## P-1 CALCULATION EXERCISESInstructor's notes and "solutions."
Assuming the students have read the assignment it will only be necessary to do for them a few new examples of mathematical operations with exponentials, and answer any questions they may have over the assignment. If the students did not have the time or the materials to read the assignment, you will have to at least summarize the theory of exponential notation and significant figures. This may take half an hour of laboratory time.
Three sets of problems are provided to be done during lab, A, B, and C. Section D has problems requiring library research, to be done as homework. The instructor may devise additional sets in the same spirit. These may be alternated in successive years, so students are less likely to use solutions of a friend who took the course the previous year.
Request that students
Express all answers in exponential notation in standard form, rounded to the appropriate number of significant figures.
Answer: 207.06 m (2) Calculate the number of cubic centimeters in a cubic meter. (The answer is
Answer: (100 cm/m) (3) Use the result of problem 2 to calculate the number of cubic centimeters
in the room described in problem 1. Then check this answer by first converting the room
dimensions to centimeters, Answer: 207 x 10 (4) How many cubic centimeters are in a cubic kilometer? Since this is a large number, express it in exponential notation. (10 (5) One light year is the distance light travels in one year. The speed of light is
2.9979250 x l0 One year = 3.156 x 10 Answer: 9.46 x 10
(A1) For expressing very large numbers, the unit 1 duo = 10 Example: (2 (A2) The For angles less than 0.14 radian (8°) the approximation is good to three significant figures. (A3) Find the sine of 0.0001 degree to four significant figures. How accurately can you determine this? sin(θ) ≅ θ
= (0.0001)2π/360 = 1.745 x 10 The accuracy is limited by the value of π used and the calculating device. The student also might suggest use of a power series expansion of sinθ, in which the accuracy is limited only be one's time and patience. (A4) An article about prime numbers in the Scientific American magazine of March, 1964 mentions that "The Computer Division of Los Alamos has a magnetic tape on which 20 million prime numbers are recorded." Suppose we wanted to print all of these numbers in a book. A moderate speed computer printer can print about 200 lines per minute. If one prime number were printed per line, how long would it take to print all those prime numbers? Express the answer in days, or years, whichever is more appropriate. If the printout has 60 numbers per sheet of paper, how thick would be the stack of sheets? If reproductions were made and sold at typical "textbook" prices, how much would a copy cost? It would take 10 (A5) If we made a scale model of the earth the size of a bowling ball, how high would
Mount Everest be on the model? The earth's radius is 6.4 x 10 The circumference of the earth is 2πr. So taking
ratio of height to circumference, h/27 in = (8.85×10 h = 27×8.85×10 h = That's 2.54 × 5.94 × 10 Our scale model of the earth the size of a bowling ball smoother than an actual bowling ball!
(Bl) The estimated amount of hydrogen in interstellar space is one atom per cubic centimeter. Our galaxy is shaped somewhat like a round disk, with a diameter of 300,000 light years and a thickness of 35,000 light years. Approximately how many atoms of interstellar hydrogen are there in our galaxy? (The data is only accurate to one significant figure, so express the answer accordingly.) V = πr2h = 3.14 x (1.5 x
10 (B2) 1000 grams mass of water at room temperature (20°C) has a volume of 1000 cubic centimeters. What is the mass of one cubic centimeter of water at this temperature? [Look this up in a reference book.] This is often taken to be approximately 1 gm/cm3. What percent discrepancy will result from use of the approximate value? Will this be significant in our work?. At room temperature (20°C) the density of water is actually 0.99823 gm/cm3. The
discrepancy is 1 - 0.99823 = 0.00177 or approximately 0.002 which is (B3) Masses of atoms are measured using a unit called the unified atomic mass (u)
defined so that the mass of the isotope Answer: (1.660559 x 10 (B4) A flea can jump over a foot upward. If you could jump that well, relative to your size, how high could you jump? From the table, the flea's size is about 1mm = 10 Easier: h = (1 ft)(1.8 meter)/(0.001 m) = 1800 ft. That's over (B5) The theory of the moon according to the analytic methods of Charles Delaunay (1816-1872) contains one equation exceeding 170 pages in length. If the pages were the typical size of books of that era (9 x 6 inches), and laid side-by side in the hallway, how many feet long would the hallway have to be? Answer: 170 × 0.5 ft =
(C1) In the chart on page 3 of this experiment, the distance to the farthest known galaxy is shown. Check the consistency of this value with the value given in the footnote on page 3. (C2) The unit of mass in the metric system was chosen so that the density of water would
be 1 gm/cm At room temperature (20°C) the density of water is actually 0.99823 gm/cm3. The discrepancy is 1 - 0.99823 = 0.00177 or approximately 0.002 which is 0.2 %. This is not likely to be important in a freshman physics lab! (C3) If you drew a "picture" of an atom one inch in diameter, how large would you have to draw the nucleus? Compare your conclusion with pictures of atoms you have seen in textbooks. Consider drawing a scale model of the solar system with the orbit of Pluto five inches in diameter. How large would the sun be on this picture? Compare with the pictures you have seen. Nucleus size: 10 (C4) In elementary books, the atom is often described as a "miniature solar system." Consult the chart of page 3 of this experiment along with the results of problem C3 to determine whether this statement is an accurate analogy, at least with respect to the sizes of the components. The atomic nucleus is smaller relative to atomic size than the sun is relative to the solar system size. It is about a factor of 10 smaller. Therefore the statement "atoms are mostly empty space" is essentially correct. A related question which could be asked is: "What fraction of the atomic volume is
occupied by its nucleus? Answer: 1/10 (C5) The German philosopher Friedrich Wilhelm Nietzsche (1844-1900) said "The earth has a skin and that skin has diseases; one of its diseases is called man." Physicist Albert Abraham Michelson (1852-1931) told his students (at the University of Chicago) that mankind as a whole was rather insignificant on a universal view, merely "a skin disease upon the face of the earth." Consult the chart of sizes to determine whether this statement is an accurate comparison of relative sizes. A 6 ft man is 1.8 meters high. But a better estimate of "average" size is about 1 meter.
The earth's diameter is 12.76 x10 A microbe's size is about 2 x 10 It is a fairly good comparison, to within a factor of 10. But should one compare linear dimensions? Why not compare the surface areas? Then the comparison would only be good to a factor of 100. See problem A6 for a comparison using the numbers of microbes on man and man on earth.
(D1) Gerald Holton, Derek J. de Solla Price end others have estimated that "80 to 90 per cent of all scientists that have ever been, are alive now." Should this surprise anyone? How would this ratio compare with other professions? With the general population? Discuss, citing appropriate data from library research. Comments: Price discusses the growth of scientific activity in his book In June 1989 the population of the world was nearly 6 billion. Other benchmark values (from various sources): See Arthur H. Westling, "A Note on How Many Humans That Have Ever Lived."
Doubling periods are indicated by square brackets above. The total number of people who have ever lived has been estimated as 69 billion (through
1960). [See Keyfitz, N. "How many people have lived on the earth?" Westling improves the calculations, taking into account the change in average life span (from 20 years during early history to 50 in recent years). He concludes that 50 billion people have lived (through 1980). There were 4.4 billion people living on earth in 1980, about 9% of those who have ever lived. In 1997 the world population is estimated to be 5.8 billion (U. S. Census Bureau). This is growing at an estimated rate of 1.4% per year in 1996 (CIA World Factbook, 1996). From "Ask Marilyn" by Marilyn vos Savant ( A recent source gives the 105 billion as the number of people who have ever lived, and
adds that 5.5% of all people ever born are alive today. [Haub, Carl. So, to say that 8 or 9 of every 10 scientists who have ever lived is alive today is indeed
remarkable, for considerably less than 1 in 10 of Price concludes: "It must be recognized that the growth of science is something very much more active, much vaster in its problems, than any other sort of growth happening in the world today. ...science has been growing so rapidly that all else, by comparison, has been stationary." (D2) Theodor Rosebury, in his fascinating book Rosebury: A human being has an average of 5 x 10 Therefore, there are 10 Information Please Almanac, 1974: The [The entire area of the earth is 196,949,970 mi This gives 27.75 m We'll use the linear size of the organism to estimate the surface area it occupies, as a basis for comparison. A 6 ft tall man is 1.8 meters high. But a better measure of "average" size is about 1
meter. A microbe's size is about 2 x 10 The microbe size/area ratio: 2 x 10 Man's size/area ratio: (1 m)/(27.75 m The comparison is very good! (D3) (a) Find out the top speed of a garden snail moving across the ground in search of juicy leaves to munch on. (b) Express this in centimeters per second. (c) Express this also in the units furlongs/fortnight. The garden snail is the fastest land snail, with a speed of 0.03 miles/hr. (An internet
source gave it as 0.0313, but all these figures are unlikely to be significant.)
Someone quoted the Its speed is 1.34 cm/sec or 32 inches/minute or 53 yards/hr.
1 furlong = 1/8 mile = 220 yard. 1 fortnight = 14 days = 2 weeks. (0.03 miles/hr)(8 furlong/mile)(24 hr/day)(14 day/fortnight) = 80.64 furlong/fortnight. 80 furlong/fortnight is good enough. This document prepared in 1992, and revised in 2004. |