CHPTER 1 MESUREMENTS ND VECTORS 1 CHPTER 1 MESUREMENTS ND VECTORS
1.1 UNITS ND STNDRDS n phsical quantit must have, besides its numerical value, a standard unit. It will be meaningless to sa that the distance between Gaza and Jerusalem is 80 because 80 kilometers is different from 80 meters or 80 miles, where kilometer, meter, and mile are standards for length known all over the world. Several sstems of units are used in phsics: The most common sstem among them is the Sstém International (French for International Sstem) abbreviated SI. The SI units for the seven-fundamental phsical quantities are kilogram for mass, meter for length, second for time, kelvin for temperature, ampere for electric current, candela for luminous intensit, and mole for the amount of substance. ll other phsical quantities are derived from these basic quantities. In mechanics all quantities are derived from the three-fundamental quantities: mass, length, and time. One meter is defined as the distance traveled b light in vacuum during a time of 1/99,79,458 second. One kilogram is defined as the mass of a specific platinumiridium allo clinder kept at the International ureau of Weights and Measures in France. One second is defined as the time required for a cesium-133 atom to undergo 9,19,631,770 vibrations. The other two sstems are the Gaussian sstem and the ritish engineering sstem. The three sstems and their standard units for mass, length, and time are listed in Table 1.1. ecause of their units the first two sstems are sometimes called the mks sstem (the first letters of meter, kilogram, and second) and the cgs sstem (the first letters of centimeter, gram, and second) respectivel. If different unit sstems are used in a phsical equation one sstem should be chosen and the quantities with the other unit sstems must be converted to the chosen sstem according to the conversion factors of Table. 1..
CHPTER 1 MESUREMENTS ND VECTORS 3 Table 1.1 Units of Length, Mass, and Time in Different sstems. Sstems Length Mass Time SI meter (m) kilogram (kg) second (s) cgs Centimeter (cm) gram (g) second (s) ritish foot (ft) slug second (s) Table 1. Conversion Factors. Length Mass Time 1 m10 cm3.8 ft 1 kg 10 3 g 1 ear3.1610 7 s 1 mi580 ft1.61 km 1 slug14.6 kg 1 da8.6410 4 s 1 d3 ft36 in 1 u1.6610-7 kg 1. DIMENSIONL NLYSIS Dimension in phsics gives the phsical nature of the quantit, whether it is a length (L), mass (M), or time (T). ll other quantities in mechanics can be epressed in terms of these fundamental quantities. For eample, the dimension of velocit is length divided b time and denoted b [ v ] L T. The dimensions of some phsical quantities are listed in Table 1.3. Table 1.3 Dimensions and units of some phsical quantities Quantit Dimension Unit (SI, cgs, ritish) rea L m, cm,ft Volume L 3 m 3, cm 3, ft 3 Velocit L/T m/s, cm/s, ft/s cceleration L/T m/s, cm/s, ft/s Force ML/T Newton (N), dne, pound Energ ML /T Joul (J), erg, ft.lb (lb) Power ML /T 3 Watt (W), erg/s, horsepower (hp)
4 1 Eample 1.1 Show that the equation vt o + at dimensionall correct. L Solution Since [ ] L, [ v ] 0 T, and [ a] L T, is Then L L T T + L T T L 1.3 VECTORS ND SCLRS scalar is the phsical quantit that has magnitude onl, for eample, time, volume, mass, densit, energ, distance, temperature. On the other hand a vector is the phsical quantit that has both magnitude and direction, for eample, displacement, velocit, acceleration, force, area. - Figure 1.1 Equalit of vectors and the negative of a vector. Note that >. The vector quantit will be distinguished from the scalar quantit b tping it in boldface, like. In write handing the vector quantit is written with an arrow over the smbol, such as, r. The magnitude of the vector will be denoted b, or simpl the
CHPTER 1 MESUREMENTS ND VECTORS 5 italic tpe. Geometricall, The vector quantit is represented b a straight line and an arrow at one end of the line. The end at which the arrow is attached is called the head of the vector, while the other end is called the tail of the vector. The length of the line is proportional to the magnitude of the vector and the arrow points toward its direction (see Figure 1.1). Equalit of Two Vectors n two vectors are said to be equal if the have the same magnitude and point in the same direction, regardless of their location. + (a) ++C + (b) Figure 1. (a) The addition of vectors graphicall showing the commutation law. (b) The distribution law of addition. C +C ddition (graphical method). To add vector to another vector, (1) in a graph paper, draw vector with its magnitude represented b a proper scale. () Draw vector according to the same scale such that its tail starts at the head of vector. (3) Draw the resultant vector R(+) from the tail of vector to the head of vector, (see Figure 1.). This method is known as the triangle method. nother method is the parallelogram method. In this method the tail of the two vectors and are coincident and the resultant vector is the diagonal of the parallelogram formed b the two vectors and. s it is clear from Figure 1., the addition of vectors obe the commutation and association laws, i.e., respectivel
6 ++ (1.1) and +(+C)(+)+C (1.) Negative of a Vector The vector - is a vector with the same magnitude as the vector but points in opposite direction. Subtraction Subtracting vector from vector is the same as adding - to, i.e., -+(-) (1.3) The arithmetic processes of vectors using the graphical method described above are not eas. In the following sections we describe easier techniques that involve algebra. 1.4 UNIT VECTOR j i k Z Figure 1.3 The unit vectors i, j, and k are pointed toward the positive,, and z aes, respectivel. From its name, a unit vector is a vector, with specific direction, having a magnitude of unit without an units or dimensions. In Cartesian coordinates three unit vectors are adopted to specif the
CHPTER 1 MESUREMENTS ND VECTORS 7 positive directions of the three-aes. These are: i for the positive - ais, j for the positive -ais and k for the positive z-ais, as shown in Figure 1.3. This means that if a vector is directed a long the positive -ais with a magnitude of, this vector can be written as i. Similarl, a vector, directed a long the positive -ais and has a magnitude is written as j. The vector CCk means that it has a magnitude of C and directed along the positive z-ais. The minus sign in front of an vector indicates the opposite direction of that vector, i.e. -i refers to the negative -ais, and so for the other two unit vectors. 1.5 COMPONENTS OF VECTORS j θ i Figure 1.4 vector can be resolved into two perpendicular components, the -component and the -component. n vector in a plane can be represented b the sum of two vectors, one parallel to the -ais ( ), and the other parallel to the -ais ( ) as shown in Figure 1.4, i.e. +. (1.4) Since, i and j we can write Equation 1.4 as
8 i + j (1.5) where and are the -component and the -component of the vector, respectivel. From Figure 1.4 it is clear that cosθ and sinθ (1.6) The magnitude of the vector is given b + (1.7) In general an vector can be resolved into three components as i + j+ k (1.8) z where z is called the z-component of the vector. Note that if 0, then 0, 0, and z 0. Eample 1. vector ling in the - plane has a magnitude 50.0 units and is directed at an angle of 10 o to the positive - ais, as shown in Figure 1.5. What are the rectangular components of this vector? Solution From Equation (1.6) we have cosθ 50 cos10 o 5.0units and o sinθ 50sin10 43.3units 10 o Figure 1.5
CHPTER 1 MESUREMENTS ND VECTORS 9 1.6 DDING VECTORS To add vectors analticall proceed as follow: 1) Resolve each vector into its components according to suitable coordinate aes. ) dd, algebraicall, the -components of the individual vectors to obtain the -component of the resultant vector. Do the same thing for the other components, i.e., if i + j + k z and i + j + k, z then the resultant vector R is or R + R i +R j + R k z R + ) i + ( + ) j + ( + ) k (1.9) ( z z Eample 1.3 If 4i + 3 j and 3i + 7 j, find the resultant vector R +. Solution From the given information we have 4, 3, -3, and 7 Now from Equation (1.9) we have and R + 1.00,
10 R + 10.0 so we can write Ri+10j Eample 1.4 particle undergoes three consecutive displacements given b d 1 ( i + 3j- k) cm, d (i - j - 3k) cm, and d 3 ( i + j) cm. Find the resultant displacement of the particle. Solution R d1 + d + d 3, or R ( 1+ 1) i+ (3 1+ 1) j + ( 1 3 + 0) k ( i + 3j 4k) cm Eample 1.5 particle undergoes the following consecutive displacements: 4.3 m southeast,.4 m east, and 5. m north. Find the magnitude and the direction of the resultant vector. R d 3 Solution If we denote the three displacements b d 1, d,and d 3 respectivel, we get the vector diagram shown in Figure 1.6. ccording to the coordinates sstem chosen, the three vectors can be written as d 1 d Figure 1.6 d1 ( 43. cos45) i-(4.3sin45) j (. 304i-304. j) m d 4. im, and d3 5. j m
CHPTER 1 MESUREMENTS ND VECTORS 11 now, R d + d + d ( 3.04 +.40 + 0) i + ( 3.04 + 0 + 5.0) ( 5.44i +.16j) m 1 3 j The magnitude of R is ( 5. 44) + ( 16. ) 5. 85 R R m + R To find the direction of a vector it is enough to determine the angle the vector makes with a specific ais. The angle q makes with the - ais is θ R R.16 5.44 1 1 tan tan 1.66 o Eample 1.6 If 4i- j, find the vector such that + 5i. Solution Since + 5i then, 5, and 0 so we have + + and 5 5 4 1 1 This leads to i + j
1 1.7 PRODUCTS OF VECTORS 1- Multipling a Vector b a Scalar: The product of a vector and a scalar λ is a new vector with a direction similar to that of if λ is positive but opposite to the direction of if λ is negative. The magnitude of the new vector λ is equal to the magnitude of multiplied b the absolute value of λ, i.e., λ λ i + λ j + λ k (1.10) z - Scalar (Dot) Product: The scalar product of two vectors and, denoted b, is a scalar quantit defined b, cosθ (1.11) with θ is the smallest angle between the two vectors. Since i, j, and k are unit vectors perpendicular to each other, and using Equation (1.11), we can write i j i k k j 0 (1.1a) i i j j k k 1 (1.1b) Equations 1.1 impl that the scalar product of the vectors and can be written as + + (1.13) z z Eample 1.7 If i - j + 3k and i + 3j - k, find the angle between the two vectors and Solution From Equation (1.11) we have
CHPTER 1 MESUREMENTS ND VECTORS 13 cos θ. ut from Equation (1.7) we have 1 + 4 + 9 3.74, and 4 + 9 + 4 4.1. Using Equation (1.13) we find 6 6 10.0. So we now have or 10 cosθ 0.69, (3.74)(4.1) θ 1 cos ( 0.69) 134 o Eample 1.8 Consider the two vectors and given in the previous eample (eample 1.7), find Solution i-4j+6k, now from Equation 1.13, we obtain ( ) 4 1 1 0. 0 Note that if one multipl the product b, or multipl ( ) the same result will be achieved, that is ( ) ( ) ( )
14 3- Vector (cross) Product: The vector product of two vectors and, written as, is a third vector C with a magnitude given b (1.14) C sinθ The vector C is perpendicular to the plane of and. Since there are two directions perpendicular to a given plane, the right hand rule is used to decide to which direction the vector C is directed. The rule states that the four fingers of the right hand are pointed along and then curled toward through the smaller angle between and. The thumb then gives the direction of C. From this rule one can verif that and - (1.15) i ij jk k0 i jk j ki k ij (1.16a) (1.16b) (1.16c) (1.16d) From Equations (1.16) we can write ( i + j + k) ( i + j + k) or, equivalentl z z ( z z ) i + ( z - z ) j+ ( - - ) k
CHPTER 1 MESUREMENTS ND VECTORS 15 i j k z z (1.17) Eample 1.9 Find the vector product of the two vectors given in the previous eample (eample 1.7) Solution i j k i j k z z 1 3 3 (4-9)i + (6+)j + (3+4)k -5i+8j+7k
CHPTER 1 MESUREMENTS ND VECTORS 16 PROLEMS 1.1 Show that the equation v vo + ais dimensionall correct, where v and v o represent velocities, is a distance, and a is an acceleration. 1. The period of a simple pendulum with the unit of time is given l b T π where l is the length of pendulum, and g is the g acceleration due to gravit. Show that the equation is dimensionall correct. 1.3 vector in the - plane is 1 units long and directed as shown in Figure 1.7, Determine the and the components of the vector. 1.4 Vector is 5 units in length and points along the positive -ais. Vector is 7 units in length and points along the negative -ais. a) Find +. b) Find the magnitude and the direction of + 1.5 Vector has a magnitude of 4 units and makes an angle of 37 o with the positive -ais. Vector has a magnitude of 6 units and makes an angle of 10 o with the positive -ais. 7 units Figure 1.8 (Problem 1.4) 6 units 5 units 10 o 4 units 37 o Figure 1.9 (Problem 1.5) 30 o 1 units Figure 1.7 (Problem 1.3)
CHPTER 1 MESUREMENTS ND VECTORS 17 a) Find the -component and the -component of each vector. b) Find - 1.6 car is driven east for a distance of 30 km. It net driven north-east for a distance of 0 km, and then in a direction of 30 o north of west for a distance of 40 km. Draw the vector diagram of the three displacements and find their resultant (magnitude and direction). 1.7 Consider the three vectors shown in Figure 1.10. If 8 m, 5 m, and C1 m, find a) the and the -components of each vector, b) the magnitude and the direction of the resultant vector. 8 m 5 m 30 o 1 m C 1.8 Find the magnitude and the direction of the resultant of the three displacement having components (-3,3) m, (,4) m, and (5,-1) m. 1.9 Two vectors are given b 3i+3j, and -i+7j. Find a) +, b) -, c), d). 1.10 You are given the two vectors i+3j, and 6i+8j. Find the magnitudes and the directions of,, +, and -. 1.11 If i+3j-k, and i+3j, and Ci+j+k, find Figure 1.10 (Problem 1.7)
18 a) ++C b) --C 1.1 Given the two vectors 4i-3j, and i+4j. Find the vector C such that ++C0. 1.13 Find a unit vector parallel to 6i-8j. 1.14 Given the two vectors i-3j+7k, and 5i+j+k, find a) + b) -. 1.15 Find the scalar product. of the two vectors, and in problem 1.11. 1.16 Find the angle between the two vectors -i+3j, and 3i- 4j. 1.17 For what values of λ are the two vectors i+j+λk, and λi-j+λk perpendicular to each other. 1.18 If i + 6j, C,and + C find and C. 1.19 Using the scalar product, prove the law of cosine (the law is stated in appendi D). 1.0 Given i+j+k, and i+3j. Find a) b) ( ) 1.1 Three vectors are given b i+j+k, i+j, and Ci-k. Find a) (+C) b) ( C)
CHPTER 1 MESUREMENTS ND VECTORS 19 c) (+C) 1. The two vectors, and represent concurrent sides of a parallelogram as shown in Figure 1.11. Show that the area of the parallelogram is. Figure 1.11 Problems 1. 1.3 Find a unit vector perpendicular to the plane of the two vectors given in problem 1.17. 1.4 Two vectors of magnitudes and make an angle θ with each other when placed tail to tail (see Figure 1.1). Prove that the magnitude of their sum, R is given b θ Figure 1.1 Problems 1.4 R + + cosθ 1.5 Prove that for an three vectors,, and C ( C)( ) C