Chapter 1

The metric system is a decimal (base-10) system, and the British system is, in part, a duodecimal (base-12) system. Discuss the ramifications if our monetary system had a duodecimal base. What would be the possible values of our coins if this were the case?

Convert the following: (a) 40000000 bytes to MB, (b) \(0.5722 \mathrm{~mL}\) to \(\mathrm{L}\), (c) \(2.684 \mathrm{~m}\) to \(\mathrm{cm},\) and (d) 5500 bucks to kilobucks.

A sailor tells you that if his ship is traveling at 25 knots (nautical miles per hour), it is moving faster than the \(25 \mathrm{mi} / \mathrm{h}\) your car travels. How can that be?

(a) What volume in liters is a cube \(20 \mathrm{~cm}\) on a side? (b) If the cube is filled with water, what is the mass of the water?

Show that the equation \(x=x_{\mathrm{o}}+v t,\) where \(v\) is velocity, \(x\) and \(x_{\mathrm{o}}\) are lengths, and \(t\) is time, is dimensionally correct.

\- If \(x\) refers to distance, \(v_{0}\) and \(v\) to velocities, \(a\) to acceleration, and \(t\) to time, which of the following equations is dimensionally correct: (a) \(x=v_{\mathrm{o}} t+a t^{3},\) (b) \(v^{2}=v_{\mathrm{o}}^{2}+2 a t\) (c) \(x=a t+v t^{2},\) or (d) \(v^{2}=v_{\mathrm{o}}^{2}+2 a x ?\)

Use SI unit analysis to show that the equation \(A=4 \pi r^{2},\) where \(A\) is the area and \(r\) is the radius of a sphere, is dimensionally correct.

The general equation for a parabola is \(y=a x^{2}+b x+c,\) where \(a, b,\) and \(c\) are constants. What are the units of each constant if \(y\) and \(x\) are in meters?

You are told that the volume of a sphere is given by \(V=\pi d^{3} / 4,\) where \(V\) is the volume and \(d\) is the diameter of the sphere. Is this equation dimensionally correct?

The units for pressure \((p)\) in terms of SI base units are known to be \(\frac{\mathrm{kg}}{\mathrm{m} \cdot \mathrm{s}^{2}} .\) For a physics class assignment, a student derives an expression for the pressure exerted by the wind on a wall in terms of the air density \((\rho)\) and wind speed \((v)\) and her result is \(p=\rho v^{2}\). Use SI unit analysis to show that her result is dimensionally consistent. Does this prove that this relationship is physically correct?

Newton's second law of motion (Section 4.3 ) is expressed by the equation \(F=m a,\) where \(F\) represents force, \(m\) is mass, and \(a\) is acceleration. (a) The SI unit of force is, appropriately, called the newton (N). What are the units of the newton in terms of base quantities? (b) An equation for force associated with uniform circular motion (Section 7.3) is \(F=m v^{2} / r,\) where \(v\) is speed and \(r\) is the radius of the circular path. Does this equation give the same units for the newton?

The angular momentum (L) of a particle of mass \(m\) moving at a constant speed \(v\) in a circle of radius \(r\) is given by \(L=\operatorname{mvr}\) (Section 8.5 ). (a) What are the units of angular momentum in terms of SI base units? (b) The units of kinetic energy in terms of SI base units are \(\frac{\mathrm{kg} \cdot \mathrm{m}^{2}}{\mathrm{~s}^{2}}\). Using SI unit analysis, show that the expression for the kinetic energy of this particle in terms of its angular momentum, \(K=\frac{L^{2}}{2 m r^{\prime}}\), is dimensionally correct. (c) In the previous equation, the term \(m r^{2}\) is called the moment of inertia of the particle in the circle. What are the units of moment of inertia in terms of SI base units?

Einstein's famous mass-energy equivalence is expressed by the equation \(E=m c^{2},\) where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light. (a) What are the SI base units of energy? (b) Another equation for energy is \(E=m g h,\) where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is height. Does this equation give the same units as in part (a)?

(a) If you wanted to express your height with the largest number, which units would you use: (1) meters, (2) feet, (3) inches, or (4) centimeters? Why? (b) If you are \(6.00 \mathrm{ft}\) tall, what is your height in centimeters?

If the capillaries of an average adult were unwound and spread out end to end, they would extend to a length over \(40000 \mathrm{mi}\) (Fig. 1.9). If you are \(1.75 \mathrm{~m}\) tall, how many times your height would the capillary length equal?

(a) Compared with a 2-L soda bottle, a half-gallon soda bottle holds (1) more, (2) the same amount of, (3) less soda. (b) Verify your answer for part (a).

How many (a) quarts and (b) gallons are there in 10.0 L?

Driving a jet-powered car, Royal Air Force pilot Andy Green broke the sound barrier on land for the first time and achieved a record land speed of more than \(763 \mathrm{mi} / \mathrm{h}\) in Black Rock Desert, Nevada, on October 15,1997 (vFig. 1.15). (a) What is this speed expressed in \(\mathrm{m} / \mathrm{s}\) ? (b) How long would it take the jet-powered car to travel the length of a 300 - \(\mathrm{ft}\) football field at this speed?

(a) Which of the following represents the greatest speed: \((1) 1 \mathrm{~m} / \mathrm{s},(2) 1 \mathrm{~km} / \mathrm{h},(3) 1 \mathrm{ft} / \mathrm{s},\) or \((4) 1 \mathrm{mi} / \mathrm{h} ?\) (b) Express the speed \(15.0 \mathrm{~m} / \mathrm{s}\) in \(\mathrm{mi} / \mathrm{h}\).

A person weighs 170 lb. (a) What is his mass in kilograms? (b) Assuming the density of the average human body is about that of water (which is true), estimate his body's volume in both cubic meters and liters. Explain why the smaller unit of the liter is more appropriate (convenient) for describing a volume of this size.

The human heartbeat, as determined by the pulse rate, is normally about 60 beats \(/ \mathrm{min}\). If the heart pumps \(75 \mathrm{~mL}\) of blood per beat, what volume of blood is pumped in one day in liters?

Fig. 1.18 is a picture of red blood cells seen under a scanning electron microscope. Normally, women possess about 4.5 million of these cells in each cubic millimeter of blood. If the blood flow to the heart is \(250 \mathrm{~mL} / \mathrm{min}\), how many red blood cells does a woman's heart receive each second?

A student was 18 in. long when she was born. She is now \(5 \mathrm{ft} 6\) in. tall and 20 years old. How many centimeters a year did she grow on average?

How many minutes of arc does the Earth rotate in 1 min of time?

The density of metal mercury is \(13.6 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is this density as expressed in kilograms per cubic meter? (b) How many kilograms of mercury would be required to fill a 0.250 -L container?

The Roman Coliseum used to be flooded with water to re-create ancient naval battles. Assuming the circular floor be \(250 \mathrm{~m}\) in diameter and the water to have a depth of \(10 \mathrm{ft},\) (a) how many cubic meters of water are required? (b) How much mass would this water have in kilograms? (c) How much would the water weigh in pounds?

In the Bible, Noah is instructed to build an ark 300 cubits long, 50.0 cubits wide, and 30.0 cubits high (vFig. 1.19). Historical records indicate a cubit is equal to half a yard. (a) What would be the dimensions of the ark in meters? (b) What would be the ark's volume in cubic meters? To approximate, assume that the ark is to be rectangular.

Express the length \(50500 \mu \mathrm{m}\) (micrometers) in centimeters, decimeters, and meters, to three significant figures.

Determine the number of significant figures in the following measured numbers: (a) \(1.007 \mathrm{~m}\), (b) \(8.03 \mathrm{~cm}\) (c) \(16.272 \mathrm{~kg}\) (d) \(0.015 \mu\) s (microseconds).

Round the following numbers to two significant figures: (a) \(95.61,\) (b) 0.00208 , (c) 9438 , (d) 0.000344

Which of the following quantities has three significant figures: (a) \(305.0 \mathrm{~cm}\) (b) \(0.0500 \mathrm{~mm}\) (c) \(1.00081 \mathrm{~kg}\) (d) \(8.06 \times 10^{4} \mathrm{~m}^{2}\) ?

The cover of your physics book measures \(0.274 \mathrm{~m}\) long and \(0.222 \mathrm{~m}\) wide. What is its area in square meters?

The interior storage compartment of a restaurant refrigerator measures \(1.3 \mathrm{~m}\) high, \(1.05 \mathrm{~m}\) wide, and \(67 \mathrm{~cm}\) deep. Determine its volume in cubic feet.

The top of a rectangular table measures \(1.245 \mathrm{~m}\) by \(0.760 \mathrm{~m} .\) (a) The smallest division on the scale of the measurement instrument is \((1) \mathrm{m},(2) \mathrm{cm},(3) \mathrm{mm} .\) Why? (b) What is the area of the tabletop?

The outside dimensions of a cylindrical soda can are reported as \(12.559 \mathrm{~cm}\) for the diameter and \(5.62 \mathrm{~cm}\) for the height. (a) How many significant figures will the total outside area have: (1) two, (2) three, (3) four, or (4) five? Why? (b) What is the total outside surface area of the can in square centimeters?

Express the following calculations using the proper number of significant figures: (a) \(12.634+2.1\), (b) \(13.5-2.134\) (c) \(\pi(0.25 \mathrm{~m})^{2}\) (d) \(\sqrt{2.37 / 3.5}\)

In doing a problem, a student adds \(46.9 \mathrm{~m}\) and \(5.72 \mathrm{~m}\) and then subtracts \(38 \mathrm{~m}\) from the result. (a) How many decimal places will the final answer have: (1) zero, (2) one, or (3) two? Why? (b) What is the final answer?

Work this exercise by the two given procedures as directed, commenting on and explaining any difference in the answers. Use your calculator for the calculations. Compute \(p=m v,\) where \(v=x / t,\) given \(x=8.5 \mathrm{~m}\) \(t=2.7 \mathrm{~s},\) and \(m=0.66 \mathrm{~kg} .\) (a) First compute \(v\) and then (b) Compute \(p=m x / t\) without an intermediate step. \(p\) (c) Are the results the same? If not, why?

A corner construction lot has the shape of a right triangle. If the two sides perpendicular to each other are \(37 \mathrm{~m}\) long and \(42.3 \mathrm{~m}\) long, what is the length of the hypotenuse?

The lightest solid material is silica aerogel, which has a typical density of only about \(0.10 \mathrm{~g} / \mathrm{cm}^{3}\). The molecular structure of silica aerogel is typically \(95 \%\) empty space. What is the mass of \(1 \mathrm{~m}^{3}\) of silica aerogel?

A cord of wood is a volume of cut wood equal to a stack \(8.0 \mathrm{ft}\) long, \(4.0 \mathrm{ft}\) wide, and \(4.0 \mathrm{ft}\) high. How many cords are there in \(3.0 \mathrm{~m}^{3}\) ?

The thickness of the numbered pages of a textbook is measured to be \(3.75 \mathrm{~cm}\). (a) If the last page of the book is numbered 860 , what is the average thickness of a page? (b) Repeat the calculation by using order-of- magnitude calculations.

To go to a football stadium from your house, you first drive \(1000 \mathrm{~m}\) north, then \(500 \mathrm{~m}\) west, and finally \(1500 \mathrm{~m}\) south. (a) Relative to your home, the football stadium is (1) north of west, (2) south of east, (3) north of east, (4) south of west. (b) What is the straight-line distance from your house to the stadium?

Tony's Pizza Palace sells a medium 9.0 -in. (diameter) pizza for \(\$ 7.95,\) and a large 12 -in. pizza for \(\$ 13.50 .\) Which pizza is the better buy?

Two students go into Tony's Pizza Palace and order a 12-in. (diameter) pizza. Shortly thereafter, the waitress brings an 8 -in. pizza special. She explains that the 12 -in. pizza was given to someone else by mistake and they could have the 8 -in. now and she would bring another 8 in. shortly to make up for the missing 12 -in. pizza. Was this a good deal?

The Channel Tunnel, or "Chunnel," which runs under the English Channel between Great Britain and France, is 31 mi long. (There are actually three separate tunnels.) A shuttle train that carries passengers through the tunnel travels with an average speed of \(75 \mathrm{mi} / \mathrm{h}\). On average, how long, in minutes, does the shuttle take to make a one-way trip through the Chunnel?

Human adult blood contains, on average, \(7000 / \mathrm{mm}^{3}\) white blood cells (leukocytes) and \(250000 / \mathrm{mm}^{3}\) platelets (thrombocytes). If a person has a blood volume of \(5.0 \mathrm{~L}\), estimate the total number of white cells and platelets in the blood.

The average number of hairs on the normal human scalp is 125000 . A healthy person loses about 65 hairs per day. (New hair from the hair follicle pushes the old hair out.) (a) How many hairs are lost in one month? (b) Pattern baldness (top-of-the-head hair loss) affects about 35 million men in the United States. If an average of \(15 \%\) of the scalp is bald, how many hairs are lost per year by one of these "bald is beautiful" people?

A car is driven 13 mi east and then a certain distance due north, ending up at a position \(25^{\circ}\) north of east of its initial position. (a) The distance traveled by the car due north is (1) less than, (2) equal to, (3) greater than 13 mi. Why? (b) What distance due north does the car travel?

At the Indianapolis 500 time trials, each car makes four consecutive laps, with its overall or average speed determining that car's place on race day. Each lap covers \(2.5 \mathrm{mi}\) (exact). During a practice run, cautiously and gradually taking his car faster and faster, a driver records the following average speeds for each successive lap: \(160 \mathrm{mi} / \mathrm{h}, 180 \mathrm{mi} / \mathrm{h}, 200 \mathrm{mi} / \mathrm{h},\) and \(220 \mathrm{mi} / \mathrm{h}\) (a) Will his average speed be (1) exactly the average of these speeds \((190 \mathrm{mi} / \mathrm{h}),\) (2) greater than \(190 \mathrm{mi} / \mathrm{h},\) or (3) less than \(190 \mathrm{mi} / \mathrm{h}\) ? Explain. (b) To corroborate your conceptual reasoning, calculate the car's average speed.