Measurements and Vectors Chapter (2) Measurements and Vectors

2.1 Unit and Standards: The measurement of any quantity is made relative to a particular standard or unit, and this must be specified a long with the numerical value of the quantity. For example, we can measure length in units such as inches, feet, or miles, or in the metric system in centimeters, meters, or kilometers. To specify that the length of a particular object is 18.6 is meaningless. The unit must be given; for clearly, 18.6 meters is very different from 18.6 millimeters.

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 long one meter or one second is. It is important that standards be chosen that are readily reproducible so that anyone needing to make a very accurate measurement can refer to the standard in the laboratory. The laws of physics are expressed in terms of basic quantities that require a clear definition. In mechanics, the three basic quantities are length (L), mass (M), and time (T). Several systems of units are used for these three quantities: The most common system among them is the system International (French for International system) abbreviated SI. The other systems are the cgs system and the British engineering system. (Table 2-1) show the three systems and their standard units for mass, length, and time.

Table 2 – 1 Units of Length, Mass, and Time in Different System Time Mass Length Systems Second (s) Kilogram (kg) Meter (m) SI Gram (gm) Centimeter (cm) cgs Slug Foot (ft) British

Some important definitions for SI units: The meter (m) is the length of path traveled by light in vacuum during a time interval of 1/229,792,458 of a second. The kilogram (kg) is the mass of a specific platinum – iridium alloy cylinder kept at the International Bureau of weights and Measures at sevres, France. The second (s) is now defined as 9192631770 times the period of vibration of radiation from the cesium atom.

Prefixes: Sometimes the numerical value of our physical quantities is too large or in the contrary is too small, which makes the numbers bothersome to is 350000 meter, or the mean radius of earth is 637000000 meter. Look, these numbers are not easy to carry and deal with. So, it is better prefixes, which are abbreviations come in front of the units to make them handy. See (Table 2 – 2) for the commonly used a abbreviations in the field of medicine.

Table 2 – 2 : Prefixes for Powers of Ten in (SI) Abbreviation Prefix Power F femto 10-15 P pico 10-12 N nano 10-9  micro 10-6 M milli 10-3 C centi 10-2 D deci 10-1 K kilo 103 mega 106 G giga 109 T tera 1012 peta 1015

2.2Dimensional analysis Dimension is physics denotes the physical nature of a quantity whether it is a [length] = L ; [mass] = M ; [time] = T, and all other quantities are derivable from these quantities. For example the dimensions of speed is length divided by time and denoted by . As another example, the dimensions of area A are [A] = L2. The dimensions of other physical quantities are listed in (Table 2 -3).

Table 2 – 3 Dimensions and Units of some Physical Quantities Unit (SI, cgs, British) Dimension Quantity M2, cm2, ft2 L2 Area M3, cm3, ft3 L3 Volume m/s, cm/s , ft/s L/T Velocity m/s2, cm/s2, ft/s2 L/T2 Acceleration Kgm/s, gcm/s , slug ft/s ML/T Momentum Netwon(N), dyne, pound (Ib) ML/S Force Joul (J), erg, ft.Ib ML2/T2 Energy Watt(W), erg/s, hosepower (hp) ML2/T3 Power

Solution: Since , , and Then Example 2- 1: Show that the equation ( ) is dimensionally correct, where x is the displacement, is an initial velocity, a is the acceleration, and t is the time. Solution: Since , , and Then

Example 2 – 3: 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 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 for the acceleration. Solution: Let us take a to be Where k is a dimensionless constant of proportionality knowing the dimensions of a , r, and v. Since , ,

Then, Equation the powers of the left hand side units to the right hand side units, gives; n + m = 1 , and m = 2 therefore, so, we can write the acceleration expression as

2.3Vector and Scalar Quantities: Some physical quantities are scalar quantities whereas others are vector quantities. A scalar quantity is completely specified by a single value with an appropriate unite and has no direction. Other examples of scalar quantities are volume, mass, speed, and time in travels. The rules of ordinary arithmetic are used to manipulate scalar quantities.

A vector quantity is completely specified by a number and appropriate units plus a direction. Another example of a vector quantity is displacement. Suppose a particle moves from point (A) to some point (B) along a straight path, as shown in Fig. 2.1. We represent this displacement by drawing an arrow from (A) to (B), with the tip of the arrow pointing away from the starting point

The vector quantity will be distinguished from the scalar quantity by typing it in boldface, like A. I write handing the vector, quantity is written with an arrow over the symbol, such as, . The magnitude of the vector a will be denoted |A|, or simply the italic type A. the magnitude of a vector has physical units, such as meters for displacement or meters per second for velocity. The magnitude of a vector is always a positive number.   Quick Quiz 2.1: Which of the following are vector quantities and which are scalar quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass

y o X 2.3 Properties of Vectors Equality of two vectors: Any two vectors are said to be equal if they have the same magnitude and point in the same direction. For example, all vectors in Fig. 2.2 are equal even though they have different starting points. y o X

(II) Adding Vectors: Graphical Method (Geometrical Method): To add vector B to vector A, first draw vector A, with its magnitude represented by a convenient scale, on graph paper and then draw vector B to the same scale with its tail starting from the tip of A, as shown in Fig.2.3a. first translate the vector B until the tail of the vector B touches the head of the vector A, Figure 2.3a. The resultant R = A + B is the vector draw from the tail of A to the tip of B. this procedure is known as the triangle method of addition. When two vectors are added, the sum is independent of the order of the addition. This can be seen from the geometric construction in Fig.(2.4) and is known as the commutative law of addition: A + B = B + A

(III)Subtracting Vectors:

Quick quiz 2.2 The magnitudes of two vectors A and B are A = 12 units and B = 8 units. Which of the following pairs of numbers represents the largest and smallest possible values for the magnitude of the resultant vector R = A + B? (a) 14.4 units, 4 units (b) 12 units, 8 units, (c) 20 units, 4 units (d) none of these answers. Quick quiz 2.3 if vector B is added vector A, which two of the following choices must be true in order for the resultant vector to be equal to zero? (a) A and B are parallel and in the same direction. (b) A and B are parallel and in opposite directions. (c) A and B have the same magnitude. (d) A and B are perpendicular.

(V) Multiplying Vector If vector A is multiplied by a positive scalar quantity m, then the product mA is a vector that has the same direction as A and magnitude mA. If vector A is multiplied by a negative scalar quantity – m, then the product – mA is directed opposite A. for example the vector 4A is four times as long as A and points in the same direction as A; the vector - 1/3 A is one-third the length of A and points in the direction opposite A.

Scalar multiplication A scalar quantity(i. e Scalar multiplication A scalar quantity(i.e. a number) can alter the magnitude of a vector but not its direction. Example - In the diagram(above) the vector of magnitude X is multiplied by 2 to become magnitude 2X. If the vector X starts at the origin and ends at the point (4,4), then the vector 2X will end at (8,8).

, 2.4 Unit Vector Vector quantities often are expressed in terms of unit vectors. A unit vector is a dimensionless vector having a magnitude of exactly 1. Unit vectors are used to specify a given direction and have no other physical significance. They are used solely as a convenience in describing a direction in space. In Cartesian coordinates we shall use the symbols ,

2.5 Components of Vectors A = Ax + Ay (2.2) Another Any vector A in a plane can be represented by the sum of two vectors, one parallel to the x –axis (Ax), and the other parallel to the y – axis (Ay) as shown A = Ax + Ay (2.2) Another

Where Ax and Ay are the x – component and the y – component of the vector A, respectively. From fig 2.8 it is clear that Ax = A cosθ (2 .4) And Ay = A sinθ (2 .5)   The magnitude and direction of the vector A are given by

Note that the signs of the components Ax and Ay depend on the angle θ Note that the signs of the components Ax and Ay depend on the angle θ. For example, it θ = 120°, then Ax is negative and Ay is positive. If θ = 225°, then both Ax and Ay are negative. In general any vector A can be resolved into three components as

2.7 Adding Vectors We use the components to add vectors when the graphical method is not sufficiently accurate. The method for adding the vectors as follow: Resolve each vector into its components according to a suitable coordinate axes. Add, algebraically, the x-components of the individual vectors to obtain the x-component of the resultant vector. Do the same thing for the other components, i.e., if

The resultant vector R = A + B is therefore , R = + the resultant vector are;  

The vector sum is R = A + B it is clear that, Example 2.3: Find the sum of two vectors A and B lying in the xy plane and given by: A = (2.0 + 2.0 )m and B = (2.0 + -4.0 ) m. Then find the magnitude and direction of the vector sum. Solution: The vector sum is R = A + B it is clear that, Ax = 2 , Ay = 2 , Bx = 2 , By = -4 So, from equation (2.9); R = (2.0 + 2.0) m + (2.0 – 4.0) m = (4.0 - 2.0 ) m Or Rx = 4.0m , Ry = - 2.0 m The magnitude of R is The direction of R   The result vector is below the x- axis by 27° degrees (clockwise rotation).

Example and Find the sum of two vectors A and B given by Solution: Note that Ax=3, Ay=4, Bx=2, and By=-5 The magnitude of vector R is The direction of R with respect to x-axis is

Example (2.4): A particle undergoes three consecutive displacements:d1= (15 + 30 +12 )cm, d2= (23 - 14 - 5.0 )cm and d3= (-13 + 15 )cm. Find the components of the resultant displacement and its magnitude. Solution: Let, R = d1 + d2 + d3 = (15 + 23 – 13) + (30 – 14 – 15) + (12 – 5.0 + 0) = (25 + 31 + 7.0 ) Then, the resultant displacement has components Rx = 25cm , Ry = 31cm , Rz = 7.0cm , its magnitude is

Example (2. 5): Consider two vectors A = 4. 0 - 3. 0 and B = -2. 0 + 7 Example (2.5): Consider two vectors A = 4.0 - 3.0 and B = -2.0 + 7.0 Calculate: A + B; A – B ; ; ; and the of A + B and A – B. Solution: A + B = (4.0 + (-2.0)) + (- 3.0 + 7.0 ) = 2.0 + 5.0 at the beginning find – B then add it to A, that is -B = 2.0 - 7.0 . A + (-B) = (4 + 2) + (-3 – 7) =6 - 10 The direction are

Example (2-6):A particle undergoes the following consecutive displacements: 3.5m southeast, 2.5m east, and 6m north. What is the resultant displacement? Solution: Figure 2.9 example 2.6 , The displacement d1 southeast, d2 west, and d3 north. If we denote the three displacements by d1 , d2 , and d3 respectively, we get the vector diagram shown in figure 2.9 . According to the coordinates system chosen, the three vectors can be written as d1 = (3.5 cos45) i - (3.5 sin 45) j = (2.5i - 2.5j )m, d2 = 2.5m, d3 = 6m .  now, R = d1 + d2 + d3 = (2.5 + 2.5 + 0)i + (2.5 + 0 + 6) j = (5 i + 8.5j)m R d3 d1 d2

2.8 Multiplication of Vectors: Like scalars, vectors of different kinds can be multiplied by one another to generate quantities of new physical dimensions. Because vectors have direction as well as magnitude, the vector multiplication cannot follow exactly the same rules as the algebraic rules of scalar multiplication. We must establish new rules of multiplication for vectors.

2.9 Multiplying a vector by a scalar The product of a vector A and a scalar is a new vector with a direction similar to that of A if is positive but opposite to the direction of A is is negative. The magnitude of the new vector A is equal to the of A multiplied by the absolute value of , i.e

1.Scalar product (Dot product): The scalar product of two vectors A and B, denoted by A, B, is a scalar quantity defined by; If The scalar product is

Example (2. 7): Show that A. B = B Example (2.7): Show that A.B = B.A using following vectors; A =i – 2j + 3k and B = 2 i + 3j – 2k . Solution: Using equation (2.14) , we get; A.B = (1× 2) + (-2 × 3) + (3× -2) = 2 – 6 – 6 = - 10 B.A = (2× 1) + (3 × -2) + (-2× 3) = 2 – 6 – 6 = - 10   Example (2.8): Consider the two vectors A and B given in the previous example (example 2.7) , find 2A.B. Solution: 2A = 2i - 4j + 6k , 2A.B = (2× 2) + (-4 × 3) + (6× -2)

Example (2.9): If A = 3i – 2j + 7k and B = 3 i + 2j– k , find the angle between the two vectors A and B. Solution: Using equation (2.14) , we find; A.B = 9 – 4 – 7 = - 2.0 So we now have

2- Cross product (Vector Product): The vector product of two vectors A and B, written an , is a third vector C with a magnitude given by;

Or equivalently

Comparing Figures 1 and 2, we notice that  A x B = - B x A