Goal Measurements and Mathematics Topic 1 Units 1 2 4 5 Units - A unit is a standard quantity that you can use to

compare other quantities to.

- For example centimeters and inches are both units

and they can be compared because 10 centimeters

equals one inch

- All measurements must have a standard quantity. - The SI system contains universally accepted units

for scientific measurements.

- There are 7 fundamental units in Physics

-Derived units are combinations of 2 or more

fundamental units. The SI System What is a Unit? Symbols for Units and Quantities - SI units are symbolized with letters. But be

careful because some of the unit symbols are

also used to symbolize formulas.

- For example: A can be ampere

or area. SI Prefixes - SI prefixes are prefixes combined

with SI base units to form new units that are larger or smaller than the base units by a multiple or sub multiple of 10. - For example a 1000 meters can become 1 km Or 0.01 meter can be expressed as 1 centimeter Tools for

Measurement

- Sin θ = a/c

- Cos θ = b/c

- Tan θ = a/b

- Knowing the length of any two sides

of a right triangle is enough to figure

out the length of the third side using the

Pythagorean theorem. The Pythagorean

theorem uses the formula:

c = a + b Measuring an Angle

- A common unit for measuring

angles is degree (°). The protractor is an

instrument used for measuring angles in degrees

Trigonometry

- - Is the branch of mathematics that treats the

relationships between angles and sides of triangles.

Basic trigonometric relationships are used to solve

some types of physics problems

- Important ratios of the sides of a

right triangle in terms of angle

θ include the following: Measuring Length

- Length is commonly measured in the Metric

System with the unit being meters (m). Occasionally

centimeters are more appropriate

Measuring Mass

- Mass is the amount of matter an object contains,

it can be measured with an electronic balance or

a triple-beam balance

- A mass determined in grams can be

converted to kilograms (kg) by

dividing by 1000 Tools for Measurement Uncertainty

in

Measurement - Before Adding/Subtracting/Multiplying/Dividing values, they must have the same units.

- After Adding/Subtracting sum or difference is rounded to the same

decimal point value as the least sensitive measurement.

- When Multiplying/Dividing values, the answer is rounded to the

same number of SigFigs as the lowest value SigFigs in the given.

ex. (200.0cm)(2.6cm)=520cm²

*The product can only have 2 SigFigs, since 2.6 is the smallest

value of SigFigs. Add, Subtract, Multiply, Divide with measured values Digits that are known with certainty plus the one digit whose value has been estimated are called Significant Digits.

Non-zero numbers are always significant.

The following rules apply in order to the zeroes in a measured value

1)Zeroes before a non-zero are not Significant. (0.002) 1SigFig.

2)Zeroes between non-zero digits are Significant. (0.706) 3SigFigs

3)Zeroes that appear after a nonzero digit are significant only if

a)followed by a decimal.

ex. (40s) 1SigFig.

(40.s) 2SigFigs

b)appear right of the decimal point.

ex. (37.0cm) 3SigFigs

(4.100m) 4SigFigs Significant Digits Every measurement has an experimental uncertainty

Uncertainty comes from:

1)Limitation of the measuring Instrument

2)Skill of the person using the instrument

3)Number of Measurements made

If several measurements are made and are nearly identical, the measurements are Precise.

If the measurements are very close to the accepted value

found in a handbook, the measurement is Accurate. Uncertainty in Measurements Scientific

Measurement Evaluating

Experimental Designs Graphing Scalar

and

Vector Quantities Scalar Quantities speed mass work power temperature energy -Physical quantities are put in two categories; scalar and vector quantities.

-A scalar quantity only has magnitude. For example, a scalar quantity would be the measurement of time or mass (ex: 5 seconds; 5 grams). A scalar quantity must have a number with an appropriate unit.

-When adding or subtracting scalar quantities, we use the rules of arithmetic.

-A vector quantity is made up of magnitude and direction. For example, a vector quantity would be the measurement of velocity because it is expressed with a proper unit as well as direction (ex: 5 meters per second, due south).

-When adding or subtracting vector quantities, we use algebraic or geometric methods. Scalar and Vector Quantities Vector Quantities Displacement Weight Force Momentum Velocity Acceleration The velocity of the ball, however, would be a vector quantity because it has magnitude and direction. For example the velocity of the football could be 23 m/s,

due north. . The speed of the ball is a scalar quantity because it doesn't have direction. Speed only measures how fast an object is going. So the speed of the ball could be 23 m/s. Solving Equations

While Using Algebra Solving Equations Using Algebra The six statements used to solve the equations for an unknown quantity are:

When equals are added, they have equal sums.

When equals are subtracted, they have equal remainders.

When equals are multiplied, they have equal products.

When equals are divided, they have equal quotients.

A quantity may be substituted for its equal.

Like powers or roots of equals are equal.

Also, the order of PEMDAS must be used to solve equations for quantities. Credit: Nishat Anjum: Power Point Presentation & Units Nakita Andrews: All Sample Problems Dimensional Analysis - Units on the left side of an equation must always be equivalent to the the units on the right side.

- Quantities can be added or subtracted only if they have the same

units. - Elapsed time can be measured with a

clock or a stopwatch because most events

measured in physics occurs quickly, the unit for

time is measured by seconds (s) Measuring Time Measuring Force - A force is a push or pull on a mass. Forces are

measured with a spring scale, being measured in

Newtons (N). Newtons is a derived

unit from Kg*m/s 2 2 2 2 Add, Subtract, Multiply, Divide

with measured values Percent Error:

- Deals with experimental value and

the most probable value or accepted value.

- Absolute error-the difference between an

experimental value and the accepted value.

- Percent error-a formula that can be

calculated as show below: (x meaning multiply)

Absolute error

Accepted value Data Analysis:

- In an experiment, multiple measurements are made

of a given or identical event.

- Range-the difference between the highest and lowest value

in the data set.

- Mean-the average of a set of “n” measurements.

- Variance-the sum of squares of the differences of the

measurements from the mean, divided by the number of

measurements.

- Standard Deviation-the square root of the variance. Evaluating Experimental

Results ***Refer to page 14 in your Physics Review Book to see the formulas for these terms*** X 100 Data Two quantities have an indirect

squared proportion if an increase in one

causes a squared decrease in the other.

Examples:

Directly proportional: y=2x

Inversely proportional: y=12/x

Constant proportion: y=6

Direct squared proportion: y=x

Indirect squared proportion: y=12/x There are several relationships that exist

between quantities measured in physics.

- Two quantities are directly proportional if an

increase in on causes an increase in the other

- Two quantities are inversely proportional if an

increase in one causes a decrease in the other.

- Two quantities have a constant proportion if an

increase in one causes no change in the other.

- Two quantities have a direct squared

proportion if an increase in one causes

a square increase in the other. Mathematical

Relationships Sometimes the line of best fit is extrapolated

which means it is extended beyond the region in

which date was taken.

The inclination of a graphed line is called the

slope. Formula: Extrapolation After you plot the data points on the graph, a line of best fit must be drawn.

Line of best fit: is a straight or curved line which approximates the relationship among a set data points. It usually does not pass through all measured points. Independent variable- the variable that the experimenter

changes. Its graphed on the x-axis.

Dependent- the one that changes as a result of the changes

made by the experimenter. Its graphed on the y-axis.

Graphs are titled as the dependent variable versus the

independent variable. Making a Graph In physics, data collected in experiments is usually represented

by graphical form. Graphing Data 2 2 Question 1

Write in scientific notation: 0.000467 and 32000000

Question 2

Express 5.43 x 10-3 as a number.

Question 3

(4.5 x 10-14) x (5.2 x 103) = ?

Question 4

(6.1 x 105)/(1.2 x 10-3) = ?

Question 5

(3.74 x 10-3)4 = ?

Question 6

The fifth root of 7.20 x 1022 = ?

Now remember, just count the zeros. It’s that easy! Questions Multiplication

- When multiplying, we multiply the numbers, add the exponents

1.2 x 104 multiplied by 5 x 106 = 6 x 1010

Division

- When dividing, we divide the numbers, subtract the

exponents 5.5 x 108 divided by 1.1 x 102 = 5 x 106 Multiplying and Dividing Scientific Notation When adding and subtracting the numbers must be expressed in the same power of ten and the same unit!

= 0.32 x 103m + 4.73 x 103 m

3.2 x 102 m + 4.73 x 103 m

= 5.05 x 103m

The same goes for subtraction… Adding and Subtracting Scientific Notation The number is written as a number between 1

and 10 multiplied by 10 raised to a power.

The power of 10 depends on:

- The number of places the decimal point is moved.

- The direction the decimal point is moved.

Left Positive exponent

Right Negative exponent Scientific notation is a way to express very large or very small values. Index Finger Width:

0.00001 km

=1.0 x 10-5 km

Representing Small Numbers Number between

1 and 10 Appropriate power

of ten Average Distance from the Earth to the Sun

is approximately 150,000,000 km = 1.5x108 km

150,000,000 = 1.5 x 100,000,000

= 1.5 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

= 1.5 x 108 Representing large numbers Radius of a Hydrogen

Atom:

0.0000000000529

= 5.29 x 1011 John Aguinaldo: Uncertainty in Measurements Simbi Akanni: Scientific Measurement Stephen Alexander Tools For Measurement Lejla Bolevic Scalar and vector quantities

& Jeopardy Alex Ab: Graphing Data Cassidy Bello Solving Equations Using Algebra Evaluating Experimental Results

& Handouts Nicole Bianco

Cook, Bernadine. "Topic 1: Measurement

and Mathematics." Brief Review in

Physics: The Physical Setting. 2013

ed. Needham, MA: Prentice Hall, 2003.

1-22. Print. Works Cited

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