Chapter 1

(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 \(410 \mathrm{mg} /\) day for males. Express this quantity in \(\mu \mathrm{g} /\) day. (b) For adults, the RDA of the amino acid lysine is \(12 \mathrm{mg}\) per \(\mathrm{kg}\) 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 \(\mathrm{RDA}\) is \(0.0030 \mathrm{~g} /\) 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 \(\mathrm{mg} /\) day.

(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 \(\mathrm{mL}\) ). Show that a milliliter is the same as a cubic centimeter. (c) How many cubic centimeters of water are there in a \(1.00 \mathrm{~L}\) bottle of drinking water?

(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 \(\mathrm{g} / \mathrm{cm}^{3} ?\) (c) How many kilograms are there in a \(1.00 \mathrm{~L}\) bottle of drinking water? How many pounds?

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

While driving in an exotic foreign land, you see a speed-limit sign on a highway that reads 180,000 furlongs per fortnight. 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.)

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

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 \(\mathrm{mi} / \mathrm{h} .\) (b) Find the speed of sound in air at \(0^{\circ} \mathrm{C}(1100 \mathrm{ft} / \mathrm{s})\) in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{mi} / \mathrm{h} .\) (c) Show that \(60 \mathrm{mi} / \mathrm{h}\) 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 \(\mathrm{ft} / \mathrm{s}^{2}\).

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

Express each of the following approximations of \(\pi\) to six significant figures: (a) \(22 / 7,\) (b) \(355 / 113 .\) (c) Are these approximations accurate to that precision?

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?

The density of aluminum is \(2.7 \mathrm{~g} / \mathrm{cm}^{3}\). What is the mass of a cube of aluminum that is \(5.656 \mathrm{~cm}\) on a side? Express your answer in \(\mathrm{SI}\) units, using the appropriate number of significant figures. (Recall that density is mass divided by volume.)

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?

A stack of printer paper is 2 inches thick and contains 500 sheets. Estimate the thickness of an individual sheet of the printer paper. Express your answer in micrometers.

You are designing a space station and want to get some idea about 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 2 weeks in case of a malfunction. (a) Estimate how many cubic meters of air an average person breathes in 2 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)?

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

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 direction. Find the magnitude and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.

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 \(60.0^{\circ},\) (b) \(75 \mathrm{~m} / \mathrm{s}\) at \(5 \pi / 6 \mathrm{rad},\) (c) \(254 \mathrm{lb}\) at \(325^{\circ},\) (d) \(69 \mathrm{~km}\) at \(1.1 \pi\) rad.

In each of the cases that follow, the components of a vector \(A\) are given. 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) \(A_{x}=-3.0 \mathrm{~km}, A_{y}=-6.0 \mathrm{~km}\) (c) \(A_{x}=9.0 \mathrm{~m} / \mathrm{s}, A_{y}=-17 \mathrm{~m} / \mathrm{s}\) (d) \(A_{x}=-8.0 \mathrm{~N}, A_{y}=12 \mathrm{~N}\)

A woman takes her dog Rover for a walk on a leash. To get the little dog 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?

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

A plane leaves Seattle, flies \(85 \mathrm{mi}\) at \(22^{\circ}\) north of east, and then changes direction to \(48^{\circ}\) south of east. After flying \(115 \mathrm{mi}\) 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.

Vector \(A\) has a magnitude of \(20 \mathrm{~m}\) and makes an angle of \(30^{\circ}\) above the positive \(x\) axis. Vector \(\vec{B}\) has a magnitude of \(15 \mathrm{~m}\) and is oriented \(60^{\circ}\) to the left of the \(y\) axis. Find (a) the magnitude and direction of \(\vec{A}-\vec{B},(b)\) the magnitude and direction of \(2 \vec{A}+\vec{B},\) and (c) the magnitude and direction of \(-\overrightarrow{\boldsymbol{A}}+3 \overrightarrow{\boldsymbol{B}}\).

I 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 and no 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. \()\)

Velocity vector \(A\) has components \(A_{x}=3 \mathrm{~m} / \mathrm{s}\) and \(A_{y}=4 \mathrm{~m} / \mathrm{s}\) A second velocity vector \(\vec{B}\) has a magnitude that is twice that of \(A\) and is pointed down the negative \(x\) axis. Find (a) the components of \(\overrightarrow{\boldsymbol{B}}\) and \((\mathrm{b})\) the components of \(\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}}\)

A typical human contains \(5.0 \mathrm{~L}\) of blood, and it takes \(1.0 \mathrm{~min}\) 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?

A woman starts from her house and begins jogging down her street. In order to keep a record of her workout, she carries a stopwatch and records the time it takes her to reach specific landmarks, such as intersections, school grounds, and churches. since she knows the distance to each of these landmarks, she is able to tabulate the data shown here: $$ \begin{array}{ll} \hline \text { Time } & \text { Distance jogged } \\ \hline 10 \mathrm{~min} & 2.1 \mathrm{~km} \\ 17 \mathrm{~min} & 3.2 \mathrm{~km} \\ 32 \mathrm{~min} & 6.5 \mathrm{~km} \\ 44 \mathrm{~min} & 8.6 \mathrm{~km} \end{array} $$ Make a plot of the distance jogged versus time. Draw a best-fit line through the data, and use this line to estimate (a) the speed at which she was jogging in \(\mathrm{m} / \mathrm{s}\) and \((\mathrm{b})\) the distance she will jog in 1 hour.

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.

In human lungs, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical human lung pair is about \(480 \times 10^{6}\) and that the average volume of a single alveolus is \(4.2 \times 10^{6} \mu \mathrm{m}^{3}\). (Recall the equation for the volume of a sphere, \(V=\frac{4}{3} \pi r^{3},\) and the equation for the area of a sphere, \(\left.A=4 \pi r^{2} .\right)\) What is the 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. \(120 \mathrm{~L}\)