Soft drinks are commonly sold in aluminum containers.To an

A shape that covers an area A and has a uniform height h has a volume V Ah. (a) Show that V Ah is dimension- ally 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.)

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

The period of a simple pendulum, defined as the time necessary for one complete oscillation, is measured in time units and is given by where is the length of the pendulum and g is the accel- eration due to gravity, in units of length divided by time squared. Show that this equation is dimensionally consis- tent. (You might want to check the formula using your keys at the end of a string and a stopwatch.)

Each of the following equations was given by a student during an examination: Do a dimensional analysis of each equation and explain why the equation cant be correct.

Newtons law of universal gravitation is represented by 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?

(a) One of the fundamental laws of motion states that the acceleration of an object is directly proportional to the re- sultant force on it and inversely proportional to its mass. If the proportionality constant is defined to have no di- mensions, 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?

How many significant figures are there in (a) 78.9 0.2, (b) 3.788 109, (c) 2.46 106, (d) 0.0032?

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.

Carry out the following arithmetic oper- ations: (a) the sum of the measured values 756, 37.2, 0.83, and 2.5; (b) the product 0.0032 356.3; (c) the product 5.620 .

The speed of light is now defined to be 2.99 7924 58 108 m/s. Express the speed of light to (a) three significant figures, (b) five significant figures, and (c) seven signifi- cant figures.

A farmer measures the perimeter of a rectangular field. The length of each long side of the rectangle is found to be 38.44 m, and the length of each short side is found to be 19.5 m. What is the perimeter of the field?

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

A fisherman catches two striped bass. The smaller of the two has a measured length of 93.46 cm (two decimal places, four significant figures), and the larger fish has a measured length of 135.3 cm (one decimal place, four sig- nificant figures). What is the total length of fish caught for the day?

(a) Using your calculator, find, in scientific notation with appropriate rounding, (a) the value of (2.437 104)(6.5211 109)/(5.37 104) and (b) the value of (3.14159 102)(27.01 104)/(1 234 106).

A fathom is a unit of length, usually reserved for measur- ing 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.

Find the height or length of these natural wonders in kilo- meters, meters, and centimeters: (a) The longest cave sys- tem 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 100 ft by 150 ft. De- termine the area of this lot in square meters (m2).

Suppose your hair grows at the rate of 1/32 inch per day. Find the rate at which it grows in nanometers per second. Since 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.

Using the data in Table 1.1 and the appropriate conver- sion factors, find the distance to the nearest star, in feet.

Using the data in Table 1.3 and the appropriate conver- sion factors, find the age of Earth in years.

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

A house is 50.0 ft long and 26 ft wide and has 8.0-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 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 43 560 ft2) and has a height of 481 ft (Fig. P1.24). 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) 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 high- way 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, es- timate the masses of a cell with a diameter of 1.0 m, a human kidney, and 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.

A billionaire offers to give you $1 billion if you can count out that sum with only $1 bills. Should you accept her of- fer? Assume that you can count at an average rate of one bill every second, and be sure to allow for the fact that you need about 8 hours a day for sleeping and eating.

Imagine that you are the equipment manager of a profes- sional 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.

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

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

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.

Estimate the number of Ping-Pong balls that would fit 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.

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.

A point is located in a polar coordinate system by the . Find the x- and y-coordinates of this point, assuming that the two coordi- nate 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 co- ordinates.

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.

For the triangle shown in Figure P1.39, what are (a) the length of the unknown side, (b) the tan- gent of , and (c) the sine of ?

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 hori- zontal, 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 circu- lar pool as shown in Figure P1.41. 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 P1.43, find (a) the side opposite , (b) the side adjacent to , (c) cos , (d) sin , and (e) tan

In a certain right triangle, the two sides that are perpen- dicular 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 44, what is the tangent of the angle for which 5.00 m is the opposite side?

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 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 35.0 . How wide is the river?

A restaurant offers pizzas in two sizes: small, with a radius of six inches; and large, with a radius of nine inches. A customer argues that if the small one sells for six dollars, the large should sell for nine dollars. Without doing any calculations, is the customer correct? Defend your answer. Calculate the area of each pizza to find out how much pie you are getting in each case. If the small one costs six dol- lars how much should the large cost?

The radius of the planet Saturn is 5.85 107 m, and its mass is 5.68 1026 kg (Fig. P1.48). (a) Find the density of Saturn (its mass divided by its volume) in grams per cubic centimeter. (The volume of a sphere is given by (4/3)r 3.) (b) Find the area of Saturn in square feet. (The surface area of a sphere is given by 4r 2.)

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

Compute the order of magnitude of the mass of (a) a bathtub filled with water and (b) a bathtub filled 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 magni- tude of the diameter of an oil molecule?

In 2003, the U.S. national debt was about $7 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, assum- ing that no interest were charged? (b) A dollar bill is about 15.5 cm long. If seven 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.)

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-of-magni- tude calculations.

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?

(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 sec- ond, 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.)

Choose a variety of objects that range in length from a few centimeters to a few meters. Try guessing the lengths in a unit appropriate to their size, and then use a meter stick supplied by your instructor to check your guesses. Keep trying until you can estimate consistently to within 20% of an objects actual length.

Choose a variety of objects that range in mass from a few grams to a few kilograms. Estimate the masses by hefting the objects, then use a balance supplied by your instructor to check your guesses. Keep trying until you can estimate consistently to within 30% of an objects actual mass.

You know that the measurements of a typical sheet of pa- per are 8.50 in. by 11.0 in. Convert these measurements to millimeters. Use your results to calculate the length of the diagonal of the sheet of paper by using the Pythagorean theorem. Measure the diagonal with a ruler to see how well you have done. Finally, use a suitable trig function to calculate the angle that the diagonal line makes with the horizontal. Use a protractor to verify your calculation.