Concepts in Physics. Friday, September 11th 2009

Transcription

1 Concepts in Physics Friday, September 11th 2009

2 Some notes Reminder: this coming Monday the tutorial (16:30-18:00) is mandatory for the LAB part of this course. Initial test solutions available on Monday Course notes and detailed syllabus available online on Monday (not sure where yet...) My (F511)

3 Chapter 1 Introduction What is physics? Concepts, models, and theories Units Power of ten notation and significant figures Reference frames and coordinate systems Chapter 2 Vectors Scalars and vectors Vector addition Components and unit vectors Scalar (dot) product

4 Ten notation, significant figures 10 0 = 1 one 10 1 =.1 tenth 10 1 = 10 ten 10 2 =.01 hundredth 10 2 = 100 hundred 10 3 =.001 thousandth 10 3 = 1000 thousand 10 6 = millionth 10 6 = million 10 9 = billion = trillion Your turn: 3min Express the following numbers in ten notation (no calculator): a) b) c) d) e) f) g) h) 1/2 i) 1/8

5 Express the following numbers in ten notation: a) b) c) d) e) f) g) h) 1/2 i) 1/8 a) *10 4 b) 5.67 *10-3 c) 2.856*10 9 d) 4*10 14 e) 3*10-3 f) *10 4 g) 4.3*10-11 h) 5*10-1 i) 1.25*10-1 Rules for using ten notation in calculations: 1.) = 10 5 To multiply powers of ten, add the exponents. 2.) 10 3 : 10 2 = 10 1 To divide powers of ten, subtract the exponents. 3.) (10 2 ) 3 = = 10 6 To raise a power of ten to a power, multiply the exponents.

6 Your turn: 3min Simplify the following expressions: a) 10 3 x 10 5 = b) 3*10 2 x 4*10 7 = c) 10 5 : 10 7 = d) 4*10 4 : 2*10 2 = e) (10 3 )4 = f) (10 2 ) 3 x 10 3 x (10 7 ) 2 g) 3.5*10 2 x 2*10 8

7 Your turn: 3min Simplify the following expressions: a) 10 3 x 10 5 = b) 3*10 2 x 4*10 7 = c) 10 5 : 10 7 = d) 4*10 4 : 2*10 2 = e) (10 3 )4 = f) (10 2 ) 3 x 10 3 x (10 7 ) 2 g) 3.5*10 2 x 2*10 8 Solutions: a) 10 3 x 10 5 = 10 8 b) 3*10 2 x 4*10 7 = 12*10 9 c) 10 5 : 10 7 = 10-2 d) (4*10 4 ) : (2*10 2 ) = 2*10 2 e) (10 3 ) 4 = 10 3 *10 3 *10 3 *10 3 = f) (10 2 ) 3 x 10 3 x (10 7 ) 2 = 10 6 x 10 3 x = g) 3.5*10 2 x 2*10 8 = 7*10 10

8 SIGNIFICANT DIGITS The number of significant digits in an answer to a calculation will depend on the number of significant digits in the given data, as discussed in the rules below. Approximate calculations (orderof-magnitude estimates) always result in answers with only one or two significant digits. When are Digits Significant? Non-zero digits are always significant. Thus, 22 has two significant digits, and 22.3 has three significant digits. With zeroes, the situation is more complicated: a.zeroes placed before other digits are not significant; has two significant digits. b.zeroes placed between other digits are always significant; 4009 kg has four significant digits. c.zeroes placed after other digits but behind a decimal point are significant; 7.90 has three significant digits. d.zeroes at the end of a number are significant only if they are behind a decimal point as in (c). Otherwise, it is impossible to tell if they are significant. For example, in the number 8200, it is not clear if the zeroes are significant or not. The number of significant digits in 8200 is at least two, but could be three or four. To avoid uncertainty, use scientific notation to place significant zeroes behind a decimal point: has four significant digits has three significant digits has two significant digits

9 Significant Digits in Multiplication, Division, Trig. functions, etc. In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc. Thus in evaluating sin(kx), where k = m -1 (two significant digits) and x = 4.73 m (three significant digits), the answer should have two significant digits. Note that whole numbers have essentially an unlimited number of significant digits. As an example, if a hair dryer uses 1.2 kw of power, then 2 identical hairdryers use 2.4 kw: 1.2 kw {2 sig. dig.} 2 {unlimited sig. dig.} = 2.4 kw {2 sig. dig.} Significant Digits in Addition and Subtraction When quantities are being added or subtracted, the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted. Example: 5.67 J (two decimal places) 1.1 J (one decimal place) J (four decimal place) 7.7 J (one decimal place)

10 When doing multi-step calculations, keep at least one more significant digit in intermediate results than needed in your final answer. For instance, if a final answer requires two significant digits, then carry at least three significant digits in calculations. If you round-off all your intermediate answers to only two digits, you are discarding the information contained in the third digit, and as a result the second digit in your final answer might be incorrect. (This phenomenon is known as "round-off error.") The Two Greatest Sins Regarding Significant Digits - What not to do 1.Writing more digits in an answer (intermediate or final) than justified by the number of digits in the data. 2.Rounding-off, say, to two digits in an intermediate answer, and then writing three digits in the final answer. YOUR TURN: Try these Exercises: 1. e kt =?, where k = yr -1, and t = 25 yr. 2. ab/c =?, where a = 483 J, b = J, and c = x + y + z =?, where x = 48.1, y = 77, and z = m - n - p =?, where m = 25.6, n = 21.1, and p = 2.43

11 Solutions: 1. e kt =?, where k = yr -1, and t = 25 yr. [Ans. 1.6] 2. ab/c =?, where a = 483 J, b = J, and c = [Ans. 2.27X10 3 J 2 ] 3. x + y + z =?, where x = 48.1, y = 77, and z = [Ans. 191] 4. m - n - p =?, where m = 25.6, n = 21.1, and p = 2.43 [Ans. 2.1] You will need these things for every single calculation you have to do - in this course or otherwise.

12 Reference Frames and Coordinate Systems A set of coordinate axes in terms of which position or movement may be specified or with reference to which physical laws may be mathematically stated. Also called reference frame. The concept of reference frame is applied to the sciences. People use frame of reference every day indeed, virtually every moment of their lives, without thinking about it. Rare indeed is the person who "walks a mile in another person's shoes" that is, someone who tries to see events from the viewpoint of another. (Which country, city, context of conversations,...) Physicists, on the other hand, have to be acutely aware of their frame of reference. Moreover, they must "rise above" their frame of reference in the sense that they have to take it into account in making calculations. For physicists in particular, and scientists in general, frame of reference has abundant "real-life applications."

13 There is no such thing as an absolute frame of reference that is, a frame of reference that is fixed, and not dependent on anything else. If the entire universe consisted of just two points, it would be impossible (and indeed irrelevant) to say which was to the right of the other. There would be no right and left: in order to have such a distinction, it is necessary to have a third point from which to evaluate the other two points. As long as there are just two points, there is only one dimension. The addition of a third point as long as it does not lie along a straight line drawn through the first two points creates two dimensions, length and width. From the frame of reference of any one point, then, it is possible to say which of the other two points is to the right. Of course, when someone is upside-down, the correct orientation of left and right is still fairly obvious. In certain situations observed by physicists and other scientists, however, orientation is not so simple. It then becomes necessary to assign values to various points, and for this, scientists use tools such as the Cartesian coordinate system.

14 Coordinate sytstem - cartesian A coordinate system in which the coordinates of a point are its distances from a set of perpendicular lines that intersect at an origin, such as two lines in a plane or three in space. Though it is named after the French mathematician and philosopher René Descartes ( ), who first described its principles, the Cartesian system owes at least as much to Pierre de Fermat ( ). Fermat, a brilliant French amateur mathematician amateur in the sense that he was not trained in mathematics, nor did he earn a living from that discipline greatly developed the Cartesian system. A coordinate is a number or set of numbers used to specify the location of a point on a line, on a surface such as a plane, or in space. In the Cartesian system, the x-axis is the horizontal line of reference, and the y-axis the vertical line of reference. Hence, the coordinate (0, 0) designates the point where the x-and y-axes meet. All numbers to the right of 0 on the x-axis, and above 0 on the y-axis, have a positive value, while those to the left of 0 on the x-axis, or below 0 on the y-axis have a negative value.

15 From Astronomy to Physics The human body is in an inertial frame of reference with regard to Earth, and hence experiences no relative motion when Earth rotates or moves through space. In the same way, if one were traveling in a train alongside another train at constant velocity, it would be impossible to perceive that either train was actually moving unless one referred to some fixed point, such as the trees or mountains in the background. Likewise, if two trains were sitting side by side, and one of them started to move, the relative motion might cause a person in the stationary train to believe that his or her train was the one moving. For any measurement of velocity, and hence, of acceleration (a change in velocity), it is essential to establish a frame of reference. Velocity and acceleration, as well as inertia and mass, figured heavily in the work of Galileo Galilei ( ) and Sir Isaac Newton ( ), both of whom may be regarded as "founding fathers" of modern physics. Before Galileo, however, had come Nicholas Copernicus ( ), the first modern astronomer to show that the Sun, and not Earth, is at the center of "the universe" by which people of that time meant the Solar System. In effect, Copernicus was saying that the frame of reference used by astronomers for millennia was incorrect: as long as they believed Earth to be the center, their calculations were bound to be wrong. Galileo and later Newton, through their studies in gravitation, were able to prove Copernicus's claim in terms of physics. At the same time, without the understanding of a heliocentric (Sun-centered) universe that he inherited from Copernicus, it is doubtful that Newton could have developed his universal law of gravitation. If he had used Earth as the center-point for his calculations, the results would have been highly erratic, and no universal law would have emerged.

16 Relativity For centuries, the model of the universe developed by Newton stood unchallenged, and even today it identifies the basic forces at work when speeds are well below that of the speed of light. However, with regard to the behavior of light itself which travels at 186,000 mi (299,339 km) a second Albert Einstein ( ) began to observe phenomena that did not fit with Newtonian mechanics. The result of his studies was the Special Theory of Relativity, published in 1905, and the General Theory of Relativity, published a decade later. Together these altered humanity's view of the universe, and ultimately, of reality itself. Einstein himself once offered this charming explanation of his epochal theory: "Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That's relativity." Of course, relativity is not quite as simple as that though the mathematics involved is no more challenging than that of a high-school algebra class. The difficulty lies in comprehending how things that seem impossible in the Newtonian universe become realities near the speed of light.

17 Some more Math... In preparation for the basics of mechanics, we have to understand the difference between Scalars and Vectors. We will come to the Physics on Monday, for the rest of today s lecture we will do some exercises.

18 Scalar and Vector Physics is a mathematical science. The underlying concepts and principles have a mathematical basis. While our emphasis will often be upon the conceptual nature of physics, we will give considerable and persistent attention to its mathematical aspect. The motion of objects can be described by words. Even a person without a background in physics has a collection of words which can be used to describe moving objects. Words and phrases such as going fast, stopped, slowing down, speeding up, and turning provide a sufficient vocabulary for describing the motion of objects. In physics, we use these words and many more. We will be expanding upon this vocabulary list with words such as distance, displacement, speed, velocity, and acceleration. As we will soon see, these words are associated with mathematical quantities which have strict definitions. The mathematical quantities which are used to describe the motion of objects can be divided into two categories. The quantity is either a vector or a scalar. These two categories can be distinguished from one another by their distinct definitions: Scalars are quantities which are fully described by a magnitude (or numerical value) alone. Vectors are quantities which are fully described by both a magnitude and a direction.

19 1. To test your understanding of this distinction, consider the following quantities listed below. Categorize each quantity as being either a vector or a scalar. Quantity a. 5 m b. 30 m/sec, East c. 5 mi., North d. 20 degrees Celsius e. 256 bytes f Calories

20 Vector Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude. This is shown in Panel 1.. If we denote one end of the arrow by the origin O and the tip of the arrow by Q. Then the vector may be represented algebraically by OQ. This is often simplified to just. The line and arrow above the Q are there to indicate that the symbol represents a vector. Note, that since a direction is implied,. Even though their lengths are identical, their directions are exactly opposite, in fact OQ = -QO. The magnitude of a vector is denoted by absolute value signs around the vector symbol: magnitude of Q = Q.

21 The operation of addition, subtraction and multiplication of ordinary algebra can be extended to vectors with some new definitions and a few new rules. There are two fundamental definitions. #1 Two vectors, A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points. #2 A vector having the same magnitude as A but in the opposite direction to A is denoted by -A, as shown in Panel 3.

22 We can now define vector addition. The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of B, as illustrated in Panel 4. This is sometines referred to as the "Tip-to-Tail" method. The operation of vector addition as described here can be written as C = A + B Vector subtraction is defined in the following way. The difference of two vectors, A - B, is a vector C that is, C = A - B or C = A + (-B).Thus vector subtraction can be represented as a vector addition. The graphical representation is shown in Panel 5. Inspection of the graphical representation shows that we place the initial point of the vector -B on the final point the vector A, and then draw a line from the initial point of A to the final point of -B to give the difference C.

23 Your turn... vectors "A" and "B" are added; vectors "C" and "B" are added; vectors "A" and "D" are added vectors "E" and "D" are added; vectors "E" and "F" are added; vectors "F" and "C" are added.