chapter (2) measurements and vectors. 2.1 unit and standards : the measurement of any quantity is...
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Chapter (2)Measurements and
Vectors
2.1 Unit and Standards2.1 Unit and Standards::
•The measurement of any quantity is made relative to a The measurement of any quantity is made relative to a particular standard or unit, and this must be specified a particular standard or unit, and this must be specified a long with the numerical value of the quantity. long with the numerical value of the quantity. •For example, we can measure length in units such as For example, we can measure length in units such as inches, feet, or miles, or in the metric system in inches, feet, or miles, or in the metric system in centimeters, meters, or kilometers. centimeters, meters, or kilometers. •To specify that the length of a particular object is 18.6 To specify that the length of a particular object is 18.6 is meaningless. is meaningless. •The unit must be given; for clearly, 18.6 meters is very The unit must be given; for clearly, 18.6 meters is very different from 18.6 millimeters.different from 18.6 millimeters.
For any unit we use, such as the mater for distance or the second For any unit we use, such as the mater for distance or the second for time, we need to define a standard which defines exactly how for time, we need to define a standard which defines exactly how long one meter or one second is. It is important that standards be long one meter or one second is. It is important that standards be chosen that are readily reproducible so that anyone needing to chosen that are readily reproducible so that anyone needing to make a very accurate measurement can refer to the standard in make a very accurate measurement can refer to the standard in the laboratory. the laboratory.
The laws of physics are expressed in terms of basic quantities The laws of physics are expressed in terms of basic quantities that require a clear definition. In mechanics, the three basic that require a clear definition. In mechanics, the three basic quantities are length quantities are length (L),(L), mass mass (M),(M), and time and time (T).(T). Several Several systems of units are used for these three quantities: The most systems of units are used for these three quantities: The most common system among them is the system International (French common system among them is the system International (French for International system) abbreviated SI. The other systems are for International system) abbreviated SI. The other systems are the cgs system and the British engineering system. the cgs system and the British engineering system. (Table 2-1)(Table 2-1) show the three systems and their standard units for mass, length, show the three systems and their standard units for mass, length, and time.and time.
Table 2 – 1 Units of Length, Mass, and Time
in Different System
Systems
LengthMassTime
SIMeter (m)Kilogram
(kg)Second
(s)
cgsCentimeter
(cm)Gram (gm)
Second (s)
BritishFoot (ft)SlugSecond
(s)
Some important definitions Some important definitions for SI unitsfor SI units::
The meter (m) is the length of path traveled by light The meter (m) is the length of path traveled by light in vacuum during a time interval of 1/229,792,458 of in vacuum during a time interval of 1/229,792,458 of a second. a second. The kilogram (kg) is the mass of a specific platinum The kilogram (kg) is the mass of a specific platinum – iridium alloy cylinder kept at the International – iridium alloy cylinder kept at the International Bureau of weights and Measures at sevres, France. Bureau of weights and Measures at sevres, France. The second (s) is now defined as 9192631770 times The second (s) is now defined as 9192631770 times the period of vibration of radiation from the cesium the period of vibration of radiation from the cesium atom.atom.
PrefixesPrefixes::
Sometimes the numerical value of our physical Sometimes the numerical value of our physical quantities is too large or in the contrary is too small, quantities is too large or in the contrary is too small, which makes the numbers bothersome to is 350000 which makes the numbers bothersome to is 350000 meter, or the mean radius of earth is 637000000 meter. meter, or the mean radius of earth is 637000000 meter. Look, these numbers are not easy to carry and deal Look, these numbers are not easy to carry and deal with. So, it is better prefixes, which are abbreviations with. So, it is better prefixes, which are abbreviations come in front of the units to make them handy. See come in front of the units to make them handy. See (Table 2 – 2) for the commonly used a abbreviations in (Table 2 – 2) for the commonly used a abbreviations in the field of medicine.the field of medicine.
Table 2 – 2 : Prefixes for Powers of Ten in (SI)
PowerPrefixAbbreviation
10-15femtoF
10-12picoP
10-9nanoN
10-6micro
10-3milliM
10-2centiC
10-1deciD
103kiloK
106megaM
109gigaG
1012teraT
1015petaP
2.2Dimensional analysis2.2Dimensional analysis
Dimension is physics denotes the physical Dimension is physics denotes the physical nature of a quantity whether it is a [length] = L ; nature of a quantity whether it is a [length] = L ; [mass] = M ; [time] = T, and all other quantities [mass] = M ; [time] = T, and all other quantities are derivable from these quantities. are derivable from these quantities.
For example the dimensions of speed is length For example the dimensions of speed is length divided by time and denoted by . As another divided by time and denoted by . As another example, the dimensions of area A are [A] = L2. example, the dimensions of area A are [A] = L2. The dimensions of other physical quantities are The dimensions of other physical quantities are listed in (Table 2 -3).listed in (Table 2 -3).
Table 2 – 3 Dimensions and Units of
some Physical Quantities
QuantityDimensionUnit (SI, cgs, British)
AreaL2M2, cm2, ft2
VolumeL3M3, cm3, ft3
VelocityL/Tm/s, cm/s , ft/s
AccelerationL/T2m/s2, cm/s2, ft/s2
MomentumML/TKgm/s, gcm/s , slug ft/s
ForceML/SNetwon(N), dyne, pound (Ib)
EnergyML2/T2Joul (J), erg, ft.Ib
PowerML2/T3Watt(W), erg/s, hosepower (hp)
Example 2- 1: Example 2- 1: Show that the equation ( ) is Show that the equation ( ) is dimensionallydimensionally
correct, where correct, where x x is the displacement, is an initial is the displacement, is an initial
velocity, a is the acceleration, and t is the timevelocity, a is the acceleration, and t is the time..
SolutionSolution::
Since , , andSince , , and
ThenThen
20 2
1attvx
x
0v
Lx T
Lv 0
2T
La
LTT
LT
T
LL 2
2
Example 2 – 3:Example 2 – 3: Suppose we are told that the acceleration a of a particle moving Suppose we are told that the acceleration a of a particle moving with uniform speed v in a a circle of radius r is proportional to with uniform speed v in a a circle of radius r is proportional to some power of r, say , and some power of v say . Determine some power of r, say , and some power of v say . Determine the values of n and m and write the simplest form of an equation the values of n and m and write the simplest form of an equation for the acceleration. for the acceleration. Solution: Solution:
Let us take a to beLet us take a to be
Where k is a dimensionless constant of proportionality Where k is a dimensionless constant of proportionality knowing the dimensions of a , r, and vknowing the dimensions of a , r, and v . .
SinceSince , , , ,
mnvkra
2T
La
T
Lv Lr
nr
mv
mv
Then,Then,
Equation the powers of the left hand side units to the right Equation the powers of the left hand side units to the right hand side units, gives;hand side units, gives;
n + m = 1 n + m = 1 , and, and m = 2 m = 2
therefore,therefore, so, we can write the acceleration expression asso, we can write the acceleration expression as
M
mnMn
T
L
T
LL
T
L
2
112 nn
r
vkvkra
221
2.3Vector and Scalar Quantities:2.3Vector and Scalar Quantities:
Some physical quantities are scalar quantities whereas others are vector Some physical quantities are scalar quantities whereas others are vector quantities.quantities.
A scalar quantity A scalar quantity is completely specified by a single value with an appropriate is completely specified by a single value with an appropriate unite and has no direction. unite and has no direction.
Other examples of scalar quantities are volume, mass, speed, and time in Other examples of scalar quantities are volume, mass, speed, and time in travels. The rules of ordinary arithmetic are used to manipulate scalar travels. The rules of ordinary arithmetic are used to manipulate scalar quantities. quantities.
A vector quantity A vector quantity is completely specified by a number and is completely specified by a number and appropriate units plus a direction. appropriate units plus a direction. Another example of a vector quantity is displacement. Another example of a vector quantity is displacement. Suppose a particle moves from point (A) to some point Suppose a particle moves from point (A) to some point (B) along a straight path, as shown in Fig. 2.1. We (B) along a straight path, as shown in Fig. 2.1. We represent this displacement by drawing an arrow from (A) represent this displacement by drawing an arrow from (A) to (B), with the tip of the arrow pointing away from the to (B), with the tip of the arrow pointing away from the starting pointstarting point
The vector quantity The vector quantity will be distinguished from the scalar quantity by will be distinguished from the scalar quantity by typing it in boldface, like typing it in boldface, like AA. . I write handinI write handing the vector, quantity isg the vector, quantity is
written with an arrow over the symbol, such as, . The magnitude written with an arrow over the symbol, such as, . The magnitude of the vector a will be denoted of the vector a will be denoted |A||A|, or simply the italic type , or simply the italic type AA. the . the magnitude of a vector has physical units, such as meters for magnitude of a vector has physical units, such as meters for displacement or meters per second for velocity. The magnitude of a displacement or meters per second for velocity. The magnitude of a vector is always a positive number.vector is always a positive number.
Quick Quiz 2.1:Quick Quiz 2.1: Which of the following are vector quantities and which are scalar Which of the following are vector quantities and which are scalar
quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass
2.32.3 Properties of VectorsProperties of Vectors
Equality of two vectors:Equality of two vectors:Any two vectors are said to be equal if they have the same Any two vectors are said to be equal if they have the same
magnitude and point in the same direction. For example, all magnitude and point in the same direction. For example, all vectors in Fig. 2.2 are equal even though they have different vectors in Fig. 2.2 are equal even though they have different
starting pointsstarting points..
o
y
X
(II) Adding Vectors:(II) Adding Vectors:Graphical Method (Geometrical Method):Graphical Method (Geometrical Method):
To add vector To add vector BB to vector to vector AA, first draw vector , first draw vector AA, with its , with its magnitude represented by a convenient scale, on graph paper magnitude represented by a convenient scale, on graph paper and then draw vector and then draw vector BB to the same scale with its tail starting to the same scale with its tail starting from the tip of from the tip of AA, as shown in Fig.2.3a. first translate the vector , as shown in Fig.2.3a. first translate the vector BB until the tail of the vector until the tail of the vector BB touches the head of the vector touches the head of the vector AA, , Figure 2.3a. The resultant Figure 2.3a. The resultant R = A + BR = A + B is the vector draw from the is the vector draw from the tail of tail of AA to the tip of to the tip of BB. this procedure is known as the triangle . this procedure is known as the triangle method of addition. method of addition.
When two vectors are added, the sum is independent of the When two vectors are added, the sum is independent of the order of the addition. This can be seen from the geometric order of the addition. This can be seen from the geometric construction in Fig.(2.4) and is known as the commutative law of construction in Fig.(2.4) and is known as the commutative law of addition: addition:
A + B = B + AA + B = B + A
(III)Subtracting Vectors:(III)Subtracting Vectors:
Quick quiz 2.2Quick quiz 2.2 The magnitudes of two vectors The magnitudes of two vectors AA and and BB are are A A = 12 units and = 12 units and BB = 8 units. Which of the following pairs of = 8 units. Which of the following pairs of numbers represents the largest and smallest possible values numbers represents the largest and smallest possible values for the magnitude of the resultant vector for the magnitude of the resultant vector R = A + B?R = A + B? (a) 14.4 (a) 14.4 units, 4 units (b) 12 units, 8 units, (c) 20 units, 4 units (d) none units, 4 units (b) 12 units, 8 units, (c) 20 units, 4 units (d) none of these answers. of these answers.
Quick quiz 2.3 Quick quiz 2.3 if vectorif vector B B is is added vector A, added vector A, which two of which two of the following choices must be true in order for the resultant the following choices must be true in order for the resultant vector to be equal to zero? (a) vector to be equal to zero? (a) AA and and BB are parallel and in the are parallel and in the same direction. same direction.
(b) (b) AA and and BB are parallel and in opposite directions. (c) are parallel and in opposite directions. (c) AA and and BB have the same magnitude. (d) A and B are perpendicular. have the same magnitude. (d) A and B are perpendicular.
))VV ( (Multiplying VectorMultiplying Vector
If vector If vector AA is multiplied by a positive scalar quantity is multiplied by a positive scalar quantity mm, then , then the product the product mmAA is a vector that has the same direction as is a vector that has the same direction as AA and magnitude and magnitude mmAA. If vector . If vector AA is multiplied by a negative is multiplied by a negative scalar quantity scalar quantity – m– m, then the product , then the product – m– mAA is directed is directed opposite opposite AA. for example the vector 4A is four times as long . for example the vector 4A is four times as long as A and points in the same direction as A; the vector - 1/3 as A and points in the same direction as A; the vector - 1/3 AA is one-third the length of is one-third the length of AA and points in the direction and points in the direction
opposite opposite AA . .
Scalar multiplication Scalar multiplication A scalar quantity(i.e. a number) can alter the magnitude of A scalar quantity(i.e. a number) can alter the magnitude of a vector but a vector but notnot its direction its direction..
Example - In the diagram(above) the vector of magnitude X Example - In the diagram(above) the vector of magnitude X is multiplied by 2 to become magnitude 2X.is multiplied by 2 to become magnitude 2X.
If the vector If the vector XX starts at the origin and ends at the point (4,4), starts at the origin and ends at the point (4,4), then the vector then the vector 2X2X will end at (8,8). will end at (8,8).
2.4 Unit Vector2.4 Unit Vector
Vector quantities often are expressed in terms of unit vectorsVector quantities often are expressed in terms of unit vectors. A. A unit unit vector is a dimensionless vector having a magnitude of exactly vector is a dimensionless vector having a magnitude of exactly 11..
Unit vectors are used to specify a given direction and have no Unit vectors are used to specify a given direction and have no other physical significance. other physical significance.
They are used solely as a convenience in describing a direction in They are used solely as a convenience in describing a direction in space. In Cartesian coordinates we shall use space. In Cartesian coordinates we shall use
the symbols ,the symbols ,
,
2.52.5 Components of VectorsComponents of Vectors
Any vector A in a plane can be represented by the Any vector A in a plane can be represented by the sum of two vectors, one parallel to the x –axis sum of two vectors, one parallel to the x –axis (A(Axx), and the other parallel to the y – axis (A), and the other parallel to the y – axis (Ayy) as ) as
shownshown
A = AA = Axx + Ay (2.2) + Ay (2.2)
Another Another
Where AWhere Axx and Ay are the x – component and the y – component and Ay are the x – component and the y – component
of the vector A, respectively. From fig 2.8 it is clear that of the vector A, respectively. From fig 2.8 it is clear that
AAxx = A cosθ = A cosθ (2 .4)(2 .4)
And AAnd Ayy = A sinθ = A sinθ (2 .5)(2 .5)
The magnitude and direction of the vector A are given by The magnitude and direction of the vector A are given by
AA yxAA
22
x
y
A
A1tan
Note that the signs of the components ANote that the signs of the components Axx and Ay depend and Ay depend
on the angle θ. For example, it θ = 120°, then Aon the angle θ. For example, it θ = 120°, then Axx is is
negative and Ay is positive. If θ = 225°, then both Anegative and Ay is positive. If θ = 225°, then both Axx
and Ay are negative.and Ay are negative.
In general any vector In general any vector AA can be resolved into three can be resolved into three components as components as
2.7 Adding Vectors2.7 Adding Vectors
We use the components to add vectors when the graphical We use the components to add vectors when the graphical method is not sufficiently accurate. The method for adding method is not sufficiently accurate. The method for adding the vectors as follow:the vectors as follow:
Resolve each vector into its components according to a Resolve each vector into its components according to a suitable coordinate axes. suitable coordinate axes.
Add, algebraically, the x-components of the individual Add, algebraically, the x-components of the individual vectors to obtain the x-component of the resultant vector. vectors to obtain the x-component of the resultant vector. Do the same thing for the other components, i.e., if Do the same thing for the other components, i.e., if
The resultant vector The resultant vector R = A + BR = A + B is therefore , is therefore ,
RR = + = +
the resultant vector are;the resultant vector are;
zzz
yyy
xxx
BAR
BAR
BAR
Example 2.3:Example 2.3:
Find the sum of two vectors A and B lying in the xy plane and given by:Find the sum of two vectors A and B lying in the xy plane and given by:
A = (2.0 + 2.0 A = (2.0 + 2.0 )m and B = (2.0 + -4.0 )m and B = (2.0 + -4.0 ) m. Then find the magnitude and direction of the ) m. Then find the magnitude and direction of the
vector sum.vector sum.
Solution: Solution:
The vector sum is The vector sum is R = A + BR = A + B it is clear that, it is clear that,
AAxx = 2 , A = 2 , Ayy = 2 , B = 2 , Bxx = 2 , B = 2 , Byy = -4 = -4
So, from equation (2.9);So, from equation (2.9);
R = (2.0 + 2.0) m + (2.0 – 4.0) m = (4.0 - 2.0 ) mR = (2.0 + 2.0) m + (2.0 – 4.0) m = (4.0 - 2.0 ) mOr Rx = 4.0m , Ry = - 2.0 m
The magnitude of R is
The direction of R
mmRRR yx 5.420)0.2()0.4( 2222
270.4
0.2tantan 11
x
y
R
R
The result vector is below the x- axis by 27° degrees (clockwise rotation).
ExamplExamplee Find the sum of two vectors A and B given Find the sum of two vectors A and B given byby
andandSolution: Solution:
Note that Note that AAxx=3, =3, AAyy=4, =4, BBxx=2, and =2, and BByy=-5=-5
The magnitude of vector R is
The direction of R with respect to x-axis is
Example (2.4): Example (2.4): A particle undergoes three consecutive displacements:dA particle undergoes three consecutive displacements:d11= (15 + 30= (15 + 30
+12 )cm, d +12 )cm, d22= (23 - 14 = (23 - 14 - 5.0 )cm and d - 5.0 )cm and d33= (-13 + 15 = (-13 + 15 )cm. )cm.
Find the components of the resultant displacement and its Find the components of the resultant displacement and its magnitude.magnitude. Solution:Solution:
Let, R = dLet, R = d11 + d + d22 + d + d33
= (15 + 23 – 13) + (30 – 14 – 15) + (12 – 5.0 + 0) = (15 + 23 – 13) + (30 – 14 – 15) + (12 – 5.0 + 0)
= (25 + 31 + 7.0 )= (25 + 31 + 7.0 )Then, the resultant displacement has components Rx = 25cm , Then, the resultant displacement has components Rx = 25cm ,
Ry = 31cm , Rz = 7.0cm , its magnitude Ry = 31cm , Rz = 7.0cm , its magnitude is is
cm
RRRR zyx
40)0.7()31()25( 222
222
EXAMPLE (2.5): EXAMPLE (2.5): CONSIDER TWO VECTORS A = 4.0 - 3.0 CONSIDER TWO VECTORS A = 4.0 - 3.0 AND B = -2.0 + 7.0 AND B = -2.0 + 7.0
CALCULATE: A + B; A – B ; ; ; AND THE CALCULATE: A + B; A – B ; ; ; AND THE
OF A + B AND OF A + B AND A – B.A – B.
Solution: Solution:
A + B = (4.0 + (-2.0)) + (- 3.0 + 7.0 )A + B = (4.0 + (-2.0)) + (- 3.0 + 7.0 ) = 2.0 + 5.0 at the = 2.0 + 5.0 at the
beginning find – B then add it to A, that is -B = 2.0 - 7.0 .beginning find – B then add it to A, that is -B = 2.0 - 7.0 .
A + (-B) = (4 + 2) + (-3 – 7) =6 - 10A + (-B) = (4 + 2) + (-3 – 7) =6 - 10
The direction are The direction are
BA BA
13610036
29254
BA
BA
6
10tan
2
5tan
12
11
Example (2-6):Example (2-6):A particle undergoes the following consecutive A particle undergoes the following consecutive displacements: 3.5m displacements: 3.5m southeastsoutheast, 2.5m east, and 6m north. What is , 2.5m east, and 6m north. What is the resultant displacement?the resultant displacement?
Solution:Solution:
Figure 2.9 example 2.6 , The displacement dFigure 2.9 example 2.6 , The displacement d11 southeast, d southeast, d22 west, and d west, and d33 north. north.
If we denote the three displacements by dIf we denote the three displacements by d11 , d , d22 , and d , and d33 respectively, respectively, we get the vector we get the vector
diagram shown in figure 2.9 . According to the coordinates systemdiagram shown in figure 2.9 . According to the coordinates system
chosen, the three vectors can be written aschosen, the three vectors can be written as dd11 = (3.5 cos45) i - (3.5 sin 45) j = (3.5 cos45) i - (3.5 sin 45) j
= (2.5i - 2.5j )m,= (2.5i - 2.5j )m,
dd22 = 2.5m, = 2.5m,
dd33 = 6m . = 6m .
now, R = dnow, R = d11 + d + d22 + d + d33
= (2.5 + 2.5 + 0)i + (2.5 + 0 + 6) j = (2.5 + 2.5 + 0)i + (2.5 + 0 + 6) j ) = ) = 55 i + 8.5ji + 8.5j((mm
R
d1d2
d3
2.82.8 Multiplication of VectorsMultiplication of Vectors : :
Like scalars, vectors of different kinds can be multiplied by Like scalars, vectors of different kinds can be multiplied by one another to generate quantities of new physical one another to generate quantities of new physical dimensions. Because vectors have direction as well as dimensions. Because vectors have direction as well as magnitude, the vector multiplication cannot follow exactly magnitude, the vector multiplication cannot follow exactly the same rules as the algebraic rules of scalar the same rules as the algebraic rules of scalar multiplication. We must establish new rules of multiplication multiplication. We must establish new rules of multiplication for vectors. for vectors.
2.92.9 Multiplying a vector by a scalarMultiplying a vector by a scalar
The product of a vector The product of a vector AA and a scalar is a new vector and a scalar is a new vector
with a direction similar to that of with a direction similar to that of AA if is positive but if is positive but
opposite to the direction of opposite to the direction of AA is is negative. is is negative.
The magnitude of the new vector A is equal to the The magnitude of the new vector A is equal to the
of A multiplied by the absolute value of , i.eof A multiplied by the absolute value of , i.e
kAjAiAA zyx
1.Scalar product (Dot product):1.Scalar product (Dot product):
The scalar product of two vectors A and B, denoted by A, B, is a scalar The scalar product of two vectors A and B, denoted by A, B, is a scalar quantity defined by; quantity defined by;
IfIf
The scalar product is The scalar product is
ExampleExample (2.7):(2.7):Show that A.B = B.A using following vectors; A =i – 2j Show that A.B = B.A using following vectors; A =i – 2j + 3k and B = 2 i + 3j – 2k .+ 3k and B = 2 i + 3j – 2k .Solution:Solution: Using equation (2.14) , we get;Using equation (2.14) , we get;A.B = (1A.B = (1×× 2) + (-2 2) + (-2 ×× 3) + (3 3) + (3× × -2) = 2 – 6 – 6 = - 10 -2) = 2 – 6 – 6 = - 10 B.A = (2B.A = (2×× 1) + (3 1) + (3 ×× -2) + (-2 -2) + (-2× × 3) = 2 – 6 – 6 = - 10 3) = 2 – 6 – 6 = - 10
Example (2.8):Example (2.8):Consider the Consider the two vectors A and B given in the previous two vectors A and B given in the previous example (example 2.7) , find 2A.B.example (example 2.7) , find 2A.B.Solution: Solution: 2A = 2i - 4j + 6k ,2A = 2i - 4j + 6k ,2A.B = (22A.B = (2×× 2) + (-4 2) + (-4 ×× 3) + (6 3) + (6× × -2) -2)
Example (2.9):Example (2.9):If If AA = 3i – 2j + 7k = 3i – 2j + 7k and and BB = 3 i + 2j– k , find the angle between the two = 3 i + 2j– k , find the angle between the two
vectors vectors AA and and BB..Solution:Solution:
AB
BAcos
7.314149
,9.7624949
B
andA
or
07.07.39.7
0.2cos
94)07.0(cos 1
Using equation (2.14) , we find; A.B = 9 – 4 – 7 = - 2.0 So we now have
2- Cross product (Vector Product):2- Cross product (Vector Product):
The vector product of two vectors A and B, written an , is a third vector The vector product of two vectors A and B, written an , is a third vector CC with a magnitude given by;with a magnitude given by;
Or equivalently
Comparing Figures 1 and 2, we notice that A x B = - B x A
