Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW ALGEBRA Simplifying Expressions 4𝑥 + 6 − 3(𝑥 + 2) Simpli>es to ● Write long algebraic expressions in a simpler way by reducing the # of terms. EXAMPLE: Simplify the algebraic expression. 2𝑥 + 3 + 4(𝑥 + 2) SIMPLIFYING ALG. EXPRESSIONS 1) Distribute constants/variables into parentheses (if any) 2) Group like terms by writing them next to each other 3) Combine like terms by adding/subtracting Exponents in Expressions ● Exponents represent repeated multiplication. General Form of Exponents 𝑎 ⋅ 𝑎 ⋅ … ⋅ 𝑎 ⋅ 𝑎 = 𝑎! 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 = 4" “4 to the 5th power” 5 times ___ 𝑛 times ▪ Base: Number and/or variable being multiplied ▪ Exponent or Power: How many times the base is multiplied EXPONENT RULES Name Example Rule Description Multiply same bases ⇒ ADD exponents Product Rule 4! × 4" = 4 =4 𝑎# × 𝑎$ = 𝑎#%$ Quotient Rule %! %" =4 &# = 𝑎#&$ &$ Zero Exp. Rule 4! = 4" = 1 4! 4# 1 %! = 4 = 4$ 4! Neg. Exp. Rule Power Rule =4 (4( )) = 4 =4 Divide same bases ⇒ SUBTRACT exponents ANYTHING to zero exp. = 1 𝑎" = 1 ' ' OR '$ = 𝑎& $ & & Neg exp in (𝑎# )$ = 𝑎#⋅$ Power to another power → MULTIPLY exponents 𝑎%& = top → flip to BOTTOM with pos exp Neg exp in bottom → flip to TOP with pos exp Power of a Product (3 ⋅ 4)( (𝒂 ⋅ 𝒃)# = 𝒂# ⋅ 𝒃# Distribute exponent to each term in parentheses Power of a Quotient 12 ( ( + 4 𝒂 𝒂# " # = # 𝒃 𝒃 Distribute exponent to numerator & denominator Page 1 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW ALGEBRA Solving Equations ● Use different operations (+, −, ×, ÷) to __________ 𝒙. ▪ ALWAYS do operations to _______ sides of the equation. EXAMPLE: Solve the equation. 2(𝑥 − 3) = 0 * + Simplifying 𝑥 + 5 = −3 Algebraic Expressions SOLVING LINEAR EQUATIONS 1) Distribute constants 2) Combine like terms 3) Group terms w/ 𝒙 & constants on opposite sides 4) Isolate / solve for 𝒙 5) Check solution by replacing 𝒙 in original equation Graphing ● Graphing in this course will usually involve plotting points/equations on the 2D/rectangular coordinate system. GRAPHING EQ’NS BY PLOTTING POINTS 1) Isolate 𝒚 to left side: 𝒚 = … 2) Calculate 𝑦-values from 3-5 chosen 𝒙-values 3) Plot (𝑥, 𝑦) points from Step 2 4) Connect points with line/curve EXAMPLE: Graph 𝒚 = 𝒙𝟐 − 𝟑𝒙 + 𝟐 by plotting points. 𝒚 5 4 𝒙 𝒚 = 𝒙𝟐 − 𝟑𝒙 + 𝟐 𝒚 0 (0)( − 3(0) + 2 2 ( , ) 1 (1)( − 3(1) + 2 0 ( , ) 2 (2)( − 3(2) + 2 0 ( , ) 3 (3)( − 3(3) + 2 2 ( , ) -5 -4 -3 -2 -1 -1 4 (4)( − 3(4) + 2 6 ( , ) -2 Ordered Pair 3 2 1 1 2 3 4 5 𝒙 -3 -4 -5 Page 2 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW ALGEBRA Systems of Equations – Solving ● When you have multiple equations, Tnd solution(s) that work for both by substituting one into the other. EXAMPLE: Find (𝑥, 𝑦) solutions that satisfy both equations. (A) 𝑦 = 3𝑥 − 6 2𝑥 + 𝑦 = 4 (B) SOLVING SYSTEMS OF EQ’NS BY SUBSTITUTING 1) Solve one EQ’n (A) for 𝒚 (or var easiest to solve for). 2) Plug the right side of EQ’n (A) in for 𝒚 in EQ’n (B). 3) Solve the resulting EQ’n from 2) for 𝒙 (or other var.) This is the ____ – value 4) Plug in 𝒙 – value from 3) into EQ’n (A) & solve for 𝒚. This is the ____ – value Page 3 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW ALGEBRA Slopes of Lines ● Slope: A number representing how _________ a line is; how much 𝒚 changes divided by how much 𝒙 changes. 𝒚 𝒎= 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 = 𝒚𝟐 7𝒚𝟏 𝒙𝟐 7𝒙𝟏 = 𝚫𝒚 𝚫𝒙 Δ = “change in” 5 4 (𝒙𝟐 , 𝒚𝟐 ) & (𝒙𝟏 , 𝒚𝟏 ) are 2 points Δ𝒚 = ___ 3 EXAMPLE: Find the slopes of Line A and Line B in the graph. Line A Line B 𝚫𝒚 = _______ 𝚫𝒚 = _______ 𝚫𝒙 = _______ 𝚫𝒙 = _______ 𝒎= = Δ𝒙 = ___ 𝒎= = 2 1 (𝑥( , 𝑦( ) (2,4) (𝑥* , 𝑦* ) (1,2) 𝒙 -1 1 -1 2 Line A 3 4 5 Line B Graphing Linear Equations ● A line equation in Slope-Intercept form tells you everything you need to graph it! 𝒚 = 𝒎𝒙 + 𝒃 (Slope – Intercept Form) 𝒚 EXAMPLE: For the equation 𝑦 = 2𝑥 − 1 (𝑩) Identify the 𝒚 – intercept & slope 5 𝑏 = _____ 4 𝑚 = _____ (𝑪) Graph the equation. Graphing Lines From Equations 1) Plot 𝒚 – intercept (0, 𝑏) 2) Plot next point using slope 3) Connect points with a line 3 2 1 -5 -4 -3 -2 -1 -1 1 2 3 4 5 𝒙 -2 -3 -4 -5 Page 4 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW ALGEBRA Quadratic Equations – Solving ● Here are the two most common methods to solve quadratic equations in Physics: SOLVING QUADRATIC EQUATIONS 𝑎𝑥 9 + 𝑏𝑥 + 𝑐 = 0 (Standard Form) STEPS USE IF SQ. ROOT PROPERTY QUADRATIC FORMULA ● (𝑥 + #)! = [𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭] OR ● Can’t easily factor ● No middle term (𝑏 = 0) ● Unsure what method to use 1) Isolate squared expression 1) Write eq’n in standard form 2) Take + & – square root 2) Plug 𝑎, 𝑏, 𝑐 in quad. form 3) Solve for 𝒙 𝑥= 4) (Optional) Check solutions −𝑏 ± √𝑏( − 4𝑎𝑐 2𝑎 3) Compute & simplify solutions 𝑥 ( + 2𝑥 − 3 = 0 EXAMPLE (𝑥 + 1)( = 4 Quadratic Equations – Graphing ● In physics, you’ll identify key info from the graphs of quadratic equations. 𝑦 = 𝑎(𝑥 − ℎ)9 + 𝑘 𝒚 FROM GRAPH TO GRAPH (Vertex Form) 𝒚 = −(𝒙 − 𝟏)𝟐 + 𝟒 5 1) Vertex (ℎ, 𝑘): _______ [ MIN | MAX ] 4 2) x-int(s)→ Solve 𝑦 = 0: __________ 3) y-int → Set 𝑥 = 0: _________ 3 4) Connect with smooth curve 2 Increasing when 𝑥 ______ 1 Decreasing when 𝑥 _______ -1 -1 1 2 3 4 5 𝒙 Page 5 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW ALGEBRA Proportional Reasoning ● Many questions in physics will ask how a variable changes when another variable in the equation changes. ▪ Proportional Reasoning: Analyzing how one quantity increases or decreases with another DIRECTLY PROPORTIONAL INVERSELY PROPORTIONAL 𝑦 = 2𝑥 𝑦= 𝑥 𝑦 𝑥 JOINTLY PROPORTIONAL ! 𝐹 =𝑚⋅𝑎 " 𝑦 2 1 0 −1 −2 5 4 3 2 1 As 𝑥 ↑, 𝑦 ___ As 𝑥 ↓, 𝑦 ___ As 𝑥 ↑, 𝑦___ As 𝑥 ↓, 𝑦 ___ 𝑚 5 4 3 2 1 𝑎 𝐹 2 1 0 −1 −2 As 𝑚 ↑ & 𝑎 ↑, 𝐹 ___ As 𝑚 ↓ & 𝑎 ↓, 𝐹 ___ For constant 𝐹, 𝑚 ↑ & 𝑎 ↓ For constant 𝐹, 𝑚 ↓ & 𝑎 ↑ 𝑚 1 2 4 𝑎 𝐹 20 10 5 20 20 20 Page 6 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW TRIGONOMETRY ● The functions Sine, Cosine, and Tangent relate angles & sides of a right triangle. 5 5 5 𝑢𝑠𝑒 𝑡 𝑒𝑛 𝑜 𝑝 h𝑦 a𝑑𝑗𝑎𝑐𝑒𝑛𝑡 3 𝜃 𝑢𝑠𝑒 𝑡 𝑒𝑛 𝑜 𝑝 h𝑦 a𝑑𝑗𝑎𝑐𝑒𝑛𝑡 3 𝜃 𝑢𝑠𝑒 𝑡 𝑒𝑛 𝑜 𝑝 h𝑦 a𝑑𝑗𝑎𝑐𝑒𝑛𝑡 4 4 4 sin (𝜃) = cos (𝜃) = tan (𝜃) = sin (𝜃) = EFF GHF 𝑜𝑝𝑝 = ℎ𝑦𝑝 ⋅ sin (𝜃) cos (𝜃) = NOP GHF tan (𝜃) = o𝑝𝑝𝑜𝑠𝑖𝑡𝑒 TANGENT (T.O.A.) o𝑝𝑝𝑜𝑠𝑖𝑡𝑒 COSINE (C.A.H.) o𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝜃 SINE (S.O.H.) 3 EFF NOP 𝑎𝑑𝑗 = ℎ𝑦𝑝 ⋅ cos (𝜃) ● Other helpful formulas: 𝑎( + 𝑏( = 𝑐 ( sin( (𝜃) + cos ( (𝜃) = 1 tan (𝜃) = VWX (Y) Z[V(Y) (Pythagorean Theorem) Page 7 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW CALCULUS Derivatives 𝒚 %𝒚𝟏 𝑚= 𝟐 𝒙𝟐 %𝒙𝟏 ● Derivative of a function = Instantaneous rate of change (Slope) ▪ Graphically represents the slope of a tangent line at a certain point (tangent lines touch the graph only once). 𝒚 5 4 3 2 1 -1 -1 1 2 3 4 5 𝒙 Common Derivatives ● To determine the exact derivative from a given equation or function, use the following rules: FUNCTION 𝒇(𝒙) DERIVATIVE 𝒇′(𝒙) EXAMPLE 𝑓(𝑥) = 3 𝑓(𝑥) = 𝑐 (constant) 𝑓’(𝑥) = 0 𝑓’(𝑥) = 𝑓(𝑥) = 𝑐 ⋅ 𝑥 𝑓(𝑥) = 𝑥 & 𝑓(𝑥) = 𝑔(𝑥) + ℎ(𝑥) 𝑓’(𝑥) = 𝑐 𝑓′(𝑥) = 𝑛𝑥 &%* 𝑓′(𝑥) = 𝑔/ (𝑥) + ℎ′(𝑥) 𝑓(𝑥) = −2𝑥: 𝑓’(𝑥) = 𝑓(𝑥) = 𝑥 ( 𝑓’(𝑥) = 𝑓(𝑥) = 𝑥 ( + 3𝑥 𝑓’(𝑥) = Page 8 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PHYSICS I: PRE – COURSE REVIEW CALCULUS Integrals ● Graphically, the integral of a function is the area under the curve. ▪ You can approximate integrals by adding the areas of many rectangles under the curve. 𝒚 5 4 3 2 1 -1 -1 1 2 3 4 5 𝒙 Rules for Integrals 2 ∫3 𝑓(𝑥) = 𝐹(𝑏) − 𝐹(𝑎) ● Mathematically, integrals are the reverse of derivatives. (De2nite Integral) ▪ To determine exact integrals equations/functions, use the following rules: FUNCTION 𝒇(𝒙) INTEGRAL 𝑭(𝒙) = ∫ 𝒇(𝒙) 𝑓(𝑥) = 𝑐 (constant) 𝐹(𝑥) = 𝑐𝑥 + 𝐶 𝑓(𝑥) = 𝑥 & 𝑓(𝑥) = 𝑐𝑥 & 𝐹(𝑥) = _ !"# `ab 𝐹(𝑥) = 𝑐 ⋅ ) ∫* 3 = EXAMPLE ) +𝐶 _ !"# `ab ∫* 𝑥 ( = ) +𝐶 ∫* −2𝑥 = ) 𝑓(𝑥) = 𝑔(𝑥) + ℎ(𝑥) 𝐹(𝑥) = 𝐺(𝑥) + 𝐻(𝑥) + 𝐶 ∫* 𝑥 ( + 2𝑥 = Page 9 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations CONCEPT: INTRODUCTION TO UNITS AND THE S.I. SYSTEM ● Physics = study of natural phenomena, which includes lots of measurements & equations! Physics = Math + Rules - In nature, we measure physical quantities (mass, length…), which must have ______________ & ___________ (Example: You measure the mass of a box) __________ ______________ [Number] [Unit] ● For physics equations to work, ALL units in it must be _______________ with each other. - Groups of compatible units that “work together” form a ____________ of units. - In Physics, always use S.I. units ( Système International ) Quantity S.I. Imperial Force = Mass × Acceleration MASS Kilogram [ ] Pound [ lb ] LENGTH Meter [ ] Foot [ ft ] TIME Second [ ] Second [ s ] [ ]=[ ]× [ ] → [ COMPATIBLE | INCOMPATIBLE ] FORCE Newton [ ] Foot-pound [ ]=[ ]× [ ] → [ COMPATIBLE | INCOMPATIBLE ] 𝑭 = 𝒎 × 𝒂 Page 10 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations CONCEPT: METRIC PREFIXES [Base Unit] ● A metric prefix is a letter or symbol that goes before a base unit: [Prefixes] m, g, s → km, ,mg, 𝝁s, - Each letter / prefix stands for a specific power of 10 multiplied by the base unit. Example: 5 km = _______________ m = __________ m 4.6 ms = _______________ s = __________ s Bigger Units ● 𝟏𝟎 𝟏𝟐 ● 𝟏𝟎 ● 𝟗 𝟏𝟎 𝟏𝟎𝟎𝟎 𝟏𝟎𝟎 𝟑 𝟐 𝟏𝟎 𝟏 𝟎 𝟏𝟎 = 𝟏 𝟏𝟎 [Base Unit] deci- centi- milli- micro- nano- pico- (d) (c) (m) (𝜇) (n) (p) 𝟔 𝟏𝟎 𝟏𝟎 𝟏𝟎 𝟎. 𝟏 tera- giga- mega- kilo- hecto- deca- (T) (G) (M) (k) (h) (da) −𝟏 𝟎. 𝟎𝟏 𝟏𝟎 −𝟐 𝟎. 𝟎𝟎𝟏 𝟏𝟎 −𝟑 ● 𝟏𝟎 ● −𝟔 ● 𝟏𝟎 −𝟗 𝟏𝟎−𝟏𝟐 Smaller Units EXAMPLE: Express the following measurements using the desired prefix. STEPS a) 6.5 hm to m 1) Identify starting & target prefixes 2) Move from start → target, count # of exponents moved b) 3.89 mm to m 3) Shift decimal place in the same direction moved in Step 2 c) 7.62 kg to 𝜇g ● When re-writing numbers with metric prefixes, - Shifting from a bigger to smaller unit, number becomes [ LARGER | SMALLER ] - Shifting from a smaller to bigger unit, number becomes [ LARGER | SMALLER ] Page 11 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PRACTICE: The earth’s circumference is approximately 40.1 Mm (megameters). What is this circumference in kilometers? A) 0.0401 km B) 40,100,000 km C) 40,100 km D) 0.00401 km ● ● ● 𝟏𝟎𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎 𝟎. 𝟏 𝟎. 𝟎𝟏 𝟎. 𝟎𝟎𝟏 ● ● ● 𝟏𝟎𝟏𝟐 𝟏𝟎𝟗 𝟏𝟎𝟔 𝟏𝟎𝟑 𝟏𝟎𝟐 𝟏𝟎𝟏 𝟏𝟎𝟎 = 𝟏 𝟏𝟎−𝟏 𝟏𝟎−𝟐 𝟏𝟎−𝟑 𝟏𝟎−𝟔 𝟏𝟎−𝟗 𝟏𝟎−𝟏𝟐 tera- giga- mega- kilo- hecto- deca- [Base Unit] deci- centi- milli- micro- nano- pico- (T) (G) (M) (k) (h) (da) (d) (c) (m) (𝜇) (n) (p) PRACTICE: Astronomers often detect radio waves with wavelengths of 3,000,000,000 nm. What is this wavelength expressed in decameters (dam)? A) 3 dam B) 0.3 dam C) 30 dam D) 0.03 dam ● ● ● 𝟏𝟎𝟎𝟎 𝟏𝟎𝟎 𝟏𝟎 𝟎. 𝟏 𝟎. 𝟎𝟏 𝟎. 𝟎𝟎𝟏 ● ● ● 𝟏𝟎𝟏𝟐 𝟏𝟎𝟗 𝟏𝟎𝟔 𝟏𝟎𝟑 𝟏𝟎𝟐 𝟏𝟎𝟏 𝟏𝟎𝟎 = 𝟏 𝟏𝟎−𝟏 𝟏𝟎−𝟐 𝟏𝟎−𝟑 𝟏𝟎−𝟔 𝟏𝟎−𝟗 𝟏𝟎−𝟏𝟐 tera- giga- mega- kilo- hecto- deca- [Base Unit] deci- centi- milli- micro- nano- pico- (T) (G) (M) (k) (h) (da) (d) (c) (m) (𝜇) (n) (p) Page 12 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations CONCEPT: SCIENTIFIC NOTATION ● We use Scientific Notation to ________________ very LONG, inconvenient numbers into SHORTER ones. Mass of Earth = 5,972,000,000,000,000,000,000,000 kg General Format for Scientific Notation 𝐀. 𝐁𝐂 × 𝟏𝟎𝐃 = _________________ [ # ≥ ___ but < ___ ] STANDARD FORM → SCIENTIFIC NOTATION a) 304,605.27 kg b) 0.000102 m [ _________ ] Standard Form → Scientific Notation 1) Move decimal to get # ≥ 1 but < 10 2) Round long numbers with many non-zero numbers to 2 decimal places 3) # of decimal places moved = Exponent - If original number > 10, exponent is + - If original number < 1, exponent is – c) 7 s Standard Form SCIENTIFIC NOTATION → STANDARD FORM → Scientific Notation 8 kg a)1)5.45×10 Move decimal to get # ≥ 1 but < 10 Scientific Notation → Standard Form 1) Exponent = # of decimal places moved - If exponent is +, number becomes larger - If exponent is –, number becomes smaller 2) Round to 2 places, if needed -5 b)3)9.62×10 # of decimals places moved = Exponent - If original number > 10, exponent is + - If original number 2 kg < 1, exponent is – a) 5.45×10 a) 304,605.27 kg b) 9.62×10-5 s b) 0.000102 m c) 38,900 s Page 13 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PRACTICE: Rewrite 0.00016 kg in scientific notation. A) 1.6×10-4 kg B) 16×10-3 kg C) 1.6×104 kg D) 1.6×10-3 kg PRACTICE: Rewrite 299,800,000 m/s in scientific notation. A) 2.998×105 m/s B) 3.00×108 m/s C) 3.00×105 m/s D) 2.998×10-8 m/s EXAMPLE: Express 0.0000529 × 10−6 m in scientific notation. Page 14 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PRACTICE: Rewrite 3.41 × 10−4 in standard form: A) 0.000341 B) 34,100 C) 0.0000341 D) 3,410 PRACTICE: Rewrite 9.98 × 107 in standard form. A) 0.000000998 B) 0.0000000998 C) 9,980,000,000 D) 99,800,000 Page 15 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations CONCEPT: COUNTING SIGNIFICANT FIGURES (SIG FIGS) ● In Physics, measurements have PRECISION, indicated by the # of digits: 10 kg 10.27 kg [ LESS | MORE ] precision [ LESS | MORE ] precision ● Not all digits in measurements matter. Significant Figures are the # of digits that ____________________. 15 kg 015 kg # digits given: _____ # digits given: _____ # digits that matter: _____ # digits that matter: _____ EXAMPLE: Determine the number of significant figures in the number below: 0.013200972000 0 _________ 0s _________ 0s _________ 0s STEPS 1) Eliminate Leading 0’s 2) If # has no decimal, eliminate Trailing 0’s 3) Count remaining digits - Never eliminate non-zeroes or Middle 0’s # of Significant Figures: _____ EXAMPLE: How many significant figures are there in each of the following numbers? a) 100.00 b) 0.0043 c) 31000092 d) 100 e) 73917000 f) 0.00900 Page 16 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PRACTICE: How many significant figures are in each of the following numbers? a) 0.0032 b) 10790 c) 08.02 Page 17 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations CONCEPT: DIMENSIONAL ANALYSIS ● Equations work only if they are dimensionally consistent, meaning the units on both sides are ___________. - Easy way to check if equations make sense without calculations. EXAMPLE: You walk a constant speed 𝒗 = 5m/s for a time t of 2s. Which equation from below would be appropriate for determining the distance d in meters? DIMENSIONAL CONSISTENCY Distance = speed × time 𝒅 = [ ]=[ 𝒗 × ]× [ Distance = speed × time2 OR 𝒕 ] [ CONSISTENT | INCONSISTENT ] 𝒅 = [ ]=[ 𝒗 × ]× [ 𝟐𝒕 1) Replace variables with units 𝟐 2) Ignore – signs & numbers (2, ½, etc..) ] 3) Multiply & divide to cancel out units 4) Check if units on left = units on right [ CONSISTENT | INCONSISTENT ] DETERMINING UNITS OF UNKOWN VARIABLES ● You’ll also need Dimensional Analysis to figure out the units of unknown variables. EXAMPLE: Hooke’s Law states that a restoring Force F, measured in Newtons [N], in springs is related to the distance from equilibrium x by the equation F = ―kx. What are the units of the force constant k? SOLVING UNITS OF VARIABLES 1) Replace variables with units 2) Ignore – signs & numbers (2, ½, etc..) 3) Isolate unknown variable 4) Solve Page 18 Mazur - Principles & Practice of Physics, Global Edition Ch. 1 - Foundations PRACTICE: A box moving with an initial speed v is accelerated horizontally. If x is measured in [m], v in [m/s], a in [m/s2], t in [s] which of the following equations is correct for solving the distance x? 𝑎 A) x = 𝑡 2 B) x = v + ½ at C) x = vt + ½ at2 PRACTICE: Newton’s Law of Gravitation describes the attraction force between two masses. The equation is 𝑭=𝐆 𝒎 𝟏 𝒎𝟐 𝒓𝟐 , where F is in [ 𝐤𝐠⋅𝐦 𝐬𝟐 ], m1 and m2 are masses in [ kg ], and r is the distance in [ m ] between them. Determine the units of the Universal Constant G. A) B) C) D) 𝑘𝑔⋅𝑠2 𝑚3 𝑚3 𝑘𝑔⋅𝑠2 𝑚 𝑠2 𝑚3 𝑠2 Page 19
0
advertisement
Related documents
Download
advertisement
Add this document to collection(s)
You can add this document to your study collection(s)
Sign in Available only to authorized usersAdd this document to saved
You can add this document to your saved list
Sign in Available only to authorized users