 g / 4.4 cm3

Answers

Answers:

1. 1.6 km 2. 2.5 g

3. 13.6 km2 4. 4.1g/cm3

Sig Figs summarized

►There are no significant digits for counting.

►Only measurements have uncertainty.

►Significant digits are important to determine meaning in your calculations.

Visualizing Data►Graphs should tell the whole story in a

picture format.

►Line graphs can be linear or non-linear.

►A variable that is changed or manipulated is an independent variable. (One you can control directly) {plot on x-axis of graph}

►Dependent variables change as a result of the independent variable. {plot on y-axis}

►Always draw a line of best fit to show the relationship of data measured (not dot-to-dot)

Line Graph Guidelines

►Label both axis with name (and unit)

►Plot #’s evenly distributed on each axis

►Decide if the origin (0,0) is a valid point

►Spread out the graph as much as possible

►Draw the best fit line (straight or smooth curve) that passes through as many data points as possible.

►Title your graph

Linear Relationships► In a linear relationship, two variables are directly

proportional.

►The relationship is y = mx + b

►The slope, m, is Δy / Δx (AKA rise/run)

►The 2 data points to determine m, MUST be ON the line of best fit. (and should be as far apart as possible)

►The y-intercept, b, is the point where the line crosses the y-axis when x = zero.

►When b = 0, then the equation is y = mx

►When y gets smaller if x gets bigger, then slope is negative.

Non-Linear RelationshipsIn a parabola, gentle curve upward, variables are

related by a quadratic relationship:

y = ax2 + bx + c

One variable depends on the square of the other.

In a hyperbola, gentle curve downward, variables are related by an inverse relationship:

y = a/x or xy = a

One variable depends on the inverse of the other.

Your turn to PracticeThe total distance a lab cart travels during specified

lengths of time is given in the following data table:

1. Plot the dist vs time and best fit line for the points

2. Describe the curve

3. What is the slope of

the line?

4. Write an equation

relating dist and time

for this data.

Time (s) Dist. (m)

1.0 s 0.32 m

2.0 s 0.60 m

3.0 s 0.95 m

4.0 s 1.18 m

5.0 s 1.45 m

Key Equations

►y = mx + b

►m = rise / run; = Δy / Δx

►y = ax2 + bx + c

►xy = a