CHAPTER 1 Introduction to Physics

Transcription

1 CHAPTER 1 Introduction to Physics Physics (from the Greek, φυσικός (phusikos), "natural", and φύσις (phusis), "nature") is the science of Nature in the broadest sense. Physicists study the behavior and properties of matter in a wide variety of contexts, ranging from the sub-nuclear particles from which all ordinary matter is made (particle physics) to the behavior of the material Universe as a whole (cosmology). Physics: the study of the fundamental laws of nature these laws can be expressed as mathematical equations much complexity can arise from relatively simple laws. Physics is often described as the study of matter and energy. It is concerned with how matter and energy relate to each other, and how they affect each other over time and through space. Physicists ask the fundamental questions how did the universe begin, what is it made, how does it change and what rules govern its behavior? Physicists may be divided into two categories: experimental physicists and theoretical physicists. Experimental physicists design and run careful investigations on a broad range of phenomena in nature, often under conditions which are atypical of our everyday lives. They may, for example, investigate what happens to the electrical properties of materials at temperatures very near absolute zero ( 460 degrees Fahrenheit) or measure the characteristics of energy emitted by very hot gases. Theoretical physicists propose and develop models and theories to explain mathematically the results of experimental observations. Experiment and theory therefore have a broad overlap. Accordingly, an experimental physicist remains keenly aware of the current theoretical work in his or her field, while the theoretical physicist must know the experimenter's results and the context in which the results need be interpreted. It is also useful to distinguish classical physics and modern physics. Classical physics has its origins approximately four hundred years ago in the studies of Galileo and Newton on mechanics, and similarly, in the work of Ampere, Faraday, Maxwell and Oersted one hundred fifty years ago in the fields of electricity and magnetism. This physics handles objects which are neither too large nor too small, which move at relatively slow speeds (at least compared to the speed of light: 186,000 miles per second!). The emergence of modern physics at the beginning of the twentieth century 1

2 was marked by three achievements. The first, in 1905, was Einstein's brilliant model of light as a stream of particles (photons). The second, which followed a few months later, was his revolutionary theory of relativity which described objects moving at speeds close to the speed of light. The third breakthrough came in 1910 with Rutherford's discovery of the nucleus of the atom. Rutherford's work was followed by Bohr's model of the atom, which in turn stimulated the work of de Broglie, Heisenberg, Schrödinger, Born, Pauli, Dirac and others on the quantum theory. The avalanche of exciting discoveries in modern physics continues today. Given these distinctions within the field of physics experimental and theoretical, classical and modern it is useful to further subdivide physics into various disciplines, including astrophysics, atomic and molecular physics, biophysics, solid state physics, optical and laser physics, fluid and plasma physics, nuclear physics, and particle physics. Physicists study the behavior and properties of matter in a wide variety of contexts, ranging from the sub-nuclear particles from which all ordinary matter is made (particle physics) to the behavior of the material Universe as a whole (cosmology). Physics: the study of the fundamental laws of nature these laws can be expressed as mathematical equations much complexity can arise from relatively simple laws. Physics and Its Relation to Other Fields Physics is needed in architecture and engineering, medicine, chemistry, sports, and other fields such as: physiology, zoology, life sciences Physics is the science of measurement Physics is used in any field where problem solving is needed. That could be in medicine, law, business, etc. You must first define the problem at hand by examining the information available, and then determine the steps that must be followed in order to solve it. Physics does just that! Physics is the number one recommended science by the UC system, private colleges/universities and the Cal State Universities. Accuracy vs. Precision The dictionary definitions of these two words do not clearly make the distinction as it is used in the science of measurement. Accurate means "capable of providing a correct reading or measurement." In physical science it means 'correct'. A measurement is accurate if it correctly reflects the size of the thing being measured. 2

3 Precise means "exact, as in performance, execution, or amount. In physical science it means "repeatable, reliable, getting the same measurement each time." We can never make a perfect measurement. The best we can do is to come as close as possible within the limitations of the measuring instruments. Let's use a model to demonstrate the difference. Suppose you are aiming at a target, trying to hit the bull's eye (the center of the target) with each of five darts. Here are some representative patterns of darts in the target. Neither Precise nor Accurate This is a random like pattern, neither precise nor accurate. The darts are not clustered together and are not near the bull's eye. Precise, Not Accurate This is a precise pattern, but not accurate. The darts are clustered together but did not hit the intended mark. 3

4 Accurate, Not Precise This is an accurate pattern, but not precise. The darts are not clustered, but their 'average' position is the center of the bull's eye. Precise and Accurate This pattern is both precise and accurate. The darts are tightly clustered and their average position is the center of the bull's eye. Quantifying the Uncertainty The number we write as the uncertainty tells the reader about the instrument used to make the measurement. (We assume that the instrument has been used correctly.) Consider the following examples. Example 1: Using a ruler 4

5 The length of the object being measured is obviously somewhere near 4.3cm (but it is certainly not exactly 4.4cm). The result could therefore be stated as 4.3cm ± half the smallest division on the ruler. In choosing an uncertainty equal to half the smallest division on the ruler, we are accepting a range of possible results equal to the size of the smallest division on the ruler. length = 4 3cm ± 0 1cm Example 2: Using a Stop-Watch In general, we state a result as reading ± the smallest division on the measuring instrument Consider using a stop-watch which measures to 1/100 of a second to find the time for a pendulum to oscillate once. Suppose that this time is about 1s. Then, the smallest division on the watch is only about 1% of the time being measured. We could write the result as T = 1s ± 0.01s which is equivalent to saying that the time T is between 0.99s and 1.01s. This sounds quite good until you remember that the reaction-time of the person using the watch might be about 0.15s. Now considering the measurement again, with a possible 0.15s at the starting and stopping time of the watch, we should now state the result as T = 1s ± ( )s In other words, T is between about 0.7s and 1.3s. Conclusions If we accept that an uncertainty (sometimes called an indeterminacy) of about 1% of the measurement being made is reasonable, then a) a ruler, marked in mm, is useful for making measurements of distances of about 10cm or greater. b) a manually operated stop-watch is useful for measuring times of about 30s or more (for precise measurements of shorter times, an electronically operated watch must be used). Percent Error Physics students often assume that each measurement that they make in the laboratory is true and accurate. Likewise, they often assume that the values that they derive through experimentation are very accurate. However, sources of error often prevent students from being as accurate as they would like. Percent error calculations are used to determine how close to the true values, or how accurate, their experimental values really are. The value that the student comes up with is usually called the observed value, or the experimental value. A value that can be found in reference tables is usually called the true value, or the accepted value. The percent error can be determined when the true value is compared to the observed value according to the equation below: (observed value - true value) Percent error = x 100 true value 5

6 Example. 3 A student measures the mass and volume of a piece of copper in the laboratory and uses his data to calculate the density o the metal. According to his results, the copper has a density of 8.37 g/cm 3. Curious about the accuracy of his results, the student consults a reference table and finds that the accepted value for the density of copper is 8.92 g/cm 3. What would be the student's percent error? Solution - Step 1. Determine which values are known. The students result, or the observed value = 8.37 g/cm 3. The accepted, or true value = 8.92 g/cm 3 Step 2. Substitute these values in the percent error calculation, as shown below: (observed value - true value) Percent error = x 100 true value 3 3 (8.37 g/cm g/cm ) Percent error = x 100 = -6.17% g/cm Note that the negative sign does not mean that the error was less than zero, which would be impossible. It shows that the student's calculated value was actually too low. The Atlantic/Pacific Rule for Determining Significant Figures The Atlantic-Pacific Rule states: If a decimal point is Present, ignore zeros on the Pacific (left) side. If the decimal point is Absent, ignore zeros on the Atlantic (right) side. Everything is significant Pacific Atlantic If a decimal point is present, count from this side starting with the first non-zero digit and keep counting until you run out of digits. Side Side If a decimal point is absent, count from this side starting with the first non-zero digit and keep counting until you run out of digits. This rule is simple to use and to remember and it lets you count significant digits without having the slightest idea what they are. 6

7 Examples using the Atlantic/Pacific Rule: Number Atlantic/Pacific Rule The decimal is Present: start on the Pacific side (left) and ignore the zeros. The first non-zero number is 2. Every number including zeros from that point on is significant. Therefore there are 4 sig figs The decimal point is Present: start on the Pacific side (left) and ignore the zeros. The first non-zero number is 4. Every number including zeros from that point on is significant. Therefore there are 4 sig figs The decimal point is Present: start on the Pacific side (left). There are starting zeros so all of the numbers are significant. There are 5 sig figs The decimal point is Absent: start on the Atlantic side (right) and ignore the zeros. The remaining numbers are significant. There are 3 sig figs. Scientific Notation Scientific notation is used to express very large or very small numbers. A number in scientific notation is written as the product of a number (integer or decimal) and a power of 10. The number has one digit to the left of the decimal point. That number must be 1 or greater but less than 10. The power of ten indicates how many places the decimal point was moved. For example the decimal number written in scientific notation would be 6.5x10-7 because the decimal point was moved 7 places to the right to form the number 6.5. The number written in scientific notation would be 2.9x10 8 because the decimal point was moved 8 places to the left. A decimal number smaller than 1 can be converted to scientific notation by decreasing the power of ten by one for each place the decimal point is moved to the right. Scientific notation numbers may be written in different forms. The number 6.5x10-7 could also be written as 6.5e-7, and the number 2.9x10 8 could also be written as 2.9e+8. The "Best-Fit" Line The "best-fit" line is the straight line which passes as near to as many of the points as possible. By drawing such a line, we are attempting to minimize the effects of random errors in the measurements. For example, if our points look like this 7

8 Notice that the best-fit line does not necessarily pass through any of the points plotted. Error Bars Instead of plotting points on a graph we sometimes plot lines representing the uncertainty in the measurements. These lines are called error bars and if we plot both vertical and horizontal bars we have what might be called "error rectangles", as shown below The best-fit line could be any line which passes through all of the rectangles. x was measured to ±0 5s y was measured to ±0 3m Measuring the Slope at a Point on a Curved Graph Usually we will plot results which we expect to give us a straight line. If we plot a graph which we expect to give us a smooth curve, we might want to find the slope of the curve at a given point; for example, the slope of a displacement against time graph tells us the (instantaneous) velocity of the object. To find the slope at a given point, draw a tangent to the curve at that point and then find the slope of the tangent in the usual way. This method is illustrated on the graph on the next page. A tangent to the curve has been drawn at x = 3s. The slope of the graph at this point is given by Dy/Dx = (approximately) 6ms-1. 8

9 Metric (SI) Prefixes Quantity Unit Abbreviation Prefix Abbreviation Value Length meter m tera T Time second s giga G 10 9 Mass kilogram kg mega M 10 6 Electric current ampere A kilo k 10 3 Temperature kelvin K hecto h 10 2 Volume cubic meter m 3 deka da 10 1 Area square meter m 2 deci d 10-1 Force newton N centi c 10-2 Energy/work joule J milli m 10-3 Power watt W micro 10-6 Pressure pascal Pa nano n 10-9 Frequency hertz Hz pico p Units, Standards, and the SI System We will be working in the SI system, where the basic units are kilograms, meters, and seconds. Other systems: cgs; units are grams, centimeters, and seconds. British engineering system has force instead of mass as one of its basic quantities, which are feet, pounds, and seconds. 9

10 Metric Conversions There are several methods available to convert from one metric unit to another. For example, converting from millimeters to hectometers. One such method is to use the following phrase: (K H Da m D C M) King Henry Died many Deaths Counting Money. K = kilo H = hector Da = deca m = many (liters, grams, meters) K H Da m D C M L g m 30 centimeters = hectometers Since centimeters is on the right and hectometers on the left we must move the decimal to the left. We move four steps (don t count the centi spot) 30.0 centimeters = hectometers 3 kilograms = milligrams Since kilograms is on the left and milligrams is on the right we must move the decimal to the right. We move five steps (don t count the kilo spot) 3.0 kilograms = milligrams Example. 4 How many significant figures are there in: (a) (b) 28 (c) 2.8 x 10-4 (d) 2.80 x 10-5 Solution: (a) 4 (b) 2 (c) 2 (d) 3 Example. 5 A storm drops 0.01 m of rain in California which covers about 70 square kilometers (10 8 m 2 ). Estimate the number of raindrops. Assume that the diameter of each raindrop is 4 mm. (r = 2mm) V Ad (10 m )(0.01 m) 10 m water (0.002 ) Vdrop r m m Vwater 10 m number of raindrops V 10 m drop 14 10

11 DEFINITION OF SIN, COS, AND TAN The sine, cosine, and tangent of an angle are numbers without units, because each is the ratio of the lengths of two sides of a right triangle. Example 6. On a sunny day, a tall tower casts a shadow that is 60.m m long. The angle between the sun s rays and the ground is 50.0 o. Determine the height of the tower. o 50.0 and h 60m h tan h o a o h h tan (60.0 m)(tan 50 ) o a (60.0 m)(1.19) 71.4m a END OF CHAPTER 1 11