Chapter 1

(a) The maximum sodium intake for a person on a 2000 calorie diet should be 2400 mg/day. How many grams of sodium is this per day? (b) The recommended daily allowance (RDA) of the trace element chromium is 120\(\mu g /\) day. Express this dose in considered safe (although the safety of higher doses has not yet been established. Express this intake in grams per day. (d) The electrical resistance of the human body is approximately 1500 ohms when it is dry. Express this resistance in kilohms. (e) An electrical current of about 0.020 amp can cause muscular spasms so that a person cannot, for example, let go of a wire with that amount of current. Express this current in milliamps.

(a) How many ohms are there in a 7.85 -megohm resistor? (b) Typical laboratory capacitors are around 5 picofarads. How many farads are they? (c) The speed of light in vacuum is \(3.00 \times 10^{8} \mathrm{m} / \mathrm{s} .\) Express this speed in gigameters per second. (d) The wavelength of visible light is between 400 \(\mathrm{nm}\) and 700 \(\mathrm{nm} .\) Express this wavelength in meters. (e) The diameter of a typical atomic nucleus is about 2 femtometers. Express this diameter in meters.

(a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 \(\mathrm{mg} / \mathrm{day}\) for males. Express this quantity in \(\mu \mathrm{g} / \mathrm{day} .\) (b) For adults, the RDA of the amino acid lysine is 12 \(\mathrm{mg}\) per kg of body weight. How many grams per day should a 75 \(\mathrm{kg}\) adult receive? (c) A typical multivitamin tablet can contain 2.0 \(\mathrm{mg}\) of vitamin \(\mathrm{B}_{2}\) (riboflavin), and the RDA is 0.0030 \(\mathrm{g} / \mathrm{day} .\) How many such tablets should a person take each day to get the proper amount of this vitamin, assuming that he gets none from any other sources? (d) The RDA for the trace element selenium is 0.000070 \(\mathrm{g} /\) day. Express this dose in mg/day.

\(\bullet\) (a) Starting with the definition 1.00 in. \(=2.54 \mathrm{cm},\) find the number of kilometers in 1.00 mile. (b) In medicine, volumes are often expressed in milliliters (ml or mL). Show that a milliliter is the same as a cubic centimeter. (c) How many cubic centimeters of water are there in a 1.00 L bottle of drinking water?

\(\bullet(\) a) The density (mass divided by volume) of water is 1.00 \(\mathrm{g} / \mathrm{cm}^{3} .\) What is this value in kilograms per cubic meter? (b) The density of blood is 1050 \(\mathrm{kg} / \mathrm{m}^{3} .\) What is this density in \(\mathrm{g} / \mathrm{cm}^{3} ?(\mathrm{c})\) How many kilograms are there in a 1.00 \(\mathrm{L}\) bottle of drinking water? How many pounds?

\(\bullet\) How many nanoseconds does it take light to travel 1.00 \(\mathrm{ft}\) in vacuum? (This result is a useful quantity to remember.)

Metric wrenches. (a) You have a new set of metric wrenches, but need to loosen a \(\frac{3}{8}\) inch bolt. To find out which size metric wrench to use, convert the \(\frac{3}{8}\) in. to millimeters, accurate to the nearest tenth of a millimeter. (b) If you want to tighten a 12 \(\mathrm{mm}\) bolt, what size wrench should you use in inches, accurate to the nearest hundredth of an inch? (c) English-unit wrenches often come in \(\frac{1}{8}\) inch intervals, not in decimal units. What size wrench should you use in part (b)?

\(\bullet\) While driving in an exotic foreign land, you see a speed-limit sign on a highway that reads \(180,000\) furlongs per fort-night. How many miles per hour is this? (One furlong is \(\frac{1}{8}\) mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

\(\bullet\) Bacteria. Bacteria vary somewhat in size, but a diameter of 2.0\(\mu \mathrm{m}\) is not unusual. What would be the volume (in cubic centimeters) and surface area (in square millimeters) of such a bacterium, assuming that it is spherical? (Consult Chapter 0 for relevant formulas.)

\(\cdot\) Compute the number of seconds in (a) an hour, (b) a 24 hour day, and (c) a 365 day year.

\(\bullet\) Some commonly occurring quantities. All of the quantities that follow will occur frequently in your study of physics. (a) Express the speed of light \(\left(3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\right)\) in \(\mathrm{mi} / \mathrm{s}\) and mph. (b) Find the speed of sound in air at \(0^{\circ} \mathrm{C}(1100 \mathrm{ft} / \mathrm{s})\) in \(\mathrm{m} / \mathrm{s}\) and mph. (c) Show that 60 \(\mathrm{mph}\) is the same as 88 \(\mathrm{ft} / \mathrm{s}\) . (d) Convert the acceleration of a freely falling body \(\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)\) to \(\mathrm{ft} / \mathrm{s}^{2}\)

\(\cdot\) Express each of the following numbers to three, five, and eight significant figures: (a) \(\pi=3.141592654 \ldots,\) (b) \(e=\) \(2.718281828 \ldots,(\mathrm{c}) \sqrt{13}=3.605551275 \ldots\)

\(\cdot\) Express each of the following approximations of \(\pi\) to six significant figures: (a) \(22 / 7,(\) b) 35\(/ 113 .\) (c) Are these approximations accurate to that precision?

\(\cdot\) An angle is given, to one significant figure, as \(4^{\circ},\) meaning that its value is between \(3.5^{\circ}\) and \(4.5^{\circ} .\) Find the corresponding range of possible values of \((a)\) the cosine of the angle, (b) the sine of the angle, and (c) the tangent of the angle.

\(\cdot\) Blood is thicker than water. The density (mass divided by volume) of pure water is \(1.00 \mathrm{g} / \mathrm{cm}^{3},\) that of whole blood is \(1.05 \mathrm{g} / \mathrm{cm}^{3},\) and the density of seawater is 1.03 \(\mathrm{g} / \mathrm{cm}^{3} .\) What is the mass (in grams) of 1.00 \(\mathrm{L}\) of each of these substances?

\(\bullet\) White dwarfs and neutron stars. Recall that density is mass divided by volume, and consult Chapter 0 and Appendix \(E\) as needed. (a) Calculate the average density of the earth in \(g / c m^{3},\) assuming our planet to be a perfect sphere. (b) In about 5 billion years, at the end of its lifetime, our sun will end up as a white dwarf, having about the same mass as it does now, but reduced to about \(15,000 \mathrm{km}\) in diameter. What will be its density at that stage? (c) A neutron star is the remnant left after certain supernovae (explosions of giant stars). Typically, neutron stars are about 20 \(\mathrm{km}\) in diameter and have around the same mass as our sun. What is a typical neutron star density in \(\mathrm{g} / \mathrm{cm}^{3} ?\)

Critical mass of neptunium. In the fall of \(2002,\) a group of scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about 60.0 \(\mathrm{kg}\) . (The critical mass of a fissionable material is the minimum amount that must be brought together to start a nuclear chain reaction.) Neptunium has a density of 19.5 \(\mathrm{g} / \mathrm{cm}^{3} .\) What would be the radius of a sphere made of this material that has a critical mass? (Recall that density is mass divided by volume.)

\bullet Cell walls. Although these quantities vary from one type of cell to another, a cell can be 2.0\(\mu \mathrm{m}\) in diameter with a cell wall 50.0 \(\mathrm{nm}\) thick. If the density (mass divided by volume) of the wall material is the same as that of pure water, what is the mass (in mg) of the cell wall, assuming the cell to be spherical and the wall to be a very thin spherical shell?

\bullet A brass washer has an outside diameter of 4.50 \(\mathrm{cm}\) with a hole of diameter 1.25 \(\mathrm{cm}\) and is 1.50 \(\mathrm{mm}\) thick. (See Figure \(1.21 . )\) The density of brass is 8600 \(\mathrm{kg} / \mathrm{m}^{3} .\) If you put this washer on a laboratory balance, what will it "weigh" in grams? (Recall that density is mass divided by volume and consult Chapter 0 as needed. \()\)

Space station. You are designing a space station and want to get some idea how large it should be to provide adequate air for the astronauts. Normally, the air is replenished, but for security, you decide that there should be enough to last for two weeks in case of a malfunction. (a) Estimate how many cubic meters of air an average person breathes in two weeks. A typical human breathes about 1\(/ 2 \mathrm{L}\) per breath. (b) If the space station is to be spherical, what should be its diameter to contain all the air you calculated in part (a)?

\(\cdot\) On a single diagram, carefully sketch each force vector to scale and identify its magnitude and direction on your drawing: (a) 60 lb at \(25^{\circ}\) east of north. (b) 40 lb at \(\pi / 3\) south of west. (c) 100 lb at \(40^{\circ}\) north of west. (d) 50 lb at \(\pi / 6\) east of south.

\(\bullet\) Hearing rattles from a snake, you make two rapid displacements of magnitude 1.8 \(\mathrm{m}\) and 2.4 \(\mathrm{m} .\) In sketches (roughly to scale), show how your two displacements might add up to give a resultant of magnitude (a) \(4.2 \mathrm{m} ;\) (b) \(0.6 \mathrm{m} ;\) (c) 3.0 \(\mathrm{m} .\)

A ladybug starts at the center of a 12 -in.-diameter turntable and crawls in a straight radial line to the edge. While this is happening, the turntable turns through a \(45^{\circ}\) angle. (a) Draw a sketch showing the bug's path and the displacement vector for the bug's progress. (b) Find the magnitude and direction of the ladybug's displacement vector.

A rocket fires two engines simultaneously. One produces a thrust of 725 \(\mathrm{N}\) directly forward, while the other gives a \(513-\mathrm{N}\) thrust at \(32.4^{\circ}\) above the forward, while the other gives a \(513-\mathrm{N}\) and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.

\(\cdot\) In each of the cases that follow, the magnitude of a vector is given along with the counterclockwise angle it makes with the \(+x\) axis. Use trigonometry to find the \(x\) and \(y\) components of the vector. Also, sketch each vector approximately to scale to see if your calculated answers seem reasonable. (a) 50.0 \(\mathrm{N}\) at \(60.0^{\circ},(\mathrm{b}) 75 \mathrm{m} / \mathrm{s}\) at \(5 \pi / 6 \mathrm{rad},(\mathrm{c}) 254 \mathrm{lb}\) at \(325^{\circ},\) (d) 69 \(\mathrm{km}\) at 1.1\(\pi \mathrm{rad} .\)

\(\cdot\) In each of the cases that follow, the components of a vector \(\vec{A}\) are given. Use trigonometry to find the magnitude of that vector and the counterclockwise angle it makes with the \(+x\) axis. Also, sketch each vector approximately to scale to see if your calculated answers seem reasonable. (a) \(A_{x}=4.0 \mathrm{m}, A_{y}=5.0 \mathrm{m},\) (b) \(A_{x}=-3.0 \mathrm{km}, A_{y}=-6.0 \mathrm{km},(\mathrm{c}) A_{x}=9.0 \mathrm{m} / \mathrm{s}, A_{y}=\) \(-17 \mathrm{m} / \mathrm{s},(\mathrm{d}) A_{x}=-8.0 \mathrm{N}, A_{\mathrm{y}}=12 \mathrm{N}\)

\(\bullet\) A woman takes her dog Rover for a walk on a leash. To get the little pooch moving forward, she pulls on the leash with a force of 20.0 \(\mathrm{N}\) at an angle of \(37^{\circ}\) above the horizontal. (a) How much force is tending to pull Rover forward? (b) How much force is tending to lift Rover off the ground?

If a vector \(\vec{A}\) has the following components, use trigonometry to find its magnitude and the counterclockwise angle it makes with the \(+x\) axis: (a) \(A_{x}=8.0\) lb, \(A_{y}=6.0\) lb (b) \(A_{x}=-24 \frac{m}{s}, A_{y}=-31 \frac{m}{s}\) (c) \(A_{x}=-1500\) km, \(A_{y}=2000\) km (d) \(A_{x}=71.3\) N, \(A_{y}=-54.7\) N

Vector \(A\) has components \(A_{x}=1.30 \mathrm{cm}, \quad A_{y}=\) \(2.25 \mathrm{cm} ; \quad\) vector \(\vec{B}\) has components \(B_{x}=4.10 \mathrm{cm}\) \(B_{y}=-3.75 \mathrm{cm} .\) Find \((\mathrm{a})\) the components of the vector sum \(A+B ;\) (b) the magnitude and direction of \(\vec{A}+\vec{B}\) (c) the components of the vector difference \(\vec{B}-\vec{A}\) (d) the magnitude and direction of \(\vec{\boldsymbol{B}}-\vec{\boldsymbol{A}}\)

. A plane leaves Seattle, flies 85 mi at \(22^{\circ}\) north of east, and then changes direction to \(48^{\circ}\) south of east. After flying at 115 mi in this new direction, the pilot must make an emergency landing on a field. The Seattle airport facility dispatches a rescue crew. (a) In what direction and how far should the crew fly to go directly to the field? Use components to solve this problem. (b) Check the reasonableness of your answer with a careful graphical sum.

\(\bullet\) You're hanging from a chinning bar, with your arms at right angles to each other. The magnitudes of the forces exerted by both your arms are the same, and together they exert just enough upward force to support your weight, 620 \(\mathrm{N}\) . (a) Sketch the two force vectors for your arms, along with their resultant, and (b) use components to find the magnitude of each of the two "arm" force vectors.

A disoriented physics professor drives 3.25 \(\mathrm{km}\) north, then 4.75 \(\mathrm{km}\) west, and then 1.50 \(\mathrm{km}\) south. (a) Use components to find the magnitude and direction of the resultant displacement of this professor. (b) Check the reasonableness of your answer with a graphical sum.

Baseball mass. Baseball rules specify that a regulation ball shall weigh no less than 5.00 ounces nor more than 5\(\frac{1}{4}\) ounces. What are the acceptable limits, in grams, for a regulation ball? (See Appendix D and use the fact that 16 oz \(=1\) lb.)

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 \(\mathrm{cm}\) and a thickness of 0.050 \(\mathrm{cm} .\) Find (a) the volume of a single cookie and (b) the ratio of the diameter to the thickness, and express both in the proper number of significant figures.

. Breathing oxygen. The density of air under standard laboratory conditions is \(1.29 \mathrm{kg} / \mathrm{m}^{3},\) and about 20\(\%\) of that air consists of oxygen. Typically, people breathe about \(\frac{1}{2} \mathrm{L}\) of air per breath. (a) How many grams of oxygen does a person breathe in a day? (b) If this air is stored uncompressed in a cubical tank, how long is each side of the tank?

\(\bullet\) The total mass of Earth's atmosphere is about \(5 \times 10^{15}\) metric tonnes \((1\) metric tonne \(=1000 \mathrm{kg}) .\) Suppose you breathe in about 1\(/ 3 \mathrm{L}\) of air with each breath, and the density of air at room temperature is about 1.2 \(\mathrm{kg} / \mathrm{m}^{3} .\) About how many breaths of air does the entire atmosphere contain? How does this compare to the number of atoms in one breath of air (about 1.2 \(\times 10^{22} ) ?\) It's sometimes said that every breath you take contains atoms that were also breathed by Albert Einstein, Confucius, and in fact anyone else who ever lived. Based on your calculation, could this be true?

\(\bullet\) How much blood in a heartbeat? A typical human contains 5.0 \(\mathrm{L}\) of blood, and it takes 1.0 min for all of it to pass through the heart when the person is resting with a pulse rate of 75 heartbeats per minute. On the average, what volume of blood, in liters and cubic centimeters, does the heart pump during each beat?

While surveying a cave, a spelunker follows a passage 180 \(\mathrm{m}\) straight west, then 210 \(\mathrm{m}\) in a direction \(45^{\circ}\) east of south, and then 280 \(\mathrm{m}\) at \(30.0^{\circ}\) east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use vector components to find the magnitude and direction of the fourth displacement. Then check the reasonableness of your answer with a graphical sum.

Bones and muscles. A patient in therapy has a forearm that weighs 20.5 \(\mathrm{N}\) and lifts a 112.0 \(\mathrm{N}\) weight. The only other significant forces on his forearm come from the biceps muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produce a pull of 232 \(\mathrm{N}\) when the forearm is raised \(43^{\circ}\) above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (Hint: The elbow force and the biceps force together must balance the weight of the arm and the weight it is carrying, so their vector sum must be 132.5 \(\mathrm{N}\) upward.)

What is total volume of the gas-exchanging region of the lungs? A. 2000\(\mu \mathrm{m}^{3}\) B. 2 \(\mathrm{m}^{3}\) C. 2.0 \(\mathrm{L}\) D. 120 \(\mathrm{L}\)

Assuming the alveoli are spherical, what is the diameter of a typical alveolus? A. 0.20 \(\mathrm{mm}\) B. 2 \(\mathrm{mm}\) C. 20 \(\mathrm{mm}\) D. 200 \(\mathrm{mm}\)