Chapter 1 Units, Physical Quantities, and Vectors 1.3 Standards and Units The metric system is also known as the S I system of units. (S I! Syst me International). A. Length The unit of length in the metric system is the meter. A meter is the distance that light travels in vacuum in (1 / 299,792,457) seconds. B. Mass The unit of mass in the metric system is the kilogram. A kilogram is the mass of a particular cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures near Paris, France. C. Time The unit of time in the metric system is the second. A second is the time required for 9,192,631,770 cycles of a 1
particular microwave radiation, which causes cesium atoms to undergo a transition between its two lowest energy states. The metric system is also known as the mks system of units. 1.4 Conversion of Units Units can be treated as algebraic quantities that can cancel each other. 1.6 Estimates and Order of Magnitude Fermi Questions Order of Magnitude: The point of such questions is that reasonable assumptions linked with 2
simple calculations can often narrow down the range of values within which an answer must lie. The order of magnitude refers to the power of 10 of the number that fits the value. To increase by an order of magnitude means to increase by a power of 10. Thus based on some reasonable assumptions, you approximate an answer to a given physical problem. Fermi Question 1: Estimate the number of revolutions that the tires of your car make when you drive your car from the parking lot at OCC to your home. Fermi Question 2: How many piano tuners are there in New York City? Fermi Question 3: How many hairs are on human head? etc 3
1.5 Uncertainty and Significant Figures When a value is not known precisely, the amount of uncertainty is usually called an error. Error represents uncertainty and has nothing to do with mistakes or sloppiness. A significant figure is a reliably known digit. When we say that a quantity has the value 3, we mean by convention that the value could actually be anywhere between 2.5 and 3.5. However, if we say that the value is 3.0, then we mean the value lies between 2.95 and 3.05. A. Significant Figures in Addition or Subtraction: The number of decimal places in the result should equal the smallest number of decimal places of any term in the sum (or subtraction). That is (1) 23.45 + 1.345 = 24.795! 24.80 (2) 56 34.56 = 21.44! 21 4
B. Significant Figures in Multiplication or Division: The number of significant figures in the final product is the same as the number of significant figures in the factor having the lowest number of significant figures. That is, (1) 123.56 x 7.89 = 974.8884! 975 (2) 564 / 0.0034 = 165882.352941! 1.7x10 5 1.7 Vectors and Vector Addition A. A Scalar quantity is a physical quantity that has magnitude only. Examples include: (a) mass m (in kilograms, kg) (b) time t (in seconds, s or sec) (c) temperature T (in Kelvin, K) (d) volume V (in cubic meters, m 3 ) (e) density ρ (in kg/m 3 ) (f) energy E (in Joules, J) (g) distance d (in meters, m) 5
(h) speed v (in meters per second, m/s) (i) electric charge q (in Coulombs, C) B. A vector quantity is a physical quantity that has both magnitude and direction. Examples include: (a) displacement! r ( in meters, m) (b) velocity v r (in meters per second, m/s) (c) acceleration a r (in meters per second square, m/s 2 ) (d) force F r (in Newtons, N) (e) linear momentum p r (in kg m/s) (f) angular momentum L r (in kg m 2 / sec) (g) electric field E r (in Volts per meter, V/m) (h) magnetic field B r (in Teslas, T) (i) torque! r (in Newton meter, N m) The displacement vector r r! (here they call it r A ) The displacement vector r r! of an object is defined as the vector whose magnitude is the shortest distance between the initial and final positions of the object, and whose 6
direction points from the initial position to the final position. 7
1.8 Components of a Vector A. The components of a vector are the projections of the vector along the axes of a rectangular coordinate system. A x! the x-component of vector A r. A y! the y-component of vector A r. A r or A denote the magnitude of vector A r. θ denotes the direction of vector A r. Any vector alone gives you the same physics as its two rectangular components together. 8
1.9 Unit Vectors A unit vector is a vector of magnitude one. î! a vector of magnitude one and in the direction of the positive x-axis. ĵ! a vector of magnitude one and in the direction of the positive y-axis. kˆ! a vector of magnitude one and in the direction of the positive z-axis. In general, any vector A r can be written in terms of its rectangular components and unit vectors. That is, r A = A iˆ + A ˆj + A kˆ so that its magnitude may also be written as r 2 2 2 A = A + A + A x x y y z z 9
Some Properties of Vectors A. Multiplication (or Division) of Vectors by a Scalar B. Addition of Vectors A r and B r (i) A r and B r in the same direction (ii) A r and B r in opposite directions 10
(iii) A r and B r in directions (iv) A r and B r in arbitrary directions... 1. Graphical method 2. Algebraic method 3. Head-to-Tail method C. Subtraction of Vectors A r and B r Think of A r - B r simply as A r + (-B r ) and proceed as in vector addition 1.10 Products of Vectors A. The Scalar (Dot) Product of Two Vectors One important thing to remember about the scalar (or dot) product between two vectors is that you multiply two vectors and the result is a scalar! There are two formulas worth remembering. These are: 11
(i) If you know the magnitudes of the two vectors (and the smallest angle φ between them) whose scalar product you wish to find, then use r A B r = A r B r cos(") (ii) If you know the two vectors in component form, that is, if you have r A = Ax î + Ay ĵ+ Azkˆ r B = B î + B ĵ B kˆ, x y + z and then use A r B r = A B + A B + x x y y A z B z From the first definition above, note that: 12
B. The Vector (cross) Product of Two Vectors When you multiply any two vectors A r and B r via the cross product, the result is a third vector C r r r r such that C = A! B. What are the magnitude and direction of the cross product C r? (a) The magnitude of C r is given by r C = A r B r sin (") where " is the smallest angle between the two vectors A r and B r. (b) The direction of C r is 1. perpendicular to both vectors A r and B r. This narrows down the choices for the direction of vector C r to two possible directions. 2. Use the right-hand-rule to choose between these two possible choices for the direction of C r. Simply point the 4 fingers of your right hand in the direction of the 13
first vector in the cross product, namely A r, and aim the palm of your right hand in the direction of the second vector in the cross product, namely B r. The thumb of your right hand then points in the direction of the cross product C r. The cross product can also be expressed in determinant form as r A " B r = ˆ i ˆ j ˆ k A x A y A z B x B y B z 14