Chapter 1 Problems 21

Problems

Section 1.3

1. What is your height in feet? In meters?

2. With a ruler, measure the thickness of this book, excluding the cover. Deduce the thickness of each of the sheets of paper making up the book.

3. A football field measures 100 yd X 53 1/3 yd. Express this in meters.

4. Express the last four entries in Table 1.1 in inches.

5. Express the following fractions of an inch in millimeters: 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64 in.

6. As seen from the Earth, the Sun has an angular diameter of 0.53°. The distance between the Earth and the Sun is 1.5 X 1011 m. From this, calculate the radius of the Sun.

7. Analogies can often help us to imagine the very large or very small distances that occur in astronomy or in atomic physics.

(a) If the Sun were the size of a grapefruit, how large would the Earth be? How far away would the nearest star be?

(b) If your head were the size of the Earth, how large would an atom be? How large would a red blood cell be?

8. One of the most distant objects yet observed by astronomers is the quasar Q1208+1011, at a distance of 12.4 billion light-years from the Earth. If you wanted to plot the position of this quasar on the same scale as the diagram at the top of page 9 of the Prelude, how far from the center of the diagram would you have to place this quasar?

9. On the scale of the second diagram on page 6 of the Prelude, what would have to be the size of the central dot if it is to represent the size of the Sun faithfully?

*10. The Earth is approximately a sphere of radius 6.37 X 106 m. Calculate the distance from the pole to the equator, measured along the surface of the Earth. Calculate the distance from the pole to the equator, measured along a straight line passing through the Earth.

*11. A nautical mile (nmi) equals 1.51 mi, or 1852 m. Show that a distance of 1 nmi along a meridian of the Earth corresponds to a change of latitude of 1 minute of arc.

**12. A physicist plants a vertical pole at the waterline on the shore of a calm lake. When she stands next to the pole, its top is at eye level, 175 cm above the waterline. She then rows across the lake and walks along the waterline on the opposite shore until she is so far away from the pole that her entire view of it is blocked by the curvature of the surface of the lake, that is, the entire pole is below the horizon (Figure 1.18). She finds that this happens when her distance from the pole is 9.4 km. From this information, deduce the radius of the Earth.


Section 1.4

13. What is your age in days? In seconds?

14. The age of the Earth is 4.5 X 109 years. Express this in seconds.

*15. Each day at noon a mechanical wristwatch was compared with WWV time signals. The watch was not reset. It consistently ran late, as follows: June 24, late 4s; June 25, late 20s; June 26, late 34s; June 27, late 51s.

(a) For each of the three 24-hour intervals, calculate the rate at which the wristwatch lost time. Express your answer in seconds lost per hour.

(b) What is the average of the rates of loss found in part (a)?

(c) When the wristwatch shows 10h30m on June 30, what is the correct WWV time? Do this calculation with the average rate of loss of part (b) and also with the largest of the rates of loss found in part (a). Estimate to within how many seconds the wristwatch can be trusted on June 30 after the correction for rate of loss has been made.

*16. The navigator of a sailing ship seeks to determine his longitude by observing at what time (Greenwich Mean Time) the Sun reaches the zenith at his position (local noon). Suppose that the navigator's chronometer is in error and is late by 1 s compared to Greenwich Mean Time. What will be the consequent error of longitude (in minutes of arc)? What will be the error in position (in kilometers) if the ship is on the equator?


Section 1.5

17. What is your mass in pounds? In kilograms? In atomic mass units?

18. What percentage of the mass of the Solar System is in the planets? What percentage is in the Sun? Use the data given in the table printed on the endpapers of the book.

19. The atomic mass of fissionable uranium is 235.0 g. What is the mass of a single uranium atom? Express your answer in kilograms and in atomic mass units.

20. The atom of uranium consists of 92 electrons, each of mass 9.1 X 10-31 kg, and a nucleus. What percentage of the total mass is in the electrons and what percentage is in the nucleus of the atom?

21. How many water molecules are there in 1 gallon (3.79 liters) of water? How many oxygen atoms? Hydrogen atoms?

*22. (a)How many molecules of water are there in one cup of water? A cup is about 250 cm3.

(b) How many molecules of water are there in the ocean? The total volume of the ocean is 1.3 X 1018 m3.

(c) Suppose you pour a cup of water into the ocean, allow it to become thoroughly mixed, and then take a cup of water out of the ocean. On the average, how many of the molecules originally in the cup will again be in the cup?

23. How many molecules are there in one cubic centimeter of air? Assume that the density of air is 1.3 kg/m3 and that it consists entirely of nitrogen molecules (N2).

*24. How many atoms are there in the Sun? The mass of the Sun is 1.99 X 1030 kg and its chemical composition (by mass) is approximately 70% hydrogen and 30% helium.

*25. The chemical composition of air is (by mass) 75.5% N2, 23.2% O2, and 1.3% Ar. What is the average "molecular mass" of air; that is, what is the mass of 6.02 X 1023 molecules of air?

*26. How many atoms are there in a human body of 73 kg? The chemical composition (by mass) of a human body is 65% oxygen, 18.5% carbon, 9.5% hydrogen, 3.3% nitrogen, 1.5% calcium, 1% phosphorus, and 0.35% other elements (ignore the "other elements" in your calculation).


Section 1.7

27. The distance from our Galaxy to the Andromeda galaxy is 2.2 X 106 light-years. Express this distance in meters.

28. In analogy with the light-year, we can define the light-second as the distance light travels in one second and the light-minute as the distance light travels in one minute. Express the Earth-Sun distance in light-minutes. Express the Earth-Moon distance in light-seconds.

29. Astronomers often use the astronomical unit (AU), the parsec (pc), and the light-year. The AU is the distance from the Earth to the Sun;12 1 AU = 1.496 X 1011 m. The pc is the distance at which 1 AU subtends an angle of exactly 1second of arc (Figure 1.19). The light-year is the distance that light travels in 1 year.

(a) Express the pc in AU.

(b) Express the pc in light-years.

(c) Express the pc and the light-year in meters.

30. The density of copper is 8.9 g/cm3. Express this in kg/m3, lb/ft3 , and lb/in.3.

31. The federal highway speed limit was 55 mi/h. Express this in kilometers per hour, feet per second, and meters per second.

32. What is the volume of an average human body? (Hint: The density of the body is about the same as that of water.)

33. Meteorologists usually report the amount of rain in terms of the depth in inches to which the water would accumulate on a flat surface if it did not run off. Suppose that 1 in. of rain falls during a storm. Express this in cubic meters of water per square meter of surface. How many kilograms of water per square meter of surface does this amount to?

34. A fire hose delivers 300 liters of water per minute. Express this in m3/s. How many kilograms of water per second does this amount to?

35. The total volume of the oceans of Earth is 1.3 X 1018 m3. What percentage of the mass of the Earth is in the oceans?

36. The nucleus of an iron atom is spherical and has a radius of 4.6 X 10-15 m; the mass of the nucleus is 9.5 X 10-26 kg. What is the density of the nuclear material? Express your answer in metric tons per cubic centimeter.

37. Our Sun has a radius of 7.0 X 108 m and a mass of 2.0 X 1030 kg. What is its average density? Express your answer in grams per cubic centimeter.

38. Pulsars, or neutron stars, typically have a radius of 20 km and a mass equal to that of our Sun (2.0 X 1030 kg). What is the average density of such a pulsar? Express your answer in metric tons per cubic centimeter.

39. The nuclei of all atoms have approximately the same density of mass. The nucleus of a copper atom has a mass of 1.06 X 10-25 kg and a radius of 4.8 X 10-15 m. The nucleus of a lead atom has a mass of 3.5 X 10-25 kg; what is its radius? The nucleus of an oxygen atom has a mass of 2.7 X 10-26 kg; what is its radius? Assume that the nuclei are spherical.

40. The table printed on the endpapers gives the masses and radii of the major planets. Calculate the average density of each planet and make a list of the planets in order of decreasing densities. Is there a correlation between the density of a planet and its distance from the Sun?

41. A small single-engine plane is flying at a height of 5000 m at a (horizontal) distance of 18 km from the San Francisco airport when the engine quits. The pilot knows that, without the engine, the plane will glide downward at an angle of 15°. Can she reach San Francisco?

*42. You are crossing the Atlantic in a sailboat and hoping to make a landfall in the Azores. The highest peak on the Azores has a height of 2300 m. From what distance can you see this peak just emerging over the horizon?

*43. In the Galapagos (on the equator) the small island of Marchena is 60 km west of the small island Genovesa. If the sun sets at 8:00 P.M. at Genovesa when will it set at Marchena?

*44. For tall trees, the diameter at the base (or the diameter at any given point of the trunk, such as the midpoint) is roughly proportional to the 3/2 power of the length. The tallest sequoia in Sequoia National Park in California has a length of 81 m, a diameter of 7.6 m at the base, and a mass of 6100 metric tons. A petrified sequoia found in Nevada has a length of 90 m. Estimate its diameter at the base, and estimate the mass it had when it was still alive.


*The asterisks indicate the levels of difficulty of the problems (see the Preface).

12Strictly, it is the semimajor axis of the Earth's orbit.