Notes, Unit 1

Scientific Thinking Conversions, Graphing,

A variable is any factor that might affect the behavior of an experimental setup.

It is the key ingredient when it comes to plotting data on a graph. The independent variable is the factor that is changed or manipulated during the experiment. • It is graphed on the x axis The dependent variable is the factor that depends on the independent variable. • It is graphed on the y-axis

Identifying Variables

Graphing Data De

pend

ent v

aria

ble

Independent variable

x y

Mathematical Model using y=mx+b • When writing the mathematical model:

– For “y”, substitute only the name of the dependent variable (label for the y axis)

– For “x”, substitute only the name of the independent variable (label for the x axis)

– For m, substitute the number and units of the slope. – For b, substitute the number & units of the y-intercept

• Example: y= stretch, x= mass m= 0.30 (cm/g) b = 3.2 cm y m x b

Stretch = 0.30 cm/g · mass + 3.2 cm

Mathematical Model using y=mx+b

slope (m) = 𝒓𝒓𝒓𝒓𝒓𝒓𝒓

= ∆𝒚∆𝒙

= 0.30

• Units of the slope =𝒓𝒓𝒓𝒖𝒓 𝒐𝒐 𝒚 (𝒓𝒓𝒓𝒓)𝒓𝒓𝒓𝒖𝒓 𝒐𝒐 𝒙 (𝒓𝒓𝒓)

• Slope units

• y-intercept = 3.2 cm • Write m & b with numbers and

units • m = [0.30 (cm/g)], b = 3.2 cm y = [0.30 (cm/g)] • x + 3.2 cm Substitute names of variables from experiment

y = x =

Stretch = 0.30 cm/g · mass + 3.2 cm

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

0 1 2 3

Stre

tch

(cm

)

Mass (g)

Stretch vs. Mass of a Spring

= 𝒄𝒄𝒈

Slope = 0.30 (cm/g)

Stretch Mass y = mx + b

Unit 1 1. What is meant by a positive slope? As x increases, y increases ( or as x decreases y decreases). 2. What is meant by a negative slope? As x increases, y decreases( or as x decreases y increases). 3. What is meant by “directly related.” A straight line with a positive slope. 4. What is meant by indirectly related? A straight line with a negative slope.

y

x

y

directly related or direct relationship

indirectly related or indirect relationship

x

Draw the graphs

The 5% Rule

0.05 * 75 = 3.75 3. Determine the y-intercept.

The y-intercept is 3.2 4. Is the y intercept = or < “5 % of y-max”? y-int = 3.2, 5% y-max = 3.75 y-int 3.2 < 5% y-max 3.75 If yes, the y intercept is negligible and can be set to zero. If no, the y-intercept is not negligible.

1. Find the largest value of y (which we will call y-max)

On this graph, y-max is 75 2. Figure out what 5% of y-

max is. 3. 5% = 0.05

y = 10.891x + 3.2 0

10

20

30

40

50

60

70

-1 0 1 2 3 4 5 6

Wt.

Supp

rt (p

enni

es)

Bridge Strength (strands)

(5, 75)

The 5% Rule 1. Is the y-intercept

insignificant? Why or why not?

y = 10.891x + 3.2 0

10

20

30

40

50

60

70

-1 0 1 2 3 4 5 6

Wt.

Supp

rt (p

enni

es)

Bridge Strength (strands)

(5, 75)

ymax = 75 5% ymax = 3.75 b = 3.2 3.2 < 3.75 b < 5% ymax b is insignificnt So b can be set to 0

𝑏 𝑦𝑚𝑚𝑚� 𝑥𝑥𝑥𝑥 ≤ 5% b is insignificant (= 0)

𝑏 𝑦𝑚𝑚𝑚� 𝑥𝑥𝑥𝑥 > 5% b is significant (cannot be zero)

3.2/ 75 * 100 = 4.26%

If b > 5%ymax then b is significant and b can not be = 0

Conversions in Physics • All measurements of length or distance

must be in meters. Convert cm m Convert kilometers m (1000 m = 1km)

• All measurements of mass must be in

kilograms. Must convert grams (g) to kilograms (kg

You probably learned in math class that it is much easier to convert meters to kilometers than feet to miles.

The ease of switching between units is another feature of the metric system.

To convert between SI units, multiply or divide by the appropriate power of 10.

SI Units

Prefixes are used to change SI units by powers of 10, as shown in the table below.

SI Units

How many m are equivalent to 3 cm? 3 cm ? m Relationship: How many m = how many cm?

Conversions: Proportional Reasoning

____cm = __m 100 1

_3_cm = __m x 100 𝑐𝑚3 𝑐𝑚

= 1 𝑚𝑚 𝑚

3 cm (1 m) = x m (100 cm)

(3 cm) •1 m 100 cm = = 0.03 m X m

1. How many m are equivalent to 3 cm? 3 cm ? m Relationship: How many m = how many cm?

Conversions: Proportional Reasoning

0.03 m ÷

____cm = __m 100 1

= 100 cm

_3_cm = __m ?

3 m

To get a smaller number for an answer, divide or multiply?

Given Relationship # Answer

Should our answer be a number bigger or smaller than 3? Should we multiply 3 by 100? Or divide 3 by 100?

Small unit big unit = smaller answer

X by number >1

÷ by number >1

Big unit small unit = bigger answer

You will often need to use different versions of a formula, or use a string of formulas, to solve a physics problem.

To check that you have set up a problem correctly, write the equation or set of equations you plan to use with the appropriate units.

Dimensional Analysis

Conversions: Proportional Reasoning

Before performing calculations, check that the answer will be in the expected units.

For example, if you are finding a speed and you see that your answer will be measured in s/m or m/s2, you know you have made an error in setting up the problem.

This method of treating the units as algebraic quantities, which can be cancelled, is called dimensional analysis.

Dimensional Analysis

Conversions: Proportional Reasoning

Dimensional analysis is also used in choosing conversion factors.

A conversion factor is a multiplier equal to 1. For example, because 1 kg = 1000 g, you can construct the following conversion factors:

Dimensional Analysis

Conversions: Proportional Reasoning

Choose a conversion factor that will make the units cancel, leaving the answer in the correct units.

For example, to convert 1.34 kg of iron ore to grams, do as shown below:

Dimensional Analysis

Conversions: Proportional Reasoning

1. How many m are equivalent to 3 cm? 3 cm ? m Relationship: How many m = how many cm?

Conversions: Dimensional Analysis

x

____cm = __m 100 1

= 1 m ` 3 cm 100 cm

𝟏𝟏𝟏 𝒄𝒄𝟏 𝒄

𝟏 𝒄

𝟏𝟏𝟏 𝒄𝒄 or Conversion factors:

Given times Conversion Factor = Answer

(𝟑)(𝟏 𝒄) 𝟏𝟏𝟏

=

Note that this term is what resulted from a setting up a proportion.

Notes, Unit 1 5. For a swinging pendulum, what is amplitude (A)? It is a measure as an angle or a distance from the rest point of the swing bob. 6. What is the period of a pendulum? The period of a pendulum (T) is the amount of time in seconds it takes the pendulum to make a complete swing back and forth.

If you are out of room on your Bell Work sheet, write this on your study guide.

Notes, Unit 1 a) What are the units for the slope of this graph? b) What does the slope of this graph tell you? c) Write a model equation for this graph d) What could the y-intercept mean in this situation?

Slope = 7 y-intercept = 3.6

Shoes (pairs)

Cos

t ($)

Online Shoe Store

Sketch the graph.

Notes, Unit 1

a. What are the units for the slope of this graph? 𝒅𝒐𝒅𝒅𝒅𝒓𝒓𝒑𝒅𝒓𝒓𝒓

b. What does the slope of this graph tell you?

The average cost of one pair of shoes in dollars. c. Write a model equation for this graph.

𝑪𝒐𝒓𝒖 = 𝟕.𝟏𝒅𝒐𝒅𝒅𝒅𝒓𝒓𝒑𝒅𝒓𝒓𝒓

𝒓𝒔𝒐𝒓𝒓 + 𝟑.𝟔 𝒅𝒐𝒅𝒅𝒅𝒓𝒓

d. What could the y-intercept could mean in this situation? The shipping cost for the order.

Notes, Unit 1 1. What is meant by precision or precise measurements? • Precision refers to the closeness of a set of measurements • Precision is the ability reproduce a measurement.

• If we weigh a 1.5 kilograms bag of sugar five times, each

weight is close to 1.5 kg: • Example: 1.51 kg, 1.51 kg, 1.50kg, 1.49 kg, 1.49 kg 2. What is meant by accuracy or accurate measurements? Accuracy refers to the closeness of measurements to the correct or accepted value of the quantity measured. • Example: the true value of mass of the sugar is 1.5 kilograms

so a measurement close to 1.5 kg is accurate.

(ex: 1.49, 1.48, 1.51, 1.52 are close to 1.50)

Notes, Unit 1 3. Draw the targets. Label as precise, accurate, both

or neither. Bull’s eye (center) = the true value

A. B.

C. D.

B. Accurate but not precise

D. Neither precise nor accurate

C. Precise but not accurate

A. Both precise & accurate

A meterstick is used to measure a pen and the measurement is recorded as 14.3 cm.

This measurement has three valid digits: two you are sure of, and one you estimated.

The valid digits in a measurement are called significant digits.

However, the last digit given for any measurement is the uncertain digit.

Significant Digits

All nonzero digits in a measurement are significant, but not all zeros are significant.

Consider a measurement such as 0.0860 m. Here the first two zeros serve only to locate the decimal point and are not significant.

The last zero, however, is the estimated digit and is significant.

Significant Digits

When you perform any arithmetic operation, it is important to remember that the result can never be more precise than the least-precise measurement.

To add or subtract measurements, first perform the operation, then round off the result to correspond to the least-precise value involved.

Significant Digits

To multiply or divide measurements, perform the calculation and then round to the same number of significant digits as the least-precise measurement.

Note that significant digits are considered only when calculating with measurements.

Significant Digits