Physics 1.2Physics 1.2

Measurements and MathematicsMeasurements and Mathematics

Unit: Unit: Standard unit of: time, distance, mass, etc.Standard unit of: time, distance, mass, etc.

SI systemSI system Scientific measurementsScientific measurements SI prefixes and symbolsSI prefixes and symbols

Some Units of Measure:Some Units of Measure:

Dimensional Analysis:Dimensional Analysis:

A way to examine the accuracy of a particular A way to examine the accuracy of a particular formula.formula.

Also allows one to convert and cancel units.Also allows one to convert and cancel units.

Example:Example:

Si Prefixes:Si Prefixes:

Tools for MeasurementTools for Measurement

LengthLength Metric rulerMetric ruler

MassMass Electronic balanceElectronic balance

TimeTime Stop watchStop watch

ForceForce Spring scaleSpring scale

AngleAngle Protractor Protractor trigonometrytrigonometry

TrigonometryTrigonometry

SOH CAH TOASOH CAH TOA

Sample ProblemSample Problem

A block is displaced a vertical distance of A block is displaced a vertical distance of 0.75 meters as is slides down a 1.25-meter 0.75 meters as is slides down a 1.25-meter long plane inclined to the horizontal.long plane inclined to the horizontal.

a)a) How far is the block How far is the block displaced horizontallydisplaced horizontally??

b)b) What is the measure of the What is the measure of the angle of inclinationangle of inclination of the plane of the plane to the horizontalto the horizontal??

Uncertainty in measurementUncertainty in measurement

Precision:Precision: Multiple measurements are nearly identical.Multiple measurements are nearly identical.

Accuracy:Accuracy: Measurements are very close to accepted value Measurements are very close to accepted value

found in reference table.found in reference table.

Significant FiguresSignificant Figures

0.00.04400990000 has how many sig figs? has how many sig figs? Zeros Zeros beforebefore nonzero digits are nonzero digits are not not significant.significant. Zeros Zeros betweenbetween nonzero digits are significant. nonzero digits are significant. Zeros Zeros afterafter nonzero digits are only significant if nonzero digits are only significant if

they:they:a.a. Are followed by a decimal point.Are followed by a decimal point.

b.b. Are to the right of a decimal point.Are to the right of a decimal point.

Addition, Subtraction, Addition, Subtraction, Multiplication, and Division:Multiplication, and Division:

RULE: When adding or subtracting your answer RULE: When adding or subtracting your answer can only show as many decimal places as the can only show as many decimal places as the measurement having the fewest number of measurement having the fewest number of decimal places.decimal places.

RULE: When multiplying or dividing, your RULE: When multiplying or dividing, your answer may only show as many significant digits answer may only show as many significant digits as the multiplied or divided measurement showing as the multiplied or divided measurement showing the least number of significant digits.the least number of significant digits.

Sig Fig Problems:Sig Fig Problems:

Example: When we add 3.76 g + 14.83 g + 2.1 Example: When we add 3.76 g + 14.83 g + 2.1 g = 20.69 g. We must round our answer, g = 20.69 g. We must round our answer, 20.69, to one decimal place (the tenth place). 20.69, to one decimal place (the tenth place). Our final answer is 20.7 g.Our final answer is 20.7 g.

Example: When multiplying 22.37 cm x 3.10 Example: When multiplying 22.37 cm x 3.10 cm x 85.75 cm = 5946.50525 cm3. Our final cm x 85.75 cm = 5946.50525 cm3. Our final answer becomes 5950 cm3.answer becomes 5950 cm3.

Scientific NotationScientific Notation

Used to express very large or very small Used to express very large or very small measurements.measurements.

Ex/ Ex/ The number 123,000,000,000 in scientific The number 123,000,000,000 in scientific notation is written as : notation is written as :

Rules for Scientific Notation:Rules for Scientific Notation:

Rule for Addition and SubtractionRule for Addition and Subtraction - when adding - when adding or subtracting in scientific notation, you must express or subtracting in scientific notation, you must express the numbers as the the numbers as the same power of 10same power of 10.  This will often .  This will often involve changing the decimal place of the coefficient.involve changing the decimal place of the coefficient.

Rule for MultiplicationRule for Multiplication - When you multiply - When you multiply numbers with scientific notation, multiply the numbers with scientific notation, multiply the coefficients together and coefficients together and add the exponentsadd the exponents.  The .  The base will remain 10.base will remain 10.

Rule for DivisionRule for Division - When dividing with scientific - When dividing with scientific notation, divide the coefficients and notation, divide the coefficients and subtract the subtract the exponentsexponents.  The base will remain 10. .  The base will remain 10.

Orders of Magnitude:Orders of Magnitude:

Estimating the answer allows you to check if Estimating the answer allows you to check if your calculated answer is reasonable.your calculated answer is reasonable.

Sometimes, you can steal an answer quickly Sometimes, you can steal an answer quickly and save time. and save time.

On the Regents exam, this is rare, so be On the Regents exam, this is rare, so be careful!careful!

Sample Problem:Sample Problem:

Estimate the magnitude of the gravitational Estimate the magnitude of the gravitational force between earth and the moon and force between earth and the moon and compare it with the actual value.compare it with the actual value.

gravitational constant gravitational constant GG = 6.67300 × 10-11 = 6.67300 × 10-11 m3 kg-1 s-2m3 kg-1 s-2

mass of Earth = 5.9742 × 1024 kilogramsmass of Earth = 5.9742 × 1024 kilograms mass of the Moon = 7.36 × 1022 kilogramsmass of the Moon = 7.36 × 1022 kilograms Distance (r)= 3.84E8 metersDistance (r)= 3.84E8 meters

Another Problem:Another Problem:

As the Voyager spacecraft passed the planet As the Voyager spacecraft passed the planet Uranus, it sent signals back to Earth. Uranus, it sent signals back to Earth. Determine the order of magnitude of the time Determine the order of magnitude of the time in seconds for a signal to reach Earth. The in seconds for a signal to reach Earth. The distance from Earth to Uranus is 2.71E12 m. distance from Earth to Uranus is 2.71E12 m. The speed of light in a vacuum is 3.00E8 m/s.The speed of light in a vacuum is 3.00E8 m/s.

From V=d/tFrom V=d/t

Evaluating Experimental Results:Evaluating Experimental Results:

Data AnalysisData Analysis Range: highest to lowest.Range: highest to lowest. Mean: averageMean: average Variance: sum of the squares of differences of Variance: sum of the squares of differences of

values from mean.values from mean. Standard Deviation: square root of the varianceStandard Deviation: square root of the variance

Problem:Problem:

In an experiment, a student determines that the In an experiment, a student determines that the acceleration due to gravity in the laboratory is acceleration due to gravity in the laboratory is 9.98 m/s9.98 m/s². Determine the percent error. (The ². Determine the percent error. (The accepted value for the acceleration due to accepted value for the acceleration due to gravity is 9.81 m/s².) gravity is 9.81 m/s².)

Percent Error:Percent Error:

% error =  % error =   | your result - accepted value |  | your result - accepted value |    x  100 %    x  100 %

                                 accepted value                                 accepted value

Graphing Data:Graphing Data:

Independent variable-Independent variable- One that experimenter changes, x or horizontal axis.One that experimenter changes, x or horizontal axis.

Dependent variable-Dependent variable- One measured for change due to change in independent One measured for change due to change in independent

variable, y or vertical axis.variable, y or vertical axis.

Line of best fit-Line of best fit- A straight or curved line which approximates the A straight or curved line which approximates the

relationship among a set of data points. May be relationship among a set of data points. May be extrapolated (extend beyond data).extrapolated (extend beyond data).

A common statistical function to execute is something called a linear regression.  A common statistical function to execute is something called a linear regression.  Linear regression looks for a correlation between an independent variable and a Linear regression looks for a correlation between an independent variable and a

dependent variable.  Let’s say our independent variable is time, and our dependent variable.  Let’s say our independent variable is time, and our dependent variable is a batter’s home run totals.  The following is a time series dependent variable is a batter’s home run totals.  The following is a time series

plot of home run numbers for Cal Ripken:plot of home run numbers for Cal Ripken:

More graph stuff:More graph stuff:

The The slopeslope of a graphed line is defined as a of a graphed line is defined as a ratio of ratio of ∆y/ ∆x.∆y/ ∆x.

Sample Problem:Sample Problem:

The position of a The position of a moving car was moving car was measured at one second measured at one second intervals and recorded intervals and recorded in the following table. in the following table. Graph the data on graph Graph the data on graph paper and draw the line paper and draw the line of best fit. Determine of best fit. Determine the slope of the line.the slope of the line.

Time (s)Time (s) Position (m)Position (m)

0.00.0 00

1.01.0 1818

2.02.0 4040

3.03.0 6262

4.04.0 8080

5.05.0 100100

Types of proportions:Types of proportions:

Direct proportionDirect proportion Increase in one causes increase in another.Increase in one causes increase in another.

Inverse proportionInverse proportion Increase in one causes decrease in anotherIncrease in one causes decrease in another

Direct squared proportionDirect squared proportion Increase in one causes a squared increase in another.Increase in one causes a squared increase in another.

Indirect squared proportionIndirect squared proportion Increase in one causes a squared decrease in another.Increase in one causes a squared decrease in another.

Constant proportionConstant proportion Increase in one causes no change in another.Increase in one causes no change in another.

More Graphs:More Graphs: