chapt01.pdf

Raymond A. Serway Chris Vuille Chapter 1 Introduction  Contents 1. Standards of length, mass and time 2. The building blocks of matter 3. Dimensional Analysis 4. Uncertainty in measurements and significant figures 5. Conversion of units 6. Estimate and order of magnitude calcu...

Raymond A. Serway Chris Vuille Chapter 1 Introduction  Contents 1. Standards of length, mass and time 2. The building blocks of matter 3. Dimensional Analysis 4. Uncertainty in measurements and significant figures 5. Conversion of units 6. Estimate and order of magnitude calculations 7. Coordinate systems 8. Trigonometry 9. Vectors 10. Component of Vector contents  Theories and Experiments The goal of physics is to develop theories based on experiments A physical theory, usually expressed mathematically, describes how a given system works The theory makes predictions about how a system should work Experiments check the theories’ predictions Every theory is a work in progress Introduction  Fundamental Quantities and Their Dimension Mechanics uses three fundamental quantities – Length [L] – Mass [M] – Time [T] Other physical quantities can be constructed from these three Introduction  Units To communicate the result of a measurement for a quantity, a unit must be defined Defining units allows everyone to relate to the same fundamental amount Section 1.1  SI System of Measurement SI – Systéme International – Agreed to in 1960 by an international committee – Main system used in this text Section 1.1  Length Units – meter, m The meter is currently defined in terms of the distance traveled by light in a vacuum during a given time – Also establishes the value for the speed of light in a vacuum Section 1.1  Mass Units kilogram, kg In 1889, a cylinder of platinum-iridium, the International Prototype of the Kilogram (IPK), became the standard of the unit of mass for the metric system and remained so until the 2019 SI base units redefinition. As of the 2019 redefinition of the SI base units, the kilogram is defined in terms of the second and the metre, both based on fundamental physical constants. This allows a properly-equipped metrology laboratory to calibrate a mass measurement instrument such as a Kibble balance as the primary standard to determine an exact kilogram mass. Section 1.1  Time Units – seconds, s The second is currently defined in terms of the oscillation of radiation from a cesium atom Section 1.1  Approximate Values Various tables in the text show approximate values for length, mass, and time – Note the wide range of values – Lengths – Table 1.1 – Masses – Table 1.2 – Time intervals – Table 1.3 Section 1.1  Other Systems of Measurements cgs – Gaussian system – Named for the first letters of the units it uses for fundamental quantities US Customary – Everyday units – Often uses weight, in pounds, instead of mass as a fundamental quantity Section 1.1  Units in Various Systems System Length Mass Time SI meter kilogram second cgs centimeter gram second US foot slug second Customary Section 1.1  Prefixes Prefixes correspond to powers of 10 Each prefix has a specific name Each prefix has a specific abbreviation See table 1.4 Section 1.1 Prefixes Section 1.1  Expressing Numbers Numbers with more than three digits are written in groups of three digits separated by spaces – Groups appear on both sides of the decimal point 10 000 instead of 10,000 3.141 592 65 Section 1.1  Structure of Matter Matter is made up of molecules – The smallest division that is identifiable as a substance Molecules are made up of atoms – Correspond to elements Section 1.2  More structure of matter Atoms are made up of – Nucleus, very dense, contains Protons, positively charged, “heavy” Neutrons, no charge, about same mass as protons – Protons and neutrons are made up of quarks – Orbited by Electrons, negatively charges, “light” – Fundamental particle, no structure Section 1.2 Structure of Matter Section 1.2  Dimensional Analysis Technique to check the correctness of an equation Dimensions (length, mass, time, combinations) can be treated as algebraic quantities – Add, subtract, multiply, divide Both sides of equation must have the same dimensions Section 1.3  Dimensional Analysis Check if this equation is dimensionally correct: v= vot + at2 v and vo are velocities, t is time, and a is acceleration. Section 1.3  Dimensional Analysis, cont. Cannot give numerical factors: this is its limitation Dimensions of some common quantities are listed in Table 1.5 Allows a check for calculations which can show up in the units Section 1.3  Example Which one of these expressions does NOT have the dimension of lenght (L)? (Note a is acceleration, v is velocity, x is distance, and t is time) a. v t b. v2/a c. v2/ax d. a t2 e. x2/vt Section 1.3  Uncertainty in Measurements There is uncertainty in every measurement, this uncertainty carries over through the calculations – Need a technique to account for this uncertainty We will use rules for significant figures to approximate the uncertainty in results of calculations Section 1.4  Significant Figures A significant figure is a reliably known digit All non-zero digits are significant Zeros are not significant when they only locate the decimal point – Using scientific notion to indicate the number of significant figures removes ambiguity when the possibility of misinterpretation is present Section 1.4 Operations with Significant Figures When multiplying or dividing two or more quantities, the number of significant figures in the final result is the same as the number of significant figures in the least accurate of the factors being combined – Least accurate means having the lowest number of significant figures When adding or subtracting, round the result to the smallest number of decimal places of any term in the sum (or difference) Section 1.4 Operations with Significant Figures 1.015 x 3.1 = 123 + 5.55 = 1.002 – 0.998 = Section 1.4  Rounding Calculators will generally report many more digits than are significant – Be sure to properly round your results Slight discrepancies may be introduced by both the rounding process and the algebraic order in which the steps are carried out – Minor discrepancies are to be expected and are not a problem in the problem-solving process In experimental work, more rigorous methods would be needed Section 1.4  Example Which one of the choices below represents the preferred practice regarding significant figures when multiplying the following: 13.2 x 7.2 x 4.98 ? a. 473.299 b. 500 c. 473.30 d. 470 e. 473.3 Section 1.4  Conversions When units are not consistent, you may need to convert to appropriate ones See the inside of the front cover for an extensive list of conversion factors Units can be treated like algebraic quantities that can “cancel” each other Example: Section 1.5  Example A cement truck can pour 20 cubic yards of cement per hour (20 yard3/hr). Express this in ft3/min. (1 yard = 3 ft and 1 hr = 60 min) a. 0.33 ft3/min b. 1 ft3/min c. 3 ft3/min d. 60 ft3/min e. 9 ft3/min Section 1.5  Estimates Can yield useful approximate answers – An exact answer may be difficult or impossible Mathematical reasons Limited information available Can serve as a partial check for exact calculations Section 1.6  Order of Magnitude Approximation based on a number of assumptions – May need to modify assumptions if more precise results are needed Order of magnitude is the power of 10 that applies Section 1.6  Order of Magnitude Find the number of times the human heart beats before it shuts down? Section 1.6  Coordinate Systems Used to describe the position of a point in space Coordinate system consists of – A fixed reference point called the origin, O – Specified axes with scales and labels – Instructions on how to label a point relative to the origin and the axes Section 1.7  Types of Coordinate Systems Cartesian (rectangular) Plane polar Section 1.7  Cartesian coordinate system x- and y- axes Points are labeled (x,y) Positive x is usually selected to be to the right of the origin Positive y is usually selected to be to upward from the origin Section 1.7  Plane polar coordinate system Origin and reference line are noted Point is distance r from the origin in the direction of angle  Positive angles are measured ccw from reference line Points are labeled (r,) The standard reference line is usually selected to be the positive x axis Section 1.7 Trigonometry Review Section 1.8  More Trigonometry Pythagorean Theorem – r2 = x2 + y2 To find an angle, you need the inverse trig function For example, if sin  = 0.707, find   = sin-1 0.707 = 45° Section 1.8  Degrees vs. Radians Be sure your calculator is set for the appropriate angular units for the problem For example: – tan -1 0.5774 = 30.0° – tan -1 0.5774 = 0.5236 rad Section 1.8  Rectangular  Polar Rectangular to polar – Given x and y, use Pythagorean theorem to find r – Use x and y and the inverse tangent to find angle Polar to rectangular – x = r cos  – y = r sin  Section 1.8  Vector vs. Scalar Review All physical quantities encountered in this text will be either a scalar or a vector A vector quantity has both magnitude (size) and direction A scalar is completely specified by only a magnitude (size) Section 1.9  Vector Notation When handwritten, use an arrow: A When printed, will be in bold print with an arrow: 𝐀 When dealing with just the magnitude of a vector in print, an italic letter will be used: A – Italics will also be used to represent scalars Section 1.9  Properties of Vectors Equality of Two Vectors – Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram – Any vector can be moved parallel to itself without being affected Section 1.9  Adding Vectors When adding vectors, their directions must be taken into account Units must be the same Two Methods: Geometric Methods Algebraic Methods The resultant vector (sum) is denoted as Section 1.9  Adding Vectors Geometrically (Triangle or Polygon Method) Draw the first vector with the appropriate length and in the direction specified. Draw the next vector with its tail at the tip of the first vector tip tail The resultant (A + B) is drawn from the origin of the first vector to the end of the last vector 𝐁 This method is called the triangle method Section 1.9 𝐀 Graphically Adding Vectors, cont. 𝐀 𝐁 Go to PHET and practice adding vectors! Section 1.9  Notes about Vector Addition Vectors obey the Commutative Law of Addition The order in which the vectors are added doesn’t affect the result Section 1.9  Notes about Vector Addition 𝐀 𝐁 𝐁 𝐀 More Practice at PHET animation. Section 1.9  Graphically Adding Vectors, cont. When you have many vectors, just keep repeating the “tip-to-tail” process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector Section 1.9  More Properties of Vectors Negative Vectors The negative of the vector is defined as the vector that gives zero when added to the original vector Two vectors are negative of each other if they have the same magnitude but are 180° apart (opposite directions) 𝐀 −𝐀 Section 1.9  Vector Subtraction Special case of vector addition Add the negative of the subtracted vector Continue with standard vector addition procedure Section 1.9  Vector Subtraction 𝐁 𝐁 𝐀 Click here PHET for more animation −𝐁 Section 1.9 Multiplying or Dividing a Vector by a Scalar The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided by the scalar If the scalar is positive, the direction of the result is the same as of the original vector If the scalar is negative, the direction of the result is opposite that of the original vector 𝐀 0.5 𝐀 −2 𝐀 Section 1.9  Components of a Vector from magnitude and angle It is useful to use rectangular components to add vectors These are the projections of the vector along the x- and y-axes Section 1.10  Components of a Vector, cont. The x-component of a vector is the projection along the x-axis 𝐴𝑥 = 𝐴 cos 𝜃 The y-component of a vector is the projection along the y-axis 𝐴𝑦 = 𝐴 sin 𝜃 Then, 𝐀 = 𝐀𝑥 +𝐀𝑦 These equations are valid only if  is measured with respect to the x-axis The components can be positive or negative and will have the same units as the original vector Section 1.10  Components of a Vector Learn using PHET Go to PHET for visualizing and animating components Section 1.10  Magnitude and angle from components The components are the legs of the right triangle whose hypotenuse is May still have to find θ with respect to the positive x-axis The value will be correct only if the angle lies in the first or fourth quadrant In the second or third quadrant, add 180° Section 1.10  Other Coordinate Systems It may be convenient to use a coordinate system other than horizontal and vertical Choose axes that are perpendicular to each other Adjust the components accordingly Section 1.10  Adding Vectors Algebraically Find the x- and y-components of all the vectors Add all the x-components and all the y- components Section 1.10