Introduction and Mathematical Concepts PHYSICS

Physics predicts how nature will behave in one situation based on the results of experimental data obtained in another situation. * Focuses on the behavior and structure of matter

Scientific Method Problem to Investigate Problem to Investigate Observations Observations Hypothesis - Educated guess Hypothesis - Educated guess Test Hypothesis Test Hypothesis Theory - Most ideas in science Theory - Most ideas in science Test Theory Test Theory Scientific Law - Mathematical proof Scientific Law - Mathematical proof

Measurement & Uncertainty Uncertainty: No measurement is absolutely precise Estimated uncertainty: – –Example: Width of a board 8.8cm +/- 0.1cm 0.1 cm represents the estimated uncertainty in the measurement Actual width  between 8.7 and 8.9 cm

Percent Uncertainty Ratio of the uncertainty to the measured value, x 100 Ratio of the uncertainty to the measured value, x 100 Example: Example: –Measurement = 8.8cm –Uncertainty = 0.1 cm –** Percent uncertainty 8.8/0.1 x 100% = 1%

Accuracy- how close a measurement comes to the actual value Precision- how close a series of measurements are to one another Percent Error- absolute value of the theoretical minus the experimental, divided by the theoretical, multiplied by 100 Theoretical- Experimental /theoretical x100

Significant Figures-all of the digits that are known plus a last digit that is estimated (all of the important/necessary numbers) 1023 sig. Figs sig. figs 1001 sig. Figs sig. figs

Rules for Significant Figures Sig Figs are used when doing measurements Sig Figs are used when doing measurements Use all measured numbers and one estimated number Use all measured numbers and one estimated number These are the rules on determining how many are in a number: These are the rules on determining how many are in a number: –Non-zero digits are always significant. –Any zeros between two significant digits are significant. –A trailing and leading zero is NOT significant. – ALL zeroes which are SIMULTANEOUSLY to the right of the decimal point AND at the end of the number are ALWAYS significant.

Significant Figures Guidelines: Guidelines: –Zeros at the beginning of a number  Not significant (Decimal point holders) m3 Significant Figures(5, 7, 8) m3 Significant Figures(5, 7, 8) –Zeros within the number  Significant m 4 Significant Figures(1, 0, 8, 7) m 4 Significant Figures(1, 0, 8, 7) –Zeros at the end of a number, after a decimal point  Significant m5 Significant Figures(8, 7, 0, 9, 0) m5 Significant Figures(8, 7, 0, 9, 0)

Significant Figures Number of reliable known digits Number of reliable known digits Examples: Examples: –56.93 = 4 significant figures –0.087 = 2 significant figures (Zero acts as a place hold, illustrating where the decimal point is located) –80 = 1 significant figure –80.0 = 2 significant figures

Rules for Counting Significant Figures - Details Nonzero integers always count as significant figures. Nonzero integers always count as significant figures has 3456 has 4 sig figs. 4 sig figs.

Rules for Counting Significant Figures - Details Zeros Zeros - Leading zeros do not count as - Leading zeros do not count as significant figures has has 3 sig figs. 3 sig figs.

Rules for Counting Significant Figures - Details Zeros Zeros - Captive zeros always count as - Captive zeros always count as significant figures has has 4 sig figs. 4 sig figs.

Rules for Counting Significant Figures - Details Zeros Zeros - Trailing zeros are significant only - Trailing zeros are significant only if the number contains a decimal point has has 4 sig figs. 4 sig figs.

Rules for Counting Significant Figures - Details Exact numbers have an infinite number of significant figures. Exact numbers have an infinite number of significant figures. 1 inch = 2.54 cm, exactly 1 inch = 2.54 cm, exactly

Rules for Significant Figures in Mathematical Operations Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation. Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation  2.0 = 6.38  2.0 =  13 (2 sig figs)  13 (2 sig figs)

Multiple/Divide Your answer must have the same number of sig figs as the original number with the least amount of sig figs. Your answer must have the same number of sig figs as the original number with the least amount of sig figs. Example: 2 x 42 = 84 (in math) Example: 2 x 42 = 84 (in math)  = 80 with sig figs

Rules for Significant Figures in Mathematical Operations Addition and Subtraction: # sig figs in the result equals the number of decimal places in the least precise measurement. Addition and Subtraction: # sig figs in the result equals the number of decimal places in the least precise measurement = =  18.7 (3 sig figs)  18.7 (3 sig figs)

Addition/Subtraction When you add or subtract, the answer must have the same number of decimal places as the original number with the least number of decimal places When you add or subtract, the answer must have the same number of decimal places as the original number with the least number of decimal places Example: = 5.1 in math, 5 with sf Example: = 5.1 in math, 5 with sf = = = =

Significant Figures HINT: HINT: –Keep extra significant figures throughout a calculation and round off only in the final result

Scientific Notation Alternative way to express very large or very small numbers. Alternative way to express very large or very small numbers. Number expressed as the product of a number between 1 and 10 and the appropriate power of 10. Number expressed as the product of a number between 1 and 10 and the appropriate power of 10. Large Number: 238,000. = Decimal placed between 1 st and 2 nd digit Small Number : = 2.38 x x 10 -4

Scientific Notation “Scientific Notation” or “Powers of Ten” “Scientific Notation” or “Powers of Ten” –Allows the number of significant figures to be clearly expressed Example: Example: –56, 800  5.68 x 10 4 –  3.4 x –6.78 x 10 4  Number is known to an accuracy of 3 significant figures –6.780 x 10 4  Number is known to an accuracy of 4 significant figures

Scientific Notation Express the following numbers as Scientific Notation Express the following numbers as Scientific Notation ,784 x 10 4

Physics experiments involve the measurement of a variety of quantities. These measurements should be accurate and reproducible. The first step in ensuring accuracy and reproducibility is defining the units in which the measurements are made.

SI units Systeme International (SI)- Specifies a consistent set of units to be utilized in the laws and equations of PhysicsSysteme International (SI)- Specifies a consistent set of units to be utilized in the laws and equations of Physics Meter (m): unit of length Kilogram (kg): unit of mass Second (s): unit of time

Metric System kilok 10 3 = 1000 hectoh 10 2 = 100 dekada 10 1 = 10 meter, liter, gram (Base) m, l, g 10 0 = 1 decid = 0.1 centic = 0.01 millim = 0.001

SI Prefixes Little Guys Big Guys

Metric System kilok 10 3 hectoh 10 2 dekada 10 1 meter, liter, gram m, l, g 10 decid centic millim 10 -3

Metric Conversions website of other prefixes website of other prefixes website of other prefixes

Volume Standard unit of volume = cubic meter (m 3 ) Standard unit of volume = cubic meter (m 3 ) Rather large unit Rather large unit More convenient to use the nonstandard unit of volume  cube 10cm per side More convenient to use the nonstandard unit of volume  cube 10cm per side Volume of a 1 Liter = 1000 cm 3 (10 cm x 10 cm x 10 cm) Volume of a 1 Liter = 1000 cm 3 (10 cm x 10 cm x 10 cm) Since 1 L = 1000 mL  1 mL= 1 cm 3 (cc) Since 1 L = 1000 mL  1 mL= 1 cm 3 (cc)

Fundamental SI Units Physical QuantityName Abbreviation Lengthmeterm Masskilogramkg Timeseconds Temperature KelvinK Electric current ampereA Amt of Substance mole mol Luminous Intensity candelacd

The units for length, mass, and time (as well as a few others), are regarded as base SI units These units are used in combination to define additional units for other important physical quantities such as force and energy  Derived Units

Derived Units Units that are created based on formulas and equations Units that are created based on formulas and equations –Volume = L·w·h = m·m·m = m 3 –Area = L·w = m·m = m 2 –Force F = m·a = kg·m·s -2 = Newton, N F = m·a = kg·m·s -2 = Newton, N –Work W = F·d = N·m = Joule, J W = F·d = N·m = Joule, J –Pressure P = F/A = N·m -2 = Pascal, Pa P = F/A = N·m -2 = Pascal, Pa

Derived Units website website Units that are created based on formulas and equations Units that are created based on formulas and equations Volume,V = l·w·h = m·m·m = m 3 Volume,V = l·w·h = m·m·m = m 3 Area, A = l·w = m·m = m 2 Area, A = l·w = m·m = m 2 Force, F = m·a = kg·m·s -2 = Newton, N Force, F = m·a = kg·m·s -2 = Newton, N Work, W = F·d = N·m = Joule, J Work, W = F·d = N·m = Joule, J Pressure, P = F/A = N·m -2 = Pascal, Pa Pressure, P = F/A = N·m -2 = Pascal, Pa

Dimensions Dimensions = fundamental (base) quantities used in physical descriptions Dimensions = fundamental (base) quantities used in physical descriptions –Example: Length, mass, and time Length, mass, and time Dimensional Analysis: Procedure utilized to check the dimensional consistency of any equation Dimensional Analysis: Procedure utilized to check the dimensional consistency of any equation

The Role of Units in Problem Solving The Role of Units in Problem Solving DIMENSIONAL ANALYSIS [L] = length [M] = mass [T] = time Is the following equation dimensionally correct?

The Role of Units in Problem Solving The Role of Units in Problem Solving Is the following equation dimensionally correct?

Derived Units Derived units are from the “base” units in the SI system. Derived units are from the “base” units in the SI system. Base unit website Base unit website Base unit website Base unit website

Dimensional Analysis (Factor-Label) Website Website Website Dimensional analysis and physics website Dimensional analysis and physics website Dimensional analysis and physics website Dimensional analysis and physics website

The Role of Units in Problem Solving THE CONVERSION OF UNITS 1 ft = m 1 in =2.54 cm 1 mi = km 1 hp = 746 W 1 liter = m 3

Problem Solving Problem Solving Hints: * Read and analyze the problem * Write down given data and what you need to find * Draw a diagram * Determine which principle(s) and equation(s) are applicable * Simplify the equations before inserting values * Check units before doing calculations * Substitute given quantities into equation(s) and perform calculations * Answer: Proper units & Proper significant figures * Is the result reasonable? * Disclaimer: This powerpoint presentation is a compilation of various works.

The Role of Units in Problem Solving The Role of Units in Problem Solving PROBLEM 1: The World’s Highest Waterfall The highest waterfall in the world is Angel Falls in Venezuela, with a total drop of m. Express this drop in feet.

Answer Since feet = 1 meter, it follows that (3.281 feet)/(1 meter) = 1 Length = (979.0 m)(3.281 feet/1 meter) = 3212 feet

The Role of Units in Problem Solving The Role of Units in Problem Solving PROBLEM 2: Interstate Speed Limit Express the speed limit of 65 miles/hour in terms of meters/second.

Answer Use 5280 feet = 1 mile and 3600 seconds = 1 hour and feet = 1 meter (65 mi/hr)(5280 ft/1 mi)(1 m/3.281 ft)(1 hr/3600 s) = 29 m/s

The Role of Units in Problem Solving The Role of Units in Problem Solving DIMENSIONAL ANALYSIS [L] = length [M] = mass [T] = time Is the following equation dimensionally correct?

The Role of Units in Problem Solving The Role of Units in Problem Solving Is the following equation dimensionally correct? * Disclaimer: This powerpoint presentation is a compilation of various works.

Trigonometry and Physics Basic Trig is needed to solve some of our physics problems. Basic Trig is needed to solve some of our physics problems.

The Law of Sines The Law of Sines

If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle

To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles.

FOUR CASES CASE 1: One side and two angles are known (SAA or ASA). CASE 2: Two sides and the angle opposite one of them are known (SSA). CASE 3: Two sides and the included angle are known (SAS). CASE 4: Three sides are known (SSS).

CASE 1: ASA or SAA S A A ASA S AA SAA

S S A CASE 2: SSA

S S A CASE 3: SAS

S S S CASE 4: SSS

The Law of Sines is used to solve triangles in which Case 1 or 2 holds. That is, the Law of Sines is used to solve SAA, ASA or SSA triangles.

Theorem Law of Sines

Not possible.

Two triangles!!

Triangle 1:

Triangle 2:

No triangle with the given measurements!

The Law of COSINES

The Law of COSINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

Use Law of COSINES when... SAS - 2 sides and their included angle SAS - 2 sides and their included angle SSS SSS you have 3 dimensions of a triangle and you need to find the other 3 dimensions. They cannot be just ANY 3 dimensions though, or you won’t have enough information to solve the Law of Cosines equations. Use the Law of Cosines if you are given:

Example 1: Given SAS Find all the missing dimensions of triangle ABC, given that angle B = 98°, side a = 13 and side c = 20. B 98° C a = 13 A c = 20 b Use the Law of Cosines equation that uses a, c and B to find side b:

Example 1: Given SAS Now that we know B and b, we can use the Law of Sines to find one of the missing angles: B 98° C a = 13 A c = 20 b = 25.3 Solution: b = 25.3, C = 51.5°, A = 30.5°

Example 2: Given SAS Find all the missing dimensions of triangle, ABC, given that angle A = 39°, side b = 20 and side c = 15. Use the Law of Cosines equation that uses b, c and A to find side a: 39° A b = 20 c = 15 B C a

Example 2: Given SAS Use the Law of Sines to find one of the missing angles: 39° A b = 20 c = 15 B C a = 12.6 Important: Notice that we used the Law of Sine equation to find angle C rather than angle B. The Law of Sine equation will never produce an obtuse angle. If we had used the Law of Sine equation to find angle B we would have gotten 87.5°, which is not correct, it is the reference angle for the correct answer, 92.5°. If an angle might be obtuse, never use the Law of Sine equation to find it.

Example 3: Given SSS Find all the missing dimensions of triangle, ABC, given that side a = 30, side b = 20 and side c = 15. A C B a = 30 c = 15 b = 20 We can use any of the Law of Cosine equations, filling in a, b & c and solving for one angle. Once we have an angle, we can either use another Law of Cosine equation to find another angle, or use the Law of Sines to find another angle.

Example 3: Given SSS Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it. For this reason, we will use the Law of Cosines to find the largest angle first (in case it happens to be obtuse). A C B a = 30 c = 15 b = 20 Angle A is largest because side a is largest:

Example 3: Given SSS Use Law of Sines to find angle B or C (its safe because they cannot be obtuse): A C B a = 30 c = 15 b = ° Solution: A = 117.3° B = 36.3° C = 26.4°

The Law of Cosines SAS SAS SSS SSS When given one of these dimension combinations, use the Law of Cosines to find one missing dimension and then use Law of Sines to find the rest. Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it.

Additional Resources Web Links: Web Links: