Measurement in Physics AP Physics I Measurement Uncertainty

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Measurement in Physics AP Physics I

Measurement in Physics AP Physics I

Measurement, Uncertainty, & Significant Figures n Accurate, precise measurements are an important part of

Measurement, Uncertainty, & Significant Figures n Accurate, precise measurements are an important part of physics, but no measurement is absolutely precise. Precision in a strict sense refers to the reproducibility of the measurement using a given instrument. q Accuracy refers to how close a measurement is to the true value. q

Measurement, Uncertainty, & Significant Figures §Sources of uncertainty: q Limited accuracy of measuring instrument

Measurement, Uncertainty, & Significant Figures §Sources of uncertainty: q Limited accuracy of measuring instrument q Inability to read an instrument beyond some fraction of the smallest division shown.

Measurement, Uncertainty, & Significant Figures n It is important to state the esti mated

Measurement, Uncertainty, & Significant Figures n It is important to state the esti mated uncertainty in a measurement. q n The uncertainty of the ruler shown is ± 0. 1 cm Uncertainty may not always be specified. q q Assumption is that uncertainty is one (or 2 -5 ) units in the last digit specified. EXAMPLE: If the width of the board above is given as 8. 8 cm, then uncertainty is assumed to be 0. 1 cm or 0. 2 cm

Measurement, Uncertainty, & Significant Figures n. The percent uncertainty in a measurement is the

Measurement, Uncertainty, & Significant Figures n. The percent uncertainty in a measurement is the ratio of the uncertainty of the measurement tool to the measurement (then multiply by 100).

Scientific Notation n A way to write and manipulate very large and very small

Scientific Notation n A way to write and manipulate very large and very small numbers in an easier manner. The first number is the coefficient (greater than or equal to 1 but less than 10). The second number is the base. q n It will be 10 in scientific notation. EXAMPLE: 300, 000 = 3 x 105 q q 3 is the coefficient 5 is the exponent.

Scientific Notation n n To write a number in scientific notation: Put the decimal

Scientific Notation n n To write a number in scientific notation: Put the decimal after the first digit and drop the zeroes. To find the exponent count the number of places from the decimal to the end of the number. q 123, 000, 000 = 1. 23 × 1011 Exponents are often expressed using other notations. The number 123, 000, 000 can also be written as: q 1. 23 E+11 or 1. 23 x 10^11

Scientific Notation n For numbers less than 1, just use a negative exponent. n

Scientific Notation n For numbers less than 1, just use a negative exponent. n A millionth of a second: 0. 000001 s 1. 0 E 6 q 1. 0^ 6 q 1. 0 x 10 6 q

Scientific Notation n Rules for Multiplication in Scientific Notation: Multiply the coefficients q Add

Scientific Notation n Rules for Multiplication in Scientific Notation: Multiply the coefficients q Add the exponents (base 10 remains) q Example: (3 x 104)(2 x 105) = q n 6 × 109

Scientific Notation n n What happens if the coefficient is 10 or more when

Scientific Notation n n What happens if the coefficient is 10 or more when using scientific notation? Example: (5 x 10 3) (6 x 103) = 30. x 106 Move the decimal point over to the left until the coefficient is between 1 and 10. For each place we move the decimal over the exponent will be raised 1 power. 30. x 106 = 3. 0 x 107 in scientific notation.

Scientific Notation n Rules for Division in Scientific Notation: Divide the coefficients q Subtract

Scientific Notation n Rules for Division in Scientific Notation: Divide the coefficients q Subtract the exponents (base 10 remains) q n Example: q 3 × 103 (6 x 106) / (2 x 103) =

Scientific Notation n n What happens if the coefficient is less than 1? Example:

Scientific Notation n n What happens if the coefficient is less than 1? Example: (2 x 10 7) / (8 x 103) = 0. 25 x 104 While the value is correct it is not correctly written in scientific notation since the coefficient is not between 1 and 10. Move the decimal point to the right until the coefficient is between 1 and 10. For each place we move the decimal, the exponent will be lowered 1 power of ten. q 0. 25 x 10 4 = 2. 5 × 103

Significant Figures/Digits The number of significant digits in an answer to a calculation will

Significant Figures/Digits The number of significant digits in an answer to a calculation will depend on the number of significant digits in the given data, as discussed in the rules to follow. n Approximate calculations (order-ofmagnitude estimates) always result in answers with only one or two significant digits. n

Significant Figures/Digits n When q are Digits Significant? Non zero digits are always significant.

Significant Figures/Digits n When q are Digits Significant? Non zero digits are always significant. n Thus, 22 has 2 significant digits, and 22. 3 has 3 significant digits.

Significant Figures/Digits The number of significant digits in an answer to a calculation will

Significant Figures/Digits The number of significant digits in an answer to a calculation will depend on the number of significant digits in the given data, as discussed in the rules to follow. n Approximate calculations (order-ofmagnitude estimates) always result in answers with only one or two significant digits. n

Significant Figures/Digits n With zeroes, the situation is more complicated: q Zeroes placed before

Significant Figures/Digits n With zeroes, the situation is more complicated: q Zeroes placed before other digits are not significant. n q Zeroes placed between other digits are always significant. n q 0. 046 has two significant digits 4009 kg has four significant digits Zeroes placed after other digits but behind a decimal point are significant. n 7. 90 has three significant digits

Significant Figures/Digits n Zeroes at the end of a number are significant only if

Significant Figures/Digits n Zeroes at the end of a number are significant only if they are behind a decimal point. q n n Otherwise, it is impossible to tell if they are significant. To avoid uncertainty, use scientific notation to place significant zeroes behind a decimal point: Example: 8200 q 8. 200× 103 has ___ significant digits 4 has ___ significant digits 3 3 q 8. 2× 10 has ___ significant digits 2 q 8. 20× 103

Significant Figures/Digits n Multiplication, Division, Trig. functions, etc. q In a calculation involving multiplication,

Significant Figures/Digits n Multiplication, Division, Trig. functions, etc. q In a calculation involving multiplication, division, trigonometric functions, etc. , the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc.

Significant Figures/Digits n n In evaluating sin(kx), where k = 0. 097 m-1 (two

Significant Figures/Digits n n In evaluating sin(kx), where k = 0. 097 m-1 (two significant digits) and x = 4. 73 m (three 2 significant digits), the answer should have __ significant digits. Note: Whole numbers have essentially an unlimited number of significant digits. q q q Example: If a hair dryer uses 1. 2 k. W of power, then 2 identical hairdryers use 2. 4 k. W: 1. 2 k. W {2 sig. dig. } × 2 {unlimited sig. dig. } = 2. 4 k. W {___ 2 sig. dig. }

Significant Figures/Digits Significant Digits in Addition & Subtraction n When quantities are being added

Significant Figures/Digits Significant Digits in Addition & Subtraction n When quantities are being added or subtracted, the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted. n Example: 5. 67 J +1. 1 J +0. 9378 J 7. 7 J (two decimal places) (one decimal place) (four decimal places) (one decimal place)

Significant Figures/Digits Keep One Extra Digit in Intermediate Answers n When doing multi-step calculations,

Significant Figures/Digits Keep One Extra Digit in Intermediate Answers n When doing multi-step calculations, keep at least one more significant digit in intermediate results than needed in your final answer. q q q For instance, if a final answer requires two significant digits, then carry at least three significant digits in calculations. If you round-off all your intermediate answers to only two digits, you are discarding the information contained in the third digit, and as a result the second digit in your final answer might be incorrect. This phenomenon is known as "round-off error. "

Significant Figures/Digits The Two Greatest Sins Regarding Significant Digits n Writing more digits in

Significant Figures/Digits The Two Greatest Sins Regarding Significant Digits n Writing more digits in an answer (intermediate or final) than justified by the number of digits in the data. n Rounding-off early q Example: Using two digits in an intermediate answer, and then writing three digits in the final answer.

SI units for Physics The SI stands for "System International”. There are 3 fundamental

SI units for Physics The SI stands for "System International”. There are 3 fundamental SI units for LENGTH, MASS, and TIME. 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 Prefix Factor Physics Symbol Mega ( mostly used for

Most commonly used prefixes in Prefix Factor Physics 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 x 10 2 distances in optics. This is the unit MOST people in AP forget to convert) 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

Example If a capacitor is labeled 2. 5μF(micro. Farads), how would it be labeled

Example If a capacitor is labeled 2. 5μF(micro. Farads), 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 XL 106. 7 transmits at a frequency of 106. 7 x 106 Hertz. How would it be written in MHz (Mega. Hertz)? A HERTZ is the fundamental unit used when discussed radio frequency! 106. 7 MHz Notice we simply drop the factor and add the prefix.

Dimensional Analysis is simply a technique you can use to convert from one unit

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

Trigonometric Functions Many concepts in physics act at angles or make right triangles. Let’s

Trigonometric Functions Many concepts in physics act at angles or make right triangles. Let’s review common functions.

Example A person attempts to measure the height of a building by walking out

Example A 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. What do I a) Find the height of the building know? b) the distance the flashlight • The angle • The adjacent beam has to travel before it side strikes the top of the building. What do I want? Course of action The opposite side USE TANGENT!

Example A van moves up a straight mountain highway, as shown above. Elevation markers

Example A van moves up a straight mountain highway, as shown above. Elevation markers at the beginning and ending points of the trip show that it has risen vertically 0. 530 km, and the mileage indicator on the van shows that it 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!