Lecture1 - General Physics (PHY 2130) Introduction...

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Unformatted text preview: General Physics (PHY 2130) Introduction Introduction • Syllabus and teaching strategy • Physics • Introduction • Mathematical review • trigonometry • vectors • Motion in one dimension http://www.physics.wayne.edu/~apetrov/PHY2130/ Chapter 1 Syllabus and teaching strategy Lecturer: Prof. Alexey A. Petrov, Room 260 Physics Building, Phone: 313­577­2739, or 313­577­2720 (for messages) e­mail: apetrov@physics.wayne.edu, Web: http://www.physics.wayne.edu/~apetrov/ Office Hours: Monday 5:00­6:00 PM, at Oakland center Tuesday 2:00­3:00 PM, on main campus, Physics Building, Room 260, or by appointment. Grading: Reading Quizzes Quiz section performance/Homework Best Hour Exam Second Best Hour Exam Final 15% 20% 15% 20% 30% PLUS: 5% online homework Reading Quizzes: Homework: Exams: It is important for you to come to class prepared! BONUS POINTS: Reading Summaries The quiz sessions meet once a week; quizzes will count towards your grade. BONUS POINTS: online homework http://webassign.net There will be THREE (3) Hour Exams (only two will count) and one Final Exam. Additional BONUS POINTS will be given out for class activity. I. Physics: Introduction I. Physics: Introduction ► Fundamental Science ► Divided into five major areas foundation of other physical sciences Mechanics Thermodynamics Electromagnetism Relativity Quantum Mechanics 1. Measurements 1. Measurements ► Basis of testing theories in science ► Need to have consistent systems of units for the measurements ► Uncertainties are inherent ► Need rules for dealing with the uncertainties Systems of Measurement Systems of Measurement ► Standardized systems ► SI ­­ Systéme International agreed upon by some authority, usually a governmental body agreed to in 1960 by an international committee main system used in this course also called mks for the first letters in the units of the fundamental quantities Systems of Measurements Systems of Measurements ► cgs ­­ Gaussian system ► US Customary named for the first letters of the units it uses for fundamental quantities everyday units (ft, etc.) often uses weight, in pounds, instead of mass as a fundamental quantity Basic Quantities and Their Dimension Basic Quantities and Their Dimension ► Length [L] ► Mass [M] ► Time [T] Why do we need standards? Length Length ► Units ► Defined in terms of a meter ­­ the distance SI ­­ meter, m cgs ­­ centimeter, cm US Customary ­­ foot, ft traveled by light in a vacuum during a given time (1/299 792 458 s) Mass Mass ► Units ► Defined in terms of kilogram, based on a SI ­­ kilogram, kg cgs ­­ gram, g USC ­­ slug, slug specific Pt­Ir cylinder kept at the International Bureau of Standards Standard Kilogram Standard Kilogram Why is it hidden under two glass domes? Time Time ► Units ► Defined in terms of the oscillation of seconds, s in all three systems radiation from a cesium atom (9 192 631 700 times frequency of light emitted) Time Measurements Time Measurements US “Official” Atomic Clock US “Official” Atomic Clock 2. Dimensional Analysis 2. Dimensional Analysis ► Dimension denotes the physical nature of a quantity ► Technique to check the correctness of an equation ► Dimensions (length, mass, time, combinations) can be treated as algebraic quantities ► Both sides of equation must have the same add, subtract, multiply, divide quantities added/subtracted only if have same units dimensions Dimensional Analysis Dimensional Analysis ► Dimensions for commonly used quantities Length Area Volume Velocity (speed) Acceleration L L2 L3 L/T L/T2 m (SI) m2 (SI) m3 (SI) m/s (SI) m/s2 (SI) Example of dimensional analysis distance = velocity · time L = (L/T) · T 3. Conversions 3. Conversions ► When units are not consistent, you may need to convert to appropriate ones ► Units can be treated like algebraic quantities that can cancel each other out 1 mile = 1609 m = 1.609 km 1m = 39.37 in = 3.281 ft 1 ft = 0.3048 m = 30.48 cm 1 in = 0.0254 m = 2.54 cm Example 1. Scotch tape: Example 1 Example 2. Trip to Canada: Legal freeway speed limit in Canada is 100 km/h. What is it in miles/h? 100 km km 1 mile miles = 100 ⋅ ≈ 62 h h 1.609 km h Prefixes Prefixes ► Prefixes correspond to powers of 10 ► Each prefix has a specific name/abbreviation Power 10 Prefix Abbrev. peta P Distance from Earth to nearest star Mean radius of Earth Length of a housefly Size of living cells Size of an atom 40 Pm 6 Mm 5 mm 10 µ m 0.1 nm Example: An aspirin tablet contains 325 mg of acetylsalicylic acid. Express this mass in grams. Given: m = 325 mg Find: m (grams)=? Solution: Recall that prefix “milli” implies 10­3, so m = 325 mg = 325 ×10−3 g = 0.325 g 4. Uncertainty in Measurements 4. Uncertainty in Measurements ► There is uncertainty in every measurement, this uncertainty carries over through the calculations ► We will use rules for significant figures to need a technique to account for this uncertainty approximate the uncertainty in results of calculations Significant Figures Significant Figures ► ► ► A significant figure is one that is reliably known All non­zero digits are significant Zeros are significant when between other non­zero digits after the decimal point and another significant figure can be clarified by using scientific notation 17400 = 1.74 × 10 4 17400. = 1.7400 × 10 4 17400.0 = 1.74000 × 10 4 3 significant figures 5 significant figures 6 significant figures Operations with Significant Figures Operations with Significant Figures ► Accuracy ­­ number of significant figures Example: meter stick: ± 0.1 cm ► When multiplying or dividing, round the result to the same accuracy as the least accurate measurement 2 significant figures Example: rectangular plate: area: 32.85 cm2 4.5 cm by 7.3 cm 33 cm2 ► When adding or subtracting, round the result to the smallest number of decimal places of any term in the sum Example: 135 m + 6.213 m = 141 m Order of Magnitude Order of Magnitude ► Approximation based on a number of assumptions may need to modify assumptions if more precise results are needed Question: McDonald’s sells about 250 million packages of fries every year. Placed back-to-back, how far would the fries reach? Solution: There are approximately 30 fries/package, thus: (30 fries/package)(250 10 packages)(3 in./fry) ~ 2 10 in ~ 5 10 m, ► Order of magnitude is the power of 10 that applies Example: John has 3 apples, Jane has 5 apples. Their numbers of apples are “of the same order of magnitude” II. Math Review: Coordinate Systems II. Math Review: ► Used to describe the position of a point in space ► Coordinate system (frame) consists of a fixed reference point called the origin specific axes with scales and labels instructions on how to label a point relative to the origin and the axes Types of Coordinate Systems Types of Coordinate Systems ► Cartesian ► Plane polar Cartesian coordinate system Cartesian coordinate system ► also called rectangular coordinate system ► x­ and y­ axes ► points are labeled (x,y) Plane polar coordinate system Plane polar coordinate system origin and reference line are noted point is distance r from the origin in the direction of angle θ , ccw from reference line points are labeled (r,θ ) II. Math Review: Trigonometry II. Math Review: opposite side sin θ = hypotenuse adjacent side cos θ = sin hypotenuse opposite side tan θ = adjacent side Pythagorean Theorem 2 c = a +b 2 2 Example: how high is the building? Example: how high is the building? Known: angle and one side Find: another side Key: tangent is defined via two sides! Fig. 1.7, p.14 Slide 13 α height of building , dist. height = dist. × tan α = (tan 39.0 )(46.0 m) = 37.3 m tan α = II. Math Review: Scalar and Vector II. Math Review: ► Scalar quantities are completely described by Quantities magnitude only (temperature, length,…) ► Vector quantities need both magnitude (size) and direction to completely describe them (force, displacement, velocity,…) Represented by an arrow, the length of the arrow is proportional to the magnitude of the vector Head of the arrow represents the direction Vector Notation Vector Notation ► When handwritten, use an arrow: A ► When printed, will be in bold print: A ► When dealing with just the magnitude of a vector in print, an italic letter will be used: A Properties of Vectors Properties of Vectors ► Equality of Two Vectors ► Movement of vectors in a diagram Two vectors are equal if they have the same magnitude and the same direction Any vector can be moved parallel to itself without being affected More Properties of Vectors More Properties of Vectors ► Negative Vectors Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) ► A = ­B ► Resultant Vector The resultant vector is the sum of a given set of vectors Adding Vectors Adding Vectors ► When adding vectors, their directions must be taken into account ► Units must be the same ► Graphical Methods ► Algebraic Methods Use scale drawings More convenient Adding Vectors Graphically Adding Vectors Graphically (Triangle or Polygon Method) ► Choose a scale ► Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate system ► Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector A and parallel to the coordinate system used for A Graphically Adding Vectors Graphically Adding Vectors ► ► ► Continue drawing the vectors “tip­to­tail” The resultant is drawn from the origin of A to the end of the last vector Measure the length of R and its angle Use the scale factor to convert length to actual magnitude Graphically Adding Vectors Graphically Adding Vectors ► When you have many vectors, just keep repeating the process until all are included ► The resultant is still drawn from the origin of the first vector to the end of the last vector Alternative Graphical Method Alternative Graphical Method ► ► When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin The remaining sides of the parallelogram are sketched to determine the diagonal, R Notes about Vector Addition Notes about Vector Addition ► Vectors obey the Commutative Law of Addition The order in which the vectors are added doesn’t affect the result Vector Subtraction Vector Subtraction ► Special case of vector addition ► If A – B, then use A+(­ B) ► Continue with standard vector addition procedure Multiplying or Dividing a Vector 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 Components of a Vector Components of a Vector ► A component is a part ► It is useful to use rectangular components These are the projections of the vector along the x­ and y­axes Components of a Vector Components of a Vector ► The x­component of a vector is the projection along the x­axis ► The y­component of a vector is the Ax = A cos θ Ay = A sin θ projection along the y­axis ► Then, u u u A = Ax + A y ► The previous equations are valid only if θ is More About Components of a More About Components of a Vector measured with respect to the x­axis ► The components can be positive or negative and will have the same units as the original vector ► The components are the legs of the right triangle whose hypotenuse is A Ax May still have to find θ with respect to the positive x­axis A= A +A 2 x 2 y and θ = tan −1 Ay Adding Vectors Algebraically Adding Vectors Algebraically ► Choose a coordinate system and sketch the vectors ► Find the x­ and y­components of all the vectors ► Add all the x­components This gives Rx: Rx = ∑ v x Adding Vectors Algebraically Adding Vectors Algebraically ► Add all the y­components ► Use the Pythagorean Theorem to find the This gives Ry: R y = ∑ v y magnitude of the Resultant: R = R 2 + R 2 x y ► Use the inverse tangent function to find the direction of R: Ry −1 θ = tan Rx III. Problem Solving Strategy III. Problem Solving Strategy Fig. 1.7, p.14 Slide 13 Known: Find: Key: tan α = angle and one side another side tangent is defined via two sides! height of building , dist. height = dist. × tan α = (tan 39.0 )(46.0 m) = 37.3 m Problem Solving Strategy Problem Solving Strategy ► Read the problem ► Draw a diagram identify type of problem, principle involved include appropriate values and coordinate system some types of problems require very specific types of diagrams Problem Solving cont. Problem Solving cont. ► Visualize the problem ► Identify information identify the principle involved list the data (given information) indicate the unknown (what you are looking for) Problem Solving, cont. Problem Solving, cont. ► Choose equation(s) ► Solve the equation(s) based on the principle, choose an equation or set of equations to apply to the problem solve for the unknown substitute the data into the equation include units Problem Solving, final Problem Solving, final ► Evaluate the answer ► Check the answer find the numerical result determine the units of the result are the units correct for the quantity being found? does the answer seem reasonable? are signs appropriate and meaningful? ► check order of magnitude IV. Motion in One Dimension Dynamics Dynamics ► The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts ► Kinematics is a part of dynamics In kinematics, you are interested in the description of motion Not concerned with the cause of the motion Position and Displacement Position and Displacement ► Position is defined in terms A of a frame of reference Frame A: x >0 and x >0 B y’ i O’ f x’ Position and Displacement Position and Displacement ► Position is defined in terms of a frame of reference One dimensional, so generally the x­ or y­axis ► Displacement measures the change in position Represented as ∆ x (if horizontal) or ∆ y (if vertical) Vector quantity indicate direction for one­ dimensional motion Units SI CGS US Cust Meters (m) Centimeters (cm) Feet (ft) ► + or ­ is generally sufficient to Displacement Displacement (example) Displacement measures represented as ∆ x or ∆ y the change in position ∆x1 = x f − xi = 80 m − 10 m = + 70 m ∆x2 = x f − xi = 20 m − 80 m = − 60 m Distance or Displacement? Distance or Displacement? ► Distance may be, but is not necessarily, the magnitude of the displacement Displacement (yellow line) Distance (blue line) Position­time graphs Position­time graphs Note: position-time graph is not necessarily a straight line, even though the motion is along x-direction ConcepTest 1 ConcepTest 1 An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. 2. 3. 4. 5. either greater than or equal to always greater than always equal to either smaller or equal to either smaller or larger than the distance it traveled. Please fill your answer as question 1 of General Purpose Answer Sheet ConcepTest 1 ConcepTest 1 An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. 2. 3. 4. 5. either greater than or equal to always greater than always equal to either smaller or equal to either smaller or larger than the distance it traveled. Please fill your answer as question 2 of General Purpose Answer Sheet ConcepTest 1 (answer) ConcepTest 1 (answer) An object (say, car) goes from one point in space to another. After it arrives to its destination, its displacement is 1. 2. 3. 4. 5. either greater than or equal to always greater than always equal to either smaller or equal to either smaller or larger than the distance it traveled. Note: displacement is a vector from the final to initial points, distance is total path traversed Average Velocity Average Velocity ► It takes time for an object to undergo a displacement ► The average velocity is rate at which the displacement occurs vaverage ∆x x f − xi = = ∆t ∆t ► It is a vector, direction will be the same as the + or ­ is sufficient for one­dimensional motion direction of the displacement (∆ t is always positive) More About Average Velocity More About Average Velocity ► Units of velocity: Units SI CGS US Customary Meters per second (m/s) Centimeters per second (cm/s) Feet per second (ft/s) ► Note: other units may be given in a problem, but generally will need to be converted to these Example: Example: Suppose that in both cases truck covers the distance in 10 seconds: ∆x1 + 70m v1 average = = ∆t 10s = +7m s ∆x2 − 60m v2 average = = ∆t 10 s = −6m s Speed Speed ► Speed is a scalar quantity ► May be, but is not necessarily, the same units as velocity speed = total distance / total time magnitude of the velocity Graphical Interpretation of Average Velocity Graphical Interpretation of Average Velocity ► Velocity can be determined from a position­ time graph vaverage ∆x + 40m = = ∆t 3 .0 s = + 13 m s ► Average velocity equals the slope of the line joining the initial and final positions Instantaneous Velocity Instantaneous Velocity ► Instantaneous velocity is defined as the limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero vinst x f − xi ∆x = lim = lim ∆t →0 ∆t ∆t →0 ∆t ► The instantaneous velocity indicates what is happening at every point of time Uniform Velocity Uniform Velocity ► Uniform velocity is constant velocity ► The instantaneous velocities are always the same All the instantaneous velocities will also equal the average velocity Graphical Interpretation of Instantaneous Graphical Interpretation of Instantaneous Velocity ► Instantaneous velocity is the slope of the tangent to the curve at the time of interest ► The instantaneous speed is the magnitude of the instantaneous velocity Average vs Instantaneous Velocity Average vs Instantaneous Velocity Average velocity Instantaneous velocity ConcepTest 2 ConcepTest 2 The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. 2. 3. 4. 5. at time tB both trains have the same velocity both trains speed up all the time both trains have the same velocity at some time before tB train A is longer than train B all of the above statements are true A B position tB time Please fill your answer as question 3 of General Purpose Answer Sheet ConcepTest 2 ConcepTest 2 The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. 2. 3. 4. 5. at time tB both trains have the same velocity both trains speed up all the time both trains have the same velocity at some time before tB train A is longer than train B all of the above statements are true A B position tB time Please fill your answer as question 4 of General Purpose Answer Sheet ConcepTest 2 (answer) ConcepTest 2 (answer) The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true: 1. 2. 3. 4. 5. at time tB both trains have the same velocity both trains speed up all the time both trains have the same velocity at some time before tB train A is longer than train B all of the above statements are true A B position Note: the slope of curve B is parallel to line A at some point t< tB tB time Average Acceleration Average Acceleration ► Changing velocity (non­uniform) means an acceleration is present ► Average acceleration is the rate of change of the velocity aaverage ∆v v f − vi = = ∆t ∆t ► Average acceleration is a vector quantity Average Acceleration Average Acceleration ► When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing ► When the sign of the velocity and the acceleration are opposite, the speed is decreasing Units SI CGS US Customary Meters per second squared (m/s2) Centimeters per second squared (cm/s2) Feet per second squared (ft/s2) Instantaneous and Uniform Instantaneous and Uniform Acceleration ► Instantaneous acceleration is the limit of the average acceleration as the time interval goes to zero ainst v f − vi ∆v = lim = lim ∆t →0 ∆t ∆t →0 ∆t ► When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration Graphical Interpretation of Graphical Interpretation of Acceleration ► Average acceleration is the slope of the line connecting the initial and final velocities on a velocity­time graph ► Instantaneous acceleration is the slope of the tangent to the curve of the velocity­time graph Example 1: Motion Diagrams Example 1: Motion Diagrams ► Uniform velocity (shown by red arrows maintaining the same size) ► Acceleration equals zero Example 2: Example 2: ► ► ► Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Example 3: Example 3: ► ► ► Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) ...
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