(a) Compute the order of magnitude of the mass of abathtub half full of water. (b)

One student uses a meterstick to measure the thickness of a textbook and obtains 4.3 cm 6 0.1 cm. Other students measure the thickness with vernier calipers and obtain four different measurements: (a) 4.32 cm 6 0.01 cm, (b) 4.31 cm 6 0.01 cm, (c) 4.24 cm 6 0.01 cm, and (d) 4.43 cm 6 0.01 cm. Which of these four measurements, if any, agree with that obtained by the first student?

A house is advertised as having 1 420 square feet under its roof. What is its area in square meters? (a) 4 660 m2 (b) 432 m2 (c) 158 m2 (d) 132 m2 (e) 40.2 m2

Answer each question yes or no. Must two quantities have the same dimensions (a) if you are adding them? (b) If you are multiplying them? (c) If you are subtracting them? (d) If you are dividing them? (e) If you are equating them?

The price of gasoline at a particular station is 1.5 euros per liter. An American student can use 33 euros to buy gasoline. Knowing that 4 quarts make a gallon and that 1 liter is close to 1 quart, she quickly reasons that she can buy how many gallons of gasoline? (a) less than 1 gallon (b) about 5 gallons (c) about 8 gallons (d) more than 10 gallons

Rank the following five quantities in order from the largest to the smallest. If two of the quantities are equal, give them equal rank in your list. (a) 0.032 kg (b) 15 g (c) 2.7 3 105 mg (d) 4.1 3 1028 Gg (e) 2.7 3 108 mg

What is the sum of the measured values 21.4 s 1 15 s 1 17.17 s 1 4.00 3 s? (a) 57.573 s (b) 57.57 s (c) 57.6 s (d) 58 s (e) 60 s

Which of the following is the best estimate for the mass of all the people living on the Earth? (a) 2 3 108 kg (b) 1 3 109 kg (c) 2 3 1010 kg (d) 3 3 1011 kg (e) 4 3 1012 kg

(a) If an equation is dimensionally correct, does that mean that the equation must be true? (b) If an equation is not dimensionally correct, does that mean that the equation cannot be true?

Newtons second law of motion (Chapter 5) says that the mass of an object times its acceleration is equal to the net force on the object. Which of the following gives the correct units for force? (a) kg ? m/s2 (b) kg ? m2/s2 (c) kg/m ? s2 (d) kg ? m2/s (e) none of those answers

A calculator displays a result as 1.365 248 0 3 107 kg. The estimated uncertainty in the result is 62%. How many digits should be included as significant when the result is written down? (a) zero (b) one (c) two (d) three (e) four

Kinetic energy K (Chapter 7) has dimensions kg ? m2/s2. It can be written in terms of the momentum p (Chapter 9) and mass m as (a) Determine the proper units for momentum using dimensional analysis. (b) The unit of force is the newton N, where 1 N 5 1 kg ? m/s2 . What are the units of momentum p in terms of a newton and another fundamental SI unit?

Newtons law of universal gravitation is represented by F 5 GMm r 2 where F is the magnitude of the gravitational force exerted by one small object on another, M and m are the masses of the objects, and r is a distance. Force has the SI units kg ? m/s2. What are the SI units of the proportionality constant G?

The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position as x 5 kamtn, where k is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if m 5 1 and n 5 2. Can this analysis give the value of k?

(a) Assume the equation x 5 At 3 1 Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B. (b) Determine the dimensions of the derivative dx/dt 5 3At2 1 B.

A solid piece of lead has a mass of 23.94 g and a volume of 2.10 cm3. From these data, calculate the density of lead in SI units (kilograms per cubic meter).

An ore loader moves 1 200 tons/h from a mine to the surface. Convert this rate to pounds per second, using 1 ton 5 2 000 lb.

A rectangular building lot has a width of 75.0 ft and a length of 125 ft. Determine the area of this lot in square meters.

Suppose your hair grows at the rate 1/32 in. per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of 0.1 nm, your answer suggests how rapidly layers of atoms are assembled in this protein synthesis.

Why is the following situation impossible? A students dormitory room measures 3.8 m by 3.6 m, and its ceiling is 2.5 m high. After the student completes his physics course, he displays his dedication by completely wallpapering the walls of the room with the pages from his copy of volume 1 (Chapters 122) of this textbook. He even covers the door and window.

A pyramid has a height of 481 ft, and its base covers an area of 13.0 acres (Fig. P1.20). The volume of a pyramid is given by the expression V 5 13 Bh, where B is the area of the base and h is the height. Find the volume of this pyramid in cubic meters. (1 acre 5 43 560 ft2)

The pyramid described in Problem 20 contains approximately 2 million stone blocks that average 2.50 tons each. Find the weight of this pyramid in pounds.

Assume it takes 7.00 min to fill a 30.0-gal gasoline tank. (a) Calculate the rate at which the tank is filled in gallons per second. (b) Calculate the rate at which the tank is filled in cubic meters per second. (c) Determine the time interval, in hours, required to fill a 1.00-m3 volume at the same rate. (1 U.S. gal 5 231 in.3)

A section of land has an area of 1 square mile and contains 640 acres. Determine the number of square meters in 1 acre.

A house is 50.0 ft long and 26 ft wide and has 8.0-fthigh ceilings. What is the volume of the interior of the house in cubic meters and in cubic centimeters?

One cubic meter (1.00 m3) of aluminum has a mass of 2.70 3 103 kg, and the same volume of iron has a mass of 7.86 3 103 kg. Find the radius of a solid aluminum sphere that will balance a solid iron sphere of radius 2.00 cm on an equal-arm balance.

Let rAl represent the density of aluminum and rFe that of iron. Find the radius of a solid aluminum sphere that balances a solid iron sphere of radius rFe on an equal-arm balance.

One gallon of paint (volume 5 3.78 3 103 m3) covers an area of 25.0 m2. What is the thickness of the fresh paint on the wall?

An auditorium measures 40.0 m 3 20.0 m 3 12.0 m. The density of air is 1.20 kg/m3. What are (a) the volume of the room in cubic feet and (b) the weight of air in the room in pounds?

(a) At the time of this books printing, the U.S. national debt is about $16 trillion. If payments were made at the rate of $1 000 per second, how many years would it take to pay off the debt, assuming no interest were charged? (b) A dollar bill is about 15.5 cm long. How many dollar bills attached end to end would it take to reach the Moon? The front endpapers give the EarthMoon distance. Note: Before doing these calculations, try to guess at the answers. You may be very surprised.

A hydrogen atom has a diameter of 1.06 3 10210 m. The nucleus of the hydrogen atom has a diameter of approximately 2.40 3 10215 m. (a) For a scale model, represent the diameter of the hydrogen atom by the playing length of an American football field (100 yards 5 300 ft) and determine the diameter of the nucleus in millimeters. (b) Find the ratio of the volume of the hydrogen atom to the volume of its nucleus.

Find the order of magnitude of the number of tabletennis balls that would fit into a typical-size room (without being crushed).

(a) Compute the order of magnitude of the mass of a bathtub half full of water. (b) Compute the order of magnitude of the mass of a bathtub half full of copper coins.

To an order of magnitude, how many piano tuners reside in New York City? The physicist Enrico Fermi was famous for asking questions like this one on oral Ph.D. qualifying examinations.

An automobile tire is rated to last for 50 000 miles. To an order of magnitude, through how many revolutions will it turn over its lifetime?

A rectangular plate has a length of (21.3 6 0.2) cm and a width of (9.8 6 0.1) cm. Calculate the area of the plate, including its uncertainty.

How many significant figures are in the following numbers? (a) 78.9 6 0.2 (b) 3.788 3 109 (c) 2.46 3 1026 (d) 0.005 3

The tropical year, the time interval from one vernal equinox to the next vernal equinox, is the basis for our calendar. It contains 365.242 199 days. Find the number of seconds in a tropical year.

Carry out the arithmetic operations (a) the sum of the measured values 756, 37.2, 0.83, and 2; (b) the product 0.003 2 3 356.3; and (c) the product 5.620 3 p.

Review. In a community college parking lot, the number of ordinary cars is larger than the number of sport utility vehicles by 94.7%. The difference between the number of cars and the number of SUVs is 18. Find the number of SUVs in the lot.

Review. While you are on a trip to Europe, you must purchase hazelnut chocolate bars for your grandmother. Eating just one square each day, she makes each large bar last for one and one-third months. How many bars will constitute a years supply for her?

Review. A child is surprised that because of sales tax she must pay $1.36 for a toy marked $1.25. What is the effective tax rate on this purchase, expressed as a percentage?

Review. The average density of the planet Uranus is 1.27 3 103 kg/m3. The ratio of the mass of Neptune to that of Uranus is 1.19. The ratio of the radius of Neptune to that of Uranus is 0.969. Find the average density of Neptune.

Review. The ratio of the number of sparrows visiting a bird feeder to the number of more interesting birds is 2.25. On a morning when altogether 91 birds visit the feeder, what is the number of sparrows?

Review. Find every angle u between 0 and 360 for which the ratio of sin u to cos u is 23.00.

Review. For the right triangle shown in Figure P1.45, what are (a) the length of the unknown side, (b) the tangent of u, and (c) the sine of f?

Review. Prove that one solution of the equation 2.00x4 2 3.00x3 1 5.00x 5 70.0 is x 5 22.22.

Review. A pet lamb grows rapidly, with its mass proportional to the cube of its length. When the lambs length changes by 15.8%, its mass increases by 17.3 kg. Find the lambs mass at the end of this process.

Review. A highway curve forms a section of a circle. A car goes around the curve as shown in the helicopter view of Figure P1.48. Its dashboard compass shows that the car is initially heading due east. After it travels d 5 840 m, it is heading u 5 35.0 south of east. Find the radius of curvature of its path. Suggestion: You may find it useful to learn a geometric theorem stated in Appendix B.3.

Review. From the set of equations p 5 3q pr 5 qs 12 pr 2 1 12 qs 2 5 12 qt 2 involving the unknowns p, q, r, s, and t, find the value of the ratio of t to r.

Review. Figure P1.50 on page 18 shows students studying the thermal conduction of energy into cylindrical blocks of ice. As we will see in Chapter 20, this process is described by the equation Q Dt 5 kpd2 1Th 2 Tc 2 4L For experimental control, in one set of trials all quantities except d and Dt are constant. (a) If d is made three times larger, does the equation predict that Dt will get larger or get smaller? By what factor? (b) What pattern of proportionality of Dt to d does the equation predict? (c) To display this proportionality as a straight line on a graph, what quantities should you plot on the horizontal and vertical axes? (d) What expression represents the theoretical slope of this graph?

Review. A student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. He cuts out various shapes in various sizes, calculates their areas, measures their masses, and prepares the graph of Figure P1.51. (a) Consider the fourth experimental point from the top. How far is it from the best-fit straight line? Express your answer as a difference in vertical-axis coordinate. (b) Express your answer as a percentage. (c) Calculate the slope of the line. (d) State what the graph demonstrates, referring to the shape of the graph and the results of parts (b) and (c). (e) Describe whether this result should be expected theoretically. (f) Describe the physical meaning of the slope.

The radius of a uniform solid sphere is measured to be (6.50 6 0.20) cm, and its mass is measured to be (1.85 6 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.

A sidewalk is to be constructed around a swimming pool that measures (10.0 6 0.1) m by (17.0 6 0.1) m. If the sidewalk is to measure (1.00 6 0.01) m wide by (9.0 6 0.1) cm thick, what volume of concrete is needed and what is the approximate uncertainty of this volume?

Collectible coins are sometimes plated with gold to enhance their beauty and value. Consider a commemorative quarter-dollar advertised for sale at $4.98. It has a diameter of 24.1 mm and a thickness of 1.78 mm, and it is completely covered with a layer of pure gold 0.180 mm thick. The volume of the plating is equal to the thickness of the layer multiplied by the area to which it is applied. The patterns on the faces of the coin and the grooves on its edge have a negligible effect on its area. Assume the price of gold is $25.0 per gram. (a) Find the cost of the gold added to the coin. (b) Does the cost of the gold significantly enhance the value of the coin? Explain your answer.

In a situation in which data are known to three significant digits, we write 6.379 m 5 6.38 m and 6.374 m 5 6.37 m. When a number ends in 5, we arbitrarily choose to write 6.375 m 5 6.38 m. We could equally well write 6.375 m 5 6.37 m, rounding down instead of rounding up, because we would change the number 6.375 by equal increments in both cases. Now consider an orderof- magnitude estimate, in which factors of change rather than increments are important. We write 500 m , 103 m because 500 differs from 100 by a factor of 5 while it differs from 1 000 by only a factor of 2. We write 437 m , 103 m and 305 m , 102 m. What distance differs from 100 m and from 1 000 m by equal factors so that we could equally well choose to represent its order of magnitude as , 102 m or as , 103 m?

(a) What is the order of magnitude of the number of microorganisms in the human intestinal tract? A typical bacterial length scale is 1026 m. Estimate the intestinal volume and assume 1% of it is occupied by bacteria. (b) Does the number of bacteria suggest whether the bacteria are beneficial, dangerous, or neutral for the human body? What functions could they serve?

The diameter of our disk-shaped galaxy, the Milky Way, is about 1.0 3 105 light-years (ly). The distance to the Andromeda galaxy (Fig. P1.57), which is the spiral galaxy nearest to the Milky Way, is about 2.0 million ly. If a scale model represents the Milky Way and Andromeda galaxies as dinner plates 25 cm in diameter, determine the distance between the centers of the two plates.

Why is the following situation impossible? In an effort to boost interest in a television game show, each weekly winner is offered an additional $1 million bonus prize if he or she can personally count out that exact amount from a supply of one-dollar bills. The winner must do this task under supervision by television show executives and within one 40-hour work week. To the dismay of the shows producers, most contestants succeed at the challenge.

A high fountain of water is located at the center of a circular pool as shown in Figure P1.59. A student walks around the pool and measures its circumference to be 15.0 m. Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation of the top of the fountain to be f 5 55.0. How high is the fountain?

A water fountain is at the center of a circular pool as shown in Figure P1.59. A student walks around the pool and measures its circumference C. Next, he stands at the edge of the pool and uses a protractor to measure the angle of elevation f of his sightline to the top of the water jet. How high is the fountain?

The data in the following table represent measurements of the masses and dimensions of solid cylinders of aluminum, copper, brass, tin, and iron. (a) Use these data to calculate the densities of these substances. (b) State how your results compare with those given in Table 14.1. Mass Diameter Length Substance (g) (cm) (cm) Aluminum 51.5 2.52 3.75 Copper 56.3 1.23 5.06 Brass 94.4 1.54 5.69 Tin 69.1 1.75 3.74 Iron 216.1 1.89 9.77

The distance from the Sun to the nearest star is about 4 3 1016 m. The Milky Way galaxy (Fig. P1.62) is roughly a disk of diameter , 1021 m and thickness , 1019 m. Find the order of magnitude of the number of stars in the Milky Way. Assume the distance between the Sun and our nearest neighbor is typical.

Assume there are 100 million passenger cars in the United States and the average fuel efficiency is 20 mi/gal of gasoline. If the average distance traveled by each car is 10 000 mi/yr, how much gasoline would be saved per year if the average fuel efficiency could be increased to 25 mi/gal?

A spherical shell has an outside radius of 2.60 cm and an inside radius of a. The shell wall has uniform thickness and is made of a material with density 4.70 g/cm3. The space inside the shell is filled with a liquid having a density of 1.23 g/cm3. (a) Find the mass m of the sphere, including its contents, as a function of a. (b) For what value of the variable a does m have its maximum possible value? (c) What is this maximum mass? (d) Explain whether the value from part (c) agrees with the result of a direct calculation of the mass of a solid sphere of uniform density made of the same material as the shell. (e) What If? Would the answer to part (a) change if the inner wall were not concentric with the outer wall?

Bacteria and other prokaryotes are found deep underground, in water, and in the air. One micron (1026 m) is a typical length scale associated with these microbes. (a) Estimate the total number of bacteria and other prokaryotes on the Earth. (b) Estimate the total mass of all such microbes.

Air is blown into a spherical balloon so that, when its radius is 6.50 cm, its radius is increasing at the rate 0.900 cm/s. (a) Find the rate at which the volume of the balloon is increasing. (b) If this volume flow rate of air entering the balloon is constant, at what rate will the radius be increasing when the radius is 13.0 cm? (c) Explain physically why the answer to part (b) is larger or smaller than 0.9 cm/s, if it is different.

A rod extending between x 5 0 and x 5 14.0 cm has uniform cross-sectional area A 5 9.00 cm2. Its density increases steadily between its ends from 2.70 g/cm3 to 19.3 g/cm3. (a) Identify the constants B and C required in the expression r 5 B 1 Cx to describe the variable density. (b) The mass of the rod is given by m 5 3 all material r dV 5 3 all x rA dx 5 3 14.0 cm 0 1B 1 Cx2 19.00 cm2 2dx Carry out the integration to find the mass of the rod.

In physics, it is important to use mathematical approximations. (a) Demonstrate that for small angles (, 20) tan a < sin a < a 5 par 1808 where a is in radians and a9 is in degrees. (b) Use a calculator to find the largest angle for which tan a may be approximated by a with an error less than 10.0%.

The consumption of natural gas by a company satisfies the empirical equation V 5 1.50t 1 0.008 00t2, where V is the volume of gas in millions of cubic feet and t is the time in months. Express this equation in units of cubic feet and seconds. Assume a month is 30.0 days.

A woman wishing to know the height of a mountain measures the angle of elevation of the mountaintop as 12.0. After walking 1.00 km closer to the mountain on level ground, she finds the angle to be 14.0. (a) Draw a picture of the problem, neglecting the height of the womans eyes above the ground. Hint: Use two triangles. (b) Using the symbol y to represent the mountain height and the symbol x to represent the womans original distance from the mountain, label the picture. (c) Using the labeled picture, write two trigonometric equations relating the two selected variables. (d) Find the height y.

A child loves to watch as you fill a transparent plastic bottle with shampoo (Fig P1.71). Every horizontal cross section of the bottle is circular, but the diameters of the circles have different values. You pour the brightly colored shampoo into the bottle at a constant rate of 16.5 cm3/s. At what rate is its level in the bottle rising (a) at a point where the diameter of the bottle is 6.30 cm and (b) at a point where the diameter is 1.35 cm?

A woman stands at a horizontal distance x from a mountain and measures the angle of elevation of the mountaintop above the horizontal as u. After walking a distance d closer to the mountain on level ground, she finds the angle to be f. Find a general equation for the height y of the mountain in terms of d, f, and u, neglecting the height of her eyes above the ground.

You stand in a flat meadow and observe two cows (Fig. P1.73). Cow A is due north of you and 15.0 m from your position. Cow B is 25.0 m from your position. From your point of view, the angle between cow A and cow B is 20.0, with cow B appearing to the right of cow A. (a) How far apart are cow A and cow B? (b) Consider the view seen by cow A. According to this cow, what is the angle between you and cow B? (c) Consider the view seen by cow B. According to this cow, what is the angle between you and cow A? Hint: What does the situation look like to a hummingbird hovering above the meadow? (d) Two stars in the sky appear to be 20.0 apart. Star A is 15.0 ly from the Earth, and star B, appearing to the right of star A, is 25.0 ly from the Earth. To an inhabitant of a planet orbiting star A, what is the angle in the sky between star B and our Sun?