Solved: Newtons law of universal gravitation is

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 significant figures. (a) 6.783 0 3 103 m2 (b) 6.783 3 103 m2 (c) 6.78 3 103 m2 (d) 6.8 3 103 m2 (e) 7 3 103 m2

Suppose two quantities, A and B, have different dimensions. Determine which of the following arithmetic operations could be physically meaningful. (a) A 1 B (b) B 2 A (c) A 2 B (d) A/B (e) AB

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

Use the rules for significant figures to find the answer to the addition problem 21.4 1 15 1 17.17 1 4.003. 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

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?

Find the polar coordinates corresponding to a point located at (25.00, 12.00) in Cartesian coordinates. (a) (13.0, 267.4) (b) (13.0, 113) (c) (14.2, 267.4) (d) (14.2, 113) (e) (19, 272.5)

Which of the following relationships is dimensionally consistent with an expression yielding a value for accel-eration? 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

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 one liter is close to one 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

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

A calculator displays a result as 1.365 248 0 3 107 kg. The estimated uncertainty in the result is 62 3 105 kg. How many digits should be included as significant when the result is written down? (a) zero (b) one (c) two (d) three (e) four

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 10210 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?

(a) If an equation is dimensionally correct, does this mean that the equation must be true? (b) If an equation is not dimensionally correct, does this mean that the equation cant be true? Explain your answers.

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, defined 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) Suppose that the displacement of an object is related to time according to the expression x 5 Bt 2. What are the dimensions of B? (b) A displacement is related to time as x 5 A sin (2pft), where A and f are constants. Find the dimensions of A. (Hint: A trigonometric function appearing in an equation must be dimensionless.)

A shape that covers an area A and has a uniform height h has a volume V 5 Ah. (a) Show that V 5 Ah is dimensionally correct. (b) Show that the volumes of a cylinder and of a rectangular box can be written in the form V 5 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: (a) 12 mv2 5 12 mv0 2 1 !mgh (b) v 5 v0 1 at2 (c) ma 5 v2. 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 carpet is to be installed in a room of length 9.72 m and width 5.3 m. Find the area of the room retaining the proper number of significant figures.

Use your calculator to determine (!8)3 to three significant figures in two ways: (a) Find !8 to four significant figures; then cube this number and round to three significant figures. (b) Find !8 to three significant figures; then cube this number and round to three significant figures. (c) Which answer is more accurate? Explain.

How many significant figures are there in (a) 78.9 6 0.2, (b) 3.788 3 109, (c) 2.46 3 1026, (d) 0.003 2

The speed of light is now defined to be 2.997 924 58 3 108 m/s. Express the speed of light to (a) three significant figures, (b) five significant figures, and (c) seven significant figures.

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 significant figures. (b) Now multiply the rounded result of part (a) by the height and again round, obtaining the volume. (c) Repeat the process, first finding 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 significant figure.

The radius of a circle is measured to be (10.5 6 0.2) m. Calculate (a) the area and (b) the circumference of the circle, and give the uncertainty in each value.

The edges of a shoebox are measured to be 11.4 cm, 17.8 cm, and 29 cm. Determine the volume of the box retaining the proper number of significant figures in your answer.

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 3 356.3; (c) the product 5.620 3 p.

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 find the distance in fathoms.

A small turtle moves at a speed of 186 furlongs per fortnight. Find the speed of the turtle in centimeters per second. Note that 1 furlong 5 220 yards and 1 fortnight 5 14 days.

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

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 car is traveling at a speed of 38.0 m/s on an interstate highway where the speed limit is 75.0 mi/h. Is the driver exceeding the speed limit? Justify your answer.

A certain car has a fuel efficiency of 25.0 miles per gallon (mi/gal). Express this efficiency in kilometers per liter (km/L).

The diameter of a sphere is measured to be 5.36 in. Find (a) the radius of the sphere in centimeters, (b) the surface area of the sphere in square centimeters, and (c) the volume of the sphere in cubic centimeters.

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 3 108 m/s. Convert this figure to miles per hour.

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?

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 to a depth of one foot. An acre is 43 560 ft2. Find the 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 5 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 5 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.)

Estimate the number of steps you would have to take to walk a distance equal to the circumference of the Earth.

Estimate the number of breaths taken by a human being during an average lifetime.

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 one micron = 1026 m. Estimate the intestinal volume and assume bacteria occupy one hundredth of it.) (b) Discuss your answer to part (a). Are these bacteria beneficial, dangerous, or neutral? What functions could they serve?

Treat a cell in a human as a sphere of radius 1.0 mm. (a) Determine the volume of a cell. (b) Estimate the volume of your body. (c) Estimate the number of cells in your body.

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 (1026 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 coordinates r 5 2.5 m and u 5 35. Find the x- and y-coordinates 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 fly 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 fly from the corner of the room?

Express the location of the fly in Problem 36 in polar coordinates.

Two points in a rectangular coordinate system have the coordinates (5.0, 3.0) and (23.0, 4.0), where the units are centimeters. Determine the distance between these points.

Two points are given in polar coordinates by (r, u) 5 (2.00 m, 50.0) and (r, u) 5 (5.00 m, 250.0), respectively. What is the distance between them?

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 1 sin2 u 5 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 un known side, (b) the tangent of u, and (c) the sine of f?

A ladder 9.00 m 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, find (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 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) 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 first solving one equation for x and substituting the result into the other equation.

A surveyor measures the distance across a straight river by the following method: Starting directly across from a tree on the opposite bank, he walks x 5 100 m along the riverbank to establish a baseline. Then he sights across to the tree. The angle from his baseline to the tree is u 5 35.0 (Fig. P1.49). How wide is the river?

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, find 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 defined 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 find 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 3 1023 kg. (a) Determine the mass of 1.0 m3 of water. Assuming that biological substances are 98% water, estimate the masses of (b) a cell with a diameter of 1.0 mm, (c) a human kidney, and (d) a fly. Take a kidney to be roughly a sphere with a radius of 4.0 cm and a fly to be roughly a cylinder 4.0 mm long and 2.0 mm in diameter.

Soft drinks are commonly sold in aluminum containers. (a) To an order of magnitude, how many such containers are thrown away or recycled each year by U.S. consumers? (b) 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 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 the analysis give the value of k?

Assume that it takes 7.00 minutes 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 = 231 in.3)

One gallon of paint (volume = 3.79 3 103 m3) covers an area of 25.0 m2. What is the thickness of the fresh paint on the wall?

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?

Assume that there are 100 million passenger cars in the United States and that the average fuel consumption 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 average fuel consumption could be increased to 25 mi/gal?

At the time of this books printing, the U.S. national debt was about $10 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 ten 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 1026 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 find how long it would take to fill 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.

The nearest neutron star (a collapsed star made primarily of neutrons) is about 3.00 3 1018 m away from Earth. Given that the Milky Way galaxy (Fig. P1.63) is roughly a disk of diameter , 1021 m and thickness , 1019 m, estimate the number of neutron stars in the Milky Way to the nearest order of magnitude.