Given points (r1, u1) and (r2, u2) in polar coordinates,

True or False. An equation that is dimensionally correct is always physically correct, up to a constant of proportionality.

True or False: Replacing v by r/t in the final answer also gives a dimensionally correct equation.

What would the answer have been if the width were given as 3.460 m?

Repeat the conversion, using the relationship 1.00 m/s 2.24 mi/h. Why is the answer slightly different? E

What time conversion factor would be used to further convert the answer to km/h2?

Would 1012 cells also be a reasonable estimate? What about 109 cells? Explain.

Based on the answer, about how many stacked pennies would reach the Moon?

Of the fourteen galaxies in the local group, only one, the Milky Way, is not a dwarf galaxy. Estimate the number of galaxies in the universe that are not dwarfs.

Starting with the answers to part (b), work backwards to recover the given radius and angle. Why are there slight differences from the original quantities?

Could the distance traveled by the light beam be found without using the Pythagorean Theorem? How?

What is the answer if both the distance traveled due east and the direct return distance are both doubled?

Newtons second law of motion (Chapter 4) 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 these

Suppose two quantities, A and B, have different dimensions. Determine which of the following arithmetic operations could be physically meaningful. (a) A + B (b) B - A (c) A - B (d) A/B (e) AB

A rectangular airstrip measures 32.30 m by 210 m, with the width measured more accurately than the length. Find the area, taking into account signifi cant fi gures. (a) 6.783 0 103 m2 (b) 6.783 103 m2 (c) 6.78 103 m2 (d) 6.8 103 m2 (e) 7 103 m2 4.

Use the rules for signifi cant fi gures to fi nd the answer to the addition problem 21.4  15  17.17  4.003. (a) 57.573 (b) 57.57 (c) 57.6 (d) 58 (e) 60

The Roman cubitus is an ancient unit of measure equivalent to about 445 mm. Convert the 2.00-m-height of a basketball forward to cubiti. (a) 2.52 cubiti (b) 3.12 cubiti (c) 4.49 cubiti (d) 5.33 cubiti (e) none of these

A house is advertised as having 1 420 square feet under roof. What is the area of this house in square meters? (a) 115 m2 (b) 132 m2 (c) 176 m2 (d) 222 m2 (e) none of these

Which of the following is the best estimate for the mass of all the people living on Earth? (a) 2 108 kg (b) 1 109 kg (c) 2 1010 kg (d) 3 1011 kg (e) 4 1012 kg 8. F

Find the polar coordinates corresponding to a point located at (5.00, 12.00) in Cartesian coordinates. (a) (13.0, -67.4) (b) (13.0, 113) (c)(14.2, -67.4) (d) (14.2, 113) (e) (19, -72.5)

At a horizontal distance of 45 m from a tree, the angle of elevation to the top of the tree is 26. How tall is the tree? (a) 22 m (b) 31 m (c) 45 m (d) 16 m (e) 11 m

What is the approximate number of breaths a person takes over a period of 70 years? (a) 3 106 breaths (b) 3 107 breaths (c) 3 108 breaths (d) 3 109 breaths (e) 3 1010 breaths 11.

Which of the following relationships is dimensionally consistent with an expression yielding a value for acceleration? Acceleration has the units of distance divided by time squared. In these equations, x is a distance, t is time, and v is velocity with units of distance divided by time. (a) v/t 2 (b) v/x2 (c) v2/t (d) v2/x (e) none of these

Estimate the order of magnitude of the length, in meters, of each of the following: (a) a mouse, (b) a pool cue, (c) a basketball court, (d) an elephant, (e) a city block.

What types of natural phenomena could serve as time standards?

Find the order of magnitude of your age in seconds.

An object with a mass of 1 kg weighs approximately 2 lb. Use this information to estimate the mass of the following objects: (a) a baseball; (b) your physics textbook; (c) a pickup truck.

(a) Estimate the number of times your heart beats in a month. (b) Estimate the number of human heartbeats in an average lifetime.

Estimate the number of atoms in 1 cm3 of a solid. (Note that the diameter of an atom is about 1010 m.)

The height of a horse is sometimes given in units of hands. Why is this a poor standard of length?

How many of the lengths or time intervals given in Tables 1.2 and 1.3 could you verify, using only equipment found in a typical dormitory room?

If an equation is dimensionally correct, does this mean that the equation must be true? If an equation is not dimensionally correct, does this mean that the equation cant be true?

Why is the metric system of units considered superior to most other systems of units?

How can an estimate be of value even when it is off by an order of magnitude? Explain and give an example.

The period of a simple pendulum, defi ned as the time necessary for one complete oscillation, is measured in time units and is given by T 5 2p , g where is the length of the pendulum and g is the acceleration due to gravity, in units of length divided by time squared. Show that this equation is dimensionally consistent. (You might want to check the formula using your keys at the end of a string and a stopwatch.)

The period of a simple pendulum, defi ned as the time necessary for one complete oscillation, is measured in time units and is given by T 5 2p , g where is the length of the pendulum and g is the acceleration due to gravity, in units of length divided by time squared. Show that this equation is dimensionally consistent. (You might want to check the formula using your keys at the end of a string and a stopwatch.)

A shape that covers an area A and has a uniform height h has a volume V Ah. (a) Show that V Ah is dimensionally correct. (b) Show that the volumes of a cylinder and of a rectangular box can be written in the form V Ah, identifying A in each case. (Note that A, sometimes called the footprint of the object, can have any shape and that the height can, in general, be replaced by the average thickness of the object.)

Each of the following equations was given by a student during an examination: 1 2mv 2 5 1 2mv0 2 1 !mgh v 5 v0 1 at 2 ma 5 v 2 Do a dimensional analysis of each equation and explain why the equation cant be correct.

Newtons law of universal gravitation is represented by F 5 G Mm r 2 where F is the gravitational force, M and m are masses, and r is a length. Force has the SI units kg m/s2. What are the SI units of the proportionality constant G?

Kinetic energy KE (Chapter 5) has dimensions kg m2/s2. It can be written in terms of the momentum p (Chapter 6) and mass m as KE 5 p2 2m (a) Determine the proper units for momentum using dimensional analysis. (b) Refer to Problem 5. Given the units of force, write a simple equation relating a constant force F exerted on an object, an interval of time t during which the force is applied, and the resulting momentum of the object, p.

A fi sherman catches two striped bass. The smaller of the two has a measured length of 93.46 cm (two decimal places, four signifi cant fi gures), and the larger fi sh has a measured length of 135.3 cm (one decimal place, four signifi cant fi gures). What is the total length of fi sh caught for the day?

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

How many signifi cant fi gures are there in (a) 78.9  0.2, (b) 3.788 109, (c) 2.46 106, (d) 0.003 2

The speed of light is now defi ned to be 2.997 924 58 108 m/s. Express the speed of light to (a) three signifi cant fi gures, (b) fi ve signifi cant fi gures, and (c) seven signifi - cant fi gures.

A block of gold has length 5.62 cm, width 6.35 cm, and height 2.78 cm. (a) Calculate the length times the width and round the answer to the appropriate number of signifi cant fi gures. (b) Now multiply the rounded result of part (a) by the height and again round, obtaining the volume. (c) Repeat the process, fi rst fi nding the width times the height, rounding it, and then obtaining the volume by multiplying by the length. (d) Explain why the answers dont agree in the third signifi cant fi gure.

The radius of a circle is measured to be (10.5  0.2) m. Calculate (a) the area and (b) the circumference of the circle, and give the uncertainty in each value.

Carry out the following arithmetic operations: (a) the sum of the measured values 756, 37.2, 0.83, and 2.5; (b) the product 0.003 2 356.3; (c) the product 5.620 p.

(a) Using your calculator, fi nd, in scientifi c notation with appropriate rounding, (a) the value of (2.437 104) (6.521 1 109)/(5.37 104) and (b) the value of (3.141 59 102)(27.01 104)/(1 234 106). SECTI

A fathom is a unit of length, usually reserved for measuring the depth of water. A fathom is approximately 6 ft in length. Take the distance from Earth to the Moon to be 250 000 miles, and use the given approximation to fi nd the distance in fathoms.

A furlong is an old British unit of length equal to 0.125 mi, derived from the length of a furrow in an acre of ploughed land. A fortnight is a unit of time corresponding to two weeks, or 14 days and nights. Find the speed of light in megafurlongs per fortnight. (One megafurlong equals a million furlongs.)

A fi rkin is an old British unit of volume equal to 9 gallons. How many cubic meters are there in 6.00 fi rkins?

Find the height or length of these natural wonders in kilometers, meters, and centimeters: (a) The longest cave system in the world is the Mammoth Cave system in Central Kentucky, with a mapped length of 348 miles. (b) In the United States, the waterfall with the greatest single drop is Ribbon Falls in California, which drops 1 612 ft. (c) At 20 320 feet, Mount McKinley in Alaska is Americas highest mountain. (d) The deepest canyon in the United States is Kings Canyon in California, with a depth of 8 200 ft.

A rectangular building lot measures 1.00 102 ft by 1.50 102 ft. Determine the area of this lot in square meters (m2).

Using the data in Table 1.3 and the appropriate conversion factors, fi nd the age of Earth in years.

Using the data in Table 1.1 and the appropriate conversion factors, fi nd the distance to the nearest star in feet.

Suppose your hair grows at the rate of 1/32 inch 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 atoms are assembled in this protein synthesis.

The speed of light is about 3.00 108 m/s. Convert this fi gure to miles per hour.

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

The amount of water in reservoirs is often measured in acre-ft. One acre-ft is a volume that covers an area of one acre-ft. One acre-ft is a volume that covers an area of one volume in SI units of a reservoir containing 25.0 acre-ft of water.

The base of a pyramid covers an area of 13.0 acres (1 acre 43 560 ft2) and has a height of 481 ft (Fig. P1.26). If the volume of a pyramid is given by the expression V bh/3, where b is the area of the base and h is the height, find the volume of this pyramid in cubic meters.

A quart container of ice cream is to be made in the form of a cube. What should be the length of a side, in centimeters? (Use the conversion 1 gallon 3.786 liter.)

A hamburger chain advertises that it has sold more than 50 billion hamburgers. Estimate how many pounds of hamburger meat must have been used by the chain and how many head of cattle were required to furnish the meat.

Estimate the number of Ping-Pong balls that would fi t into a typical-size room (without being crushed). In your solution, state the quantities you measure or estimate and the values you take for them.

Estimate the number of people in the world who are suffering from the common cold on any given day. (Answers may vary. Remember that a person suffers from a cold for about a week.)

(a) About how many microorganisms are found in the human intestinal tract? (A typical bacterial length scale is 106 m. Estimate the intestinal volume and assume one hundredth of it is occupied by bacteria.) (b) Discuss your answer to part (a). Are these bacteria benefi cial, dangerous, or neutral? What functions could they serve?

Grass grows densely everywhere on a quarter-acre plot of land. What is the order of magnitude of the number of blades of grass? Explain your reasoning. Note that 1 acre 43 560 ft2.

An automobile tire is rated to last for 50 000 miles. Estimate the number of revolutions the tire will make in its lifetime.

Bacteria and other prokaryotes are found deep underground, in water, and in the air. One micron (106 m) is a typical length scale associated with these microbes. (a) Estimate the total number of bacteria and other prokaryotes in the biosphere of the Earth. (b) Estimate the total mass of all such microbes. (c) Discuss the relative importance of humans and microbes to the ecology of planet Earth. Can Homo sapiens survive without them?

A point is located in a polar coordinate system by the co ordinates r 2.5 m and u 35. Find the x- and yco ordinates of this point, assuming that the two coordinate systems have the same origin.

A certain corner of a room is selected as the origin of a rectangular coordinate system. If a fl y is crawling on an adjacent wall at a point having coordinates (2.0, 1.0), where the units are meters, what is the distance of the fl y from the corner of the room?

Express the location of the fl y in Problem 36 in polar coordinates.

Two points in a rectangular coordinate system have the coordinates (5.0, 3.0) and (3.0, 4.0), where the units are centimeters. Determine the distance between these points.

Two points are given in polar coordinates by (r, u) (2.00 m, 50.0) and (r, u (5.00 m, 50.0), respectively. What is the distance between them? 4

Given points (r1, u1) and (r2, u2) in polar coordinates, obtain a general formula for the distance between them. Simplify it as much as possible using the identity cos2 u  sin2 u 1. Hint: Write the expressions for the two points in Cartesian coordinates and substitute into the usual distance formula.

For the triangle shown in Figure P1.41, what are (a) the length of the unknown side, (b) the tangent of u, and (c) the sine of f?

A ladder 9.00m long leans against the side of a building. If the ladder is inclined at an angle of 75.0 to the horizontal, what is the horizontal distance from the bottom of the ladder to the building?

A high fountain of water is located at the center of a circular pool as shown in Figure P1.43. Not wishing to get his feet wet, 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 at the bottom of the fountain to be 55.0. How high is the fountain?

A right triangle has a hypotenuse of length 3.00 m, and one of its angles is 30.0. What are the lengths of (a) the side opposite the 30.0 angle and (b) the side adjacent to the 30.0 angle?

In Figure P1.45, fi nd (a) the side opposite u, (b) the side adjacent to f, (c) cos u, (d) sin f, and (e) tan f.

In a certain right triangle, the two sides that are perpendicular to each other are 5.00 m and 7.00 m long. What is the length of the third side of the triangle?

In Problem 46, what is the tangent of the angle for which 5.00 m is the opposite side?

A woman measures the angle of elevation of a mountaintop as 12.0. After walking 1.00 km closer to the mountain on level ground, she fi nds the angle to be 14.0. (a) Draw a picture of the problem, neglecting the height of the woman's eyes above the ground. Hint: Use two triangles. (b) Select variable names for the mountain height (suggestion: y) and the womans original distance from the mountain (suggestion: x) and label the picture. (c) Using the labeled picture and the tangent function, write two trigonometric equations relating the two selected variables. (d) Find the height y of the mountain by fi rst solving one equation for x and substituting the result into the other equation.

A woman measures the angle of elevation of a mountaintop as 12.0. After walking 1.00 km closer to the mountain on level ground, she fi nds the angle to be 14.0. (a) Draw a picture of the problem, neglecting the height of the woman's eyes above the ground. Hint: Use two triangles. (b) Select variable names for the mountain height (suggestion: y) and the womans original distance from the mountain (suggestion: x) and label the picture. (c) Using the labeled picture and the tangent function, write two trigonometric equations relating the two selected variables. (d) Find the height y of the mountain by fi rst solving one equation for x and substituting the result into the other equation.

Refer to Problem 48. Suppose the mountain height is y, the womans original distance from the mountain is x, and the angle of elevation she measures from the horizontal to the top of the mountain is u. If she moves a distance d closer to the mountain and measures an angle of elevation f, fi nd a general equation for the height of the mountain y in terms of d, f, and u, neglecting the height of her eyes above the ground

(a) One of the fundamental laws of motion states that the acceleration of an object is directly proportional to the resultant force on it and inversely proportional to its mass. If the proportionality constant is defi ned to have no dimensions, determine the dimensions of force. (b) The newton is the SI unit of force. According to the results for (a), how can you express a force having units of newtons by using the fundamental units of mass, length, and time?

(a) Find a conversion factor to convert from miles per hour to kilometers per hour. (b) For a while, federal law mandated that the maximum highway speed would be 55 mi/h. Use the conversion factor from part (a) to fi nd the speed in kilometers per hour. (c) The maximum highway speed has been raised to 65 mi/h in some places. In kilometers per hour, how much of an increase is this over the 55-mi/h limit?

One cubic centimeter (1.0 cm3) of water has a mass of 1.0 103 kg. (a) Determine the mass of 1.0 m3 of water. (b) Assuming that biological substances are 98% water, estimate the masses of a cell with a diameter of 1.0 mm, a human kidney, and a fl y. Take a kidney to be roughly a sphere with a radius of 4.0 cm and a fl y to be roughly a cylinder 4.0 mm long and 2.0 mm in diameter.

Soft drinks are commonly sold in aluminum containers. To an order of magnitude, how many such containers are thrown away or recycled each year by U.S. consumers? How many tons of aluminum does this represent? In your solution, state the quantities you measure or estimate and the values you take for them.

The displacement of an object moving under uniform acceleration is some function of time and the acceleration. Suppose we write this displacement as s kamt n, where k is a dimensionless constant. Show by dimensional analysis that this expression is satisfi ed if m 1 and n 2. Can the analysis give the value of k? 56

Compute the order of magnitude of the mass of (a) a bathtub fi lled with water and (b) a bathtub fi lled with pennies. In your solution, list the quantities you estimate and the value you estimate for each.

You can obtain a rough estimate of the size of a molecule by the following simple experiment: Let a droplet of oil spread out on a smooth surface of water. The resulting oil slick will be approximately one molecule thick. Given an oil droplet of mass 9.00 107 kg and density 918 kg/m3 that spreads out into a circle of radius 41.8 cm on the water surface, what is the order of magnitude of the diameter of an oil molecule?

Sphere 1 has surface area A1 and volume V1, and sphere 2 has surface area A2 and volume V2. If the radius of sphere 2 is double the radius of sphere 1, what is the ratio of (a) the areas, A2/A1 and (b) the volumes, V2/V1?

Estimate the number of piano tuners living in New York City. This question was raised by the physicist Enrico Fermi, who was well known for making order-ofmagnitude calculations.

In 2007, the U.S. national debt was about $9 trillion. (a) If payments were made at the rate of $1 000 per second, how many years would it take to pay off the debt, assuming that no interest were charged? (b) A dollar bill is about 15.5 cm long. If nine trillion dollar bills were laid end to end around the Earths equator, how many times would they encircle the planet? Take the radius of the Earth at the equator to be 6 378 km. (Note: Before doing any of these calculations, try to guess at the answers. You may be very surprised.)

(a) How many seconds are there in a year? (b) If one micrometeorite (a sphere with a diameter on the order of 106 m) struck each square meter of the Moon each second, estimate the number of years it would take to cover the Moon with micrometeorites to a depth of one meter. (Hint: Consider a cubic box, 1 m on a side, on the Moon, and fi nd how long it would take to fi ll the box.)

Imagine that you are the equipment manager of a professional baseball team. One of your jobs is to keep baseballs on hand for games. Balls are sometimes lost when players hit them into the stands as either home runs or foul balls. Estimate how many baseballs you have to buy per season in order to make up for such losses. Assume that your team plays an 81-game home schedule in a season.