INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine, cosine, and tangent, and te manipulation of scalars and vectors. Te content contained in all sections of capter 1 of te textbook is included on te P Pysics B exam. QUICK EFEENCE Important Terms base SI (Systeme Internationale) units te fundamental units of meters for lengt, kilograms for mass, and seconds for time used in te metric system cosine ratio of te lengt of te adjacent side to an angle in a rigt triangle to te lengt of te ypotenuse derived unit any unit, suc as m/s, in te metric system wic is not fundamental and is a combination of meters, kilograms, and/or seconds equilibrant a vector wic is equal and opposite to te resultant vector, and can cancel te effects of te individual vectors resultant te vector sum of two or more vectors scalar component te magnitude of a component of a vector scalar quantity a quantity, suc as mass, wic can be completely specified by its magnitude or size. sine ratio of te lengt of te opposite side to an angle in a rigt triangle to te lengt of te ypotenuse tangent ratio of te lengt of te opposite side to an angle in a rigt triangle to te lengt of te adjacent side 1
vector quantity a quantity, suc as displacement, wic is specified by its magnitude (size) and its direction (angle) vector component te projection of a vector onto te x or y axis vector addition adding vectors to eac oter eiter grapically (ead-to-tail) or using vector components to find te sum (resultant) of te vectors Equations and Symbols ectangle Triangle Circle Parallelepiped b 1 b r and C V lw r Cylinder o sin cos tan a o a o a V 4 Spere V 3 igt Triangle r l and S r 3 and S 4 rl r r θ a o
were = area b = base = eigt C = circumference V = volume l = lengt w = widt S = surface area r = radius θ = angle sin = sine cos = cosine tan = tangent DISCUSSION OF SELECTED SECTIONS Te ole of Units in Problem Solving ll pysical quantities ave units so tat we can communicate teir measurement. IN te metric system, te base units are called SI units. Te base SI units for te fundamental quantities of mass, lengt, and time are te kilogram, meter, and second, respectively. Only SI units are used on te P Pysics B exam. ny unit wic is a combination of tese fundamental units is called a derived unit. n example of a derived unit would be meters/second or kilometers/our, wic are bot units for speed. Sometimes we will need to convert from one unit to anoter. Example 1 Convert 80.0 km/ to m/s. km 1000m 1 m 80.0. 1km 3600s s Note tat we multiplied 80.0 km/ by two quotients wic were eac equal to one, so we didn t cange te value of te speed, we only expressed it in different units. Dimensional nalysis Often you will need to be able to determine te validity of equations by analyzing te dimensions of te quantities involved. 3
Example Verify tat te equation below is valid by using dimensional analysis. F t m v f v 0 were F is force measured in Newtons, t is time in seconds, m is mass in kg, and v is speed in m/s. kgm We will sow later tat a Newton =. Tus te units for eac side of te equation s can be written as kgm m m s kg s s s kgm s Te unit 7. kgm s kgm s Trigonometry is a unit of momentum or impulse, and will be discussed furter in capter Trigonometry is te study of triangles, and often rigt triangles. Te lengts of te sides of a rigt triangle can be used to define some useful relationsips, called te sine, cosine, and tangent, abbreviated sin, cos, and tan, respectively. Example 3 ball on te end of a string of lengt L = 0.50 cm is ung from a ook in te ceiling. Te ball is pulled back to an angle θ = 30º from te vertical. Wat is te eigt above te lowest point of te ball? L θ L If we draw a orizontal line from te raised ball to form a rigt triangle, te portion of te vertical line above te orizontal line forms te adjacent side of te rigt triangle. Tis lengt would ten be Lcosθ. 4
L θ L Lcosθ = L - Lcosθ Tus te eigt would be te difference between L and Lcosθ: L Lcos L 1 cos Te trigonometric relationsips listed in te Equations and Symbols section will be particularly elpful wen dealing wit vectors. Scalars and Vectors scalar is a quantity wic as no direction associated wit it, suc as mass, volume, time, and temperature. We say tat scalars ave only magnitude, or size. mass may ave a magnitude of kilograms, a volume may ave a magnitude of 5 liters, and so on. But a vector is a quantity wic as bot magnitude (size) and direction (angle). For example, if someone tells you tey are going to apply a 0 pound force on you, you would want to know te direction of te force, tat is, weter it will be a pus or a pull. So, force is a vector, since direction is important in specifying a force. Te same is true of displacement, as we will see in te following sections. Te table below lists some vectors and scalars you will be using in your pysics course. Vectors displacement velocity acceleration force weigt momentum Scalars distance speed mass time volume temperature work and energy Vector ddition and Subtraction We can grapically add vectors to eac oter by placing te tail of one vector onto te tip of te previous vector: 5
B - B = + B = + (- B), or = B In te diagram on te left above, we ave added two vectors ead to tail by placing te tail of vector B on te ead (tip) of vector. Wen adding vectors grapically, we may move a vector anywere we like, but we must not cange its lengt or direction. Te resultant is drawn from te tail of te first vector to te ead of te last vector. Te resultant is also called te vector sum of and B, and can replace te two vectors and yield te same result. Example 4 Displacement is also a vector. Consider a iker wo walks 8 kilometers due east, ten 10 km due nort, ten 1 km due west. Wat is te iker s displacement from te origin? vector can be represented by an arrow wose lengt gives an indication of its magnitude (size), wit te arrow tip pointing in te direction of te vector. We represent a vector by a letter written in bold type. For tis example, we list te displacement vectors like tis: = 8 km west B = 10 km nort C = 1 km east We can grapically add te second displacement vector to te first, and te tird displacement vector to te second: 6
1 km N 10 km = 10.8 km at 68º from E W 8 km 68º E S Te resultant vector is te displacement from te origin to te tip of te last vector. In oter words, te resultant is te vector sum of te individual vectors, and can replace te individual vectors and end up wit te same result. Of course, just adding te lengts of te vectors togeter will not acieve te same result. dding 8 km, 10 km, and 1 km gives 30 km, wic is te total distance traveled, but not te straigt-line displacement from te origin. We see in te diagram above tat te resultant displacement is 10.8 km from te origin at an angle of 68º from te east axis. We could ave added te displacements in any order and acieved te same resultant. We say tat te addition of vectors is commutative. Te equilibrant is te vector wic can cancel or balance te resultant vector. In tis case, te equilibrant displacement is te vector wic can bring te iker back to te origin. Tus, te equilibrant is always equal and opposite to te resultant vector. 7
1 km N 10 km E = - = 10.8 km at 68º + 180º from E W 8 km 68º E S Te Components of a Vector We may also work wit vectors matematically by breaking tem into teir components. vector component is te projection or sadow of a vector onto te x- or y-axis. For example, let s say we ave two vectors and B sown below: y B y α B x B x We will call te projection of vector onto te x-axis its x-component, x. Similarly, te projection of onto te y-axis is y.te vector sum of x and y is, and, since te 8
magnitude of is te ypotenuse of te triangle formed by legs x and y, te Pytagorean teorem olds true: x y and from te figures above, x = sin y = cos tan y x We can write te same relationsips for vector B by simply replacing wit B and te angles wit in eac of te equations above. Example 5 Find te x- and y-components of te resultant in Example 4. Te resultant is 10.8 km long at an angle of 68º from te east (+x) axis: y 68º Since te vector component x is adjacent to te 68º angle, we ave te relationsip x = cosθ = (10.8 km)cos 68º = 4 km Similarly, te vector component y is opposite to te 68º angle, so we ave y = sinθ = (10.8 km)sin 68º = 10 km ddition of Vectors by Means of Components x Earlier we added vectors togeter grapically to find teir resultant. Using te ead-totail metod of adding vectors, we can find te resultant of and B, wic we called. We can also use components to find te resultant of any number of vectors. For example, te x-components of te resultant vector is te sum of te x-components of, B, and C. Similarly, Te y-components of te resultant vector is te sum of te y-components of and B. 9
So, we ave tat x = x + B x + C x and y = y + B y + C y and by te Pytagorean teorem, x y z y B x C Example 6: Using te diagrams above, let = 4 meters at 30º from te x-axis, B = 3 meters at 45º from te x-axis, and C = 5 m at 5º from te y-axis. Find te magnitude and direction of te resultant vector. (cos30 = 0.87, sin 30 = 0.50, cos45 = sin45 = 0.70, cos5 = 0.90, sin5 = 0.4) Solution: First, we need to find x, y, B x, B y, C x, and C y : x = cos30 = (4 m)cos30 = (4 m)(0.87) = 3.5 m. y = sin30 = (4 m)cos30 = (4 m)(0.50) =.0 m B x = Bcos45 = (3 m)cos45 = (3 m)(0.70) =.1 m B y = Bsin45 = (3 m)sin45 = (3 m)(0.70) =.1 m C x = - Ccos5 = (5 m)cos5 = - (5 m)(0.90) = - 4.5 m C y = - Csin5 = (5 m)sin5 = - (5 m)(0.43) = -. m 10
Note tat C x and C y are bot negative, since tey point to te left and down, respectively. Now we can find te x- and y-components of te resultant : x = x + B x + C x = 3.5 m +.1 m + (- 4.5 m) = 1.1 m y = y + B y + C y =.0 m +.1 m + (-. m) = 1.9 m Te magnitude of te resultant C is x y 1.1m 1.9 m. m and its angle from te x-axis can be found by tan y x tan 1 1.9 m 1.1m 60 y =. m at 60º from te +x-axis x Te properties of vectors we ve discussed ere can be applied to any vector, including velocity, acceleration, force, and momentum. 11
EVIEW QUESTIONS For eac of te multiple coice questions below, coose te best answer. 1. Te fundamental SI units for mass, lengt, and time, respectively are 4. Consider te velocity vs time grap sown. () newton, meter, minute (B) kilogram, meter, second (C) pound, foot, our (D) pound, foot, second (E) kilogram, centimeter, our. How many nanometers are in a kilometer? () 10-9 (B) 10 9 (C) 10-1 (D) 10 1 (E) 10 7 3. Te data plotted on a grap of distance on te y-axis vs time on te x- axis yields a linear grap. Te slope of te grap is d () t (B) ( d)( t) t (C) d (D) ( d) + ( t) (E) ( d) - ( t) Te area under te grap from s to 6 s is most nearly () 8 (B) 16 (C) 4 (D) 3 (E) 36 5. Data from an experiment sows tat potential energy is proportional to te square of displacement. Te plot of te grap of potential energy (y) vs. displacement (x) would be () a straigt orizontal line (B) a straigt diagonal line sloping upward (C) a parabola (D) a yperbola (E) a circle 1
6. Te cosine of an acute angle in a rigt triangle is equal to te 9. y () sine of te angle (B) tangent of te angle (C) te ypotenuse of te triangle (D) te side adjacent to te angle (E) te ratio of te side adjacent to te angle and te ypotenuse of te triangle 7. Wic of te following quantities is NOT a vector quantity? () displacement (B) mass (C) resultant (D) equilibrant (E) 10 km at 30 nort of east 8. Te resultant of te two displacement vectors 30 m east and 40 m sout is () 50 m souteast (B) 70 m norteast (C) 10 m soutwest (D) 10 m norteast (E) 10 m norteast Te x-component of te vector sown above is most nearly (sin 30 = 0.5, cos 30 = 0.87, tan 30 = 0.58) () 100 m (B) 50 m (C) 87 m (D) 58 m (E) 1000 m 10. Two displacement vectors eac aving a y-component of 100 km are added togeter vectorially to form a resultant wic forms an angle of 60 from te +x-axis. Wat is te magnitude of te resultant? (sin 60 = 0.87, cos 60 = 0.5) () 30 m (B) 400 m (C) 10 m (D) 00 m (E) 300 m 30 100 m x 13
Free esponse Question Directions: Sow all work in working te following question. Te question is wort 15 points, and te suggested time for answering te question is about 15 minutes. Te parts witin a question may not ave equal weigt. 1. (15 points) displacement is a lengt wit a particular direction, and terefore can be represented by a vector. n explorer travels a displacement of 0 km at an angle of 30 from east, ten 30 km at 45 from east, ten 0 km 10 from east, and finally, 10 km at 40 from east. Label eac of tese displacement vectors, B, C, and D, respectively. ll angles are measured counterclockwise from te east axis. (a) Determine te total distance traveled by te explorer in meters. (b) Coose a scale, and using a protractor and a ruler, add te displacement vectors eadto-tail on te grap provided below. Be sure to clearly label eac vector. Scale (c) Wat is te resultant displacement of te explorer from is original starting point? Be sure to indicate bot te magnitude of te resultant displacement and its direction from east. esultant (d) Using te sine, cosine, and/or tangent of te appropriate angles, find te x- and y- 14
components of eac of te displacement vectors. (e) Using addition of vectors by means of components, find te magnitude and direction of te resultant vector. NSWES ND EXPLNTIONS TO CHPTE 1 EVIEW QUESTIONS Multiple Coice 1. B Mass is measured in kilograms, lengt in meters, and time in seconds.. D 3 9 10 m 10 nm 1 1 km ( )( ) 10 nm km m 3. Slope equals rise over run, wic in tis case would be d. t 4. C Te total area can be calculated by adding te area bounded by 4(6-) = 16 and te area of te triangle bounded by v = 4 to v = 8 and t = to t = 6 is ½ (8 4)(6 ) = 8. Tus, te total area is 16 + 8 = 4. 5. C Since te potential energy is proportional to te square of te displacement, plotting PE vs x would produce a parabola. 6. E Te cosine is defined as te ratio of te adjacent side to te ypotenuse. 7. B Mass as no direction associated wit it, terefore it is a scalar and not a vector. 8. 15
Te two displacements represent te two legs of a rigt triangle. By te Pytagorean teorem, te ypotenuse is equal to 30 40 50m souteast. 9. C Te x-component of te vector is te side adjacent to te 30 angle and is found by (100m)cos30 = 87 m. 10. Te figure at te rigt sows tat te two y-components add up to 00 m, wic is te y-component of te resultant. Te y-component of te resultant (00 m) is te opposite side to te 60 angle, and te magnitude of te resultant can be 00m found by sin 60, 30m. y 60 y = 100 + 100 = 00 m x Free esponse Question Solution (a) 1 point Distance is a scalar quantity, so te total distance traveled by te explorer is simply te sum of te individual distances e traveled, witout regard to direction: distance = 0 km + 30 km + 0 km + 10 km = 80 km Since 1 km = 1000 meters, 80 km = 80,000 m or 8 x 10 4 m. 16
(b) 5 points Scale: 1 cm = 10 km y D C B x (east) (c) points If we measure te lengt and direction (angle) of te resultant vector, we find tat it is 4.5 cm (10 km/cm) = 45 km at an angle of 60 from te x-axis (east). (d) 4 points ll angles are measured counterclockwise from te +x-axis: x = (0 km) cos 30 = 17.3 km y = (0 km) sin 30 = 10.0 km B x = (30 km) cos 45 = 1. km B y = (30 km) sin 45 = 1. km C x = (0 km) cos 10 = - 10.0 km C y = (0 km) sin 10 = 17.3 km D x = (10 km) cos 40 = - 5.0 km D y = (10 km) sin 40 = - 8.7 km (e) 3 points Te x- and y-components of te resultant are equal to te sum of te x- and y- components of te individual vectors: x = x + B x + C x + D x = 3.5 m y = y + B y + C y + D y = 39.8 m 17
By Pytagoras s Teorem, x y 3.5m 39.8m 46. m Te angle of te resultant vector from te +x-axis is tan y x y tan 1 39.8m 3.5m 59.4 Tus, our answers for parts (d) and (e) are reasonably consistent. 60º x 18