Chapter 1

Starting with the definition 1 in. = 2.54 cm, find the number of (a) kilometers in 1.00 mile and (b) feet in 1.00 km.

According to the label on a bottle of salad dressing, the volume of the contents is 0.473 liter (L). Using only the conversions 1 (L) = 1000 cm\(^3\) and 1 in. = 2.54 cm, express this volume in cubic inches.

How many nanoseconds does it take light to travel 1.00 ft in vacuum? (This result is a useful quantity to remember.)

The density of gold is 19.3 g/cm\(^3\). What is this value in kilograms per cubic meter?

The most powerful engine available for the classic 1963 Chevrolet Corvette Sting Ray developed 360 horsepower and had a displacement of 327 cubic inches. Express this displacement in liters (L) by using only the conversions 1 L = 1000 cm\(^3\) and 1 in. = 2.54 cm.

A square field measuring 100.0 m by 100.0 m has an area of 1.00 hectare. An acre has an area of 43,600 ft\(^2\). If a lot has an area of 12.0 acres, what is its area in hectares?

How many years older will you be 1.00 gigasecond from now? (Assume a 365-day year.)

While driving in an exotic foreign land, you see a speed limit sign that reads 180,000 furlongs per fortnight. How many miles per hour is this? (One furlong is \\(\frac{1}{8}\\) mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

The following conversions occur frequently in physics and are very useful. (a) Use 1 mi = 5280 ft and 1 h = 3600 s to convert 60 mph to units of ft/s. (b) The acceleration of a freely falling object is 32 ft/s\(^2\). Use 1 ft = 30.48 cm to express this acceleration in units of m/s\(^2\). (c) The density of water is 1.0 g/cm\(^3\). Convert this density to units of kg/m\(^3\).

In the fall of 2002, scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about 60 kg. The critical mass of a fissionable material is the minimum amount that must be brought together to start a nuclear chain reaction. Neptunium-237 has a density of 19.5 g/cm\(^3\). What would be the radius of a sphere of this material that has a critical mass?

(a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 mg/day for males. Express this quantity in \(\mu\)g/day. (b) For adults, the RDA of the amino acid lysine is 12 mg per kg of body weight. How many grams per day should a 75-kg adult receive? (c) A typical multivitamin tablet can contain 2.0 mg of vitamin B\(_2\) (riboflavin), and the RDA is 0.0030 g/day. How many such tablets should a person take each day to get the proper amount of this vitamin, if he gets none from other sources? (d) The RDA for the trace element selenium is 0.000070 g/day. Express this dose in mg/day.

With a wooden ruler, you measure the length of a rectangular piece of sheet metal to be 12 mm. With micrometer calipers, you measure the width of the rectangle to be 5.98 mm. Use the correct number of significant figures: What is (a) the area of the rectangle; (b) the ratio of the rectangle's width to its length; (c) the perimeter of the rectangle; (d) the difference between the length and the width; and (e) the ratio of the length to the width?

A useful and easy-to-remember approximate value for the number of seconds in a year is \(\pi\) \\(\times\\) 10\(^7\). Determine the percent error in this approximate value. (There are 365.24 days in one year.)

Express each approximation of \(\pi\) to six significant figures: (a) 22/7 and (b) 355/113. (c) Are these approximations accurate to that precision?

A rather ordinary middle-aged man is in the hospital for a routine checkup. The nurse writes "200" on the patient's medical chart but forgets to include the units. Which of these quantities could the 200 plausibly represent? The patient's (a) mass in kilograms; (b) height in meters; (c) height in centimeters; (d) height in millimeters; (e) age in months.

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm\(^3\) of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain all this air?

A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction \(45^{\circ}\) east of south. and then \(280 \mathrm{~m}\) at \(30^{\circ}\) east of north. After a fourth displacement,she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement. (See also Problem 1.61 for a different approach.)

Let \(\theta\) be the angle that the vector \(\overrightarrow{A}\) makes with the \(+$$x\)-axis, measured counterclockwise from that axis. Find angle \(\theta\) for a vector that has these components: (a) A\(_x\) = 2.00 m, A\(_y\) = \(-\)1.00 m; (b) A\(_x\) = 2.00 m, A\(_y\) = 1.00 m; (c) A\(_x\) = \(-\)2.00 m, A\(_y\) = 1.00 m; (d) A\(_x\) = -2.00 m, A\(_y\) = -1.00 m.

Vector \(\overrightarrow{A}\) has \(y\)-component A\(_y\) = \(+\)9.60 m. \(\overrightarrow{A}\) makes anangle of 32.0\(^{\circ}\) counterclockwise from the \(+$$y\)-axis. (a) What is the \(x\)-component of \(\overrightarrow{A}\)? (b) What is the magnitude of \(\overrightarrow{A}\)?

Vector \(\overrightarrow{A}\) is in the direction 34.0\(^{\circ}\) clockwise from the \(-$$y\)-axis. The \(x\)-component of \(\overrightarrow{A}\) is A\(_x\) = -16.0 m. (a) What is the \(y\)-component of \(\overrightarrow{A}\)? (b) What is the magnitude of \(\overrightarrow{A}\)?

A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Find the magnitude and direction of the vector represented by the following pairs of components: (a) A\(x\) = \(-\)8.60 cm, A\(y\) = 5.20 cm; (b) A\(x\) = \(-\)9.70 m, A\(y\) = -2.45 m; (c) A\(x\) = 7.75 km, A\(y\) = \(-\)2.70 km.

Vector \(\overrightarrow{A}\) is 2.80 cm long and is 60.0\(^{\circ}\) above the \(x\)-axis in the first quadrant. Vector \(\overrightarrow{B}\) is 1.90 cm long and is 60.0\(^{\circ}\) below the x-axis in the fourth quadrant (Fig. E1.35).Use components to find the magnitude and direction of (a) \(\overrightarrow{A}\) \(+\) \(\overrightarrow{B}\); (b) \(\overrightarrow{A}\) \(-\) \(\overrightarrow{B}\); (c) \(\overrightarrow{B}\) \(-\) \(\overrightarrow{A}\). In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.

In each case, find the \(x\)- and \(y\)-components of vector \(\overrightarrow{A}\): (a) \(\overrightarrow{A}\) = 5.0\(\hat{\imath}\) \(-\) 6.3\(\hat{\jmath}\); (b) \(\overrightarrow{A}\) = 11.2\(\hat{\jmath}\) \(-\) 9.91\(\hat{\imath}\); (c) \(\overrightarrow{A}\) = \(-\)15.0\(\hat{\imath}\) \(+\) 22.4\(\hat{\jmath}\) ; (d) \(\overrightarrow{A}\) = 5.0\(\overrightarrow{B}\), where \(\overrightarrow{B}\) = 4\(\hat{\imath}\) \(+\) 6\(\hat{\jmath}\).

Given two vectors \(\overrightarrow{A}\) = 4.00\(\hat{\imath}\) \(+\) 7.00\(\hat{\jmath}\) and \(\overrightarrow{B}\) = 5.00\(\hat{\imath}\) \(-\) 2.00\(\hat{\jmath}\), (a) find the magnitude of each vector; (b) use unit vectors to write an expression for the vector difference \(\overrightarrow{A}\) \(-\) \(\overrightarrow{B}\); and (c) find the magnitude and direction of the vector difference \(\overrightarrow{A}\) \(-\) \(\overrightarrow{B}\). (d) In a vector diagram show \(\overrightarrow{A}\), \(\overrightarrow{B}\), and \(\overrightarrow{A}\) \(-\) \(\overrightarrow{B}\), and show that your diagram agrees qualitatively with your answer to part (c).

You are given two vectors \(\overrightarrow{A}\) = \(-\)3.00\(\hat{\imath}\) \(+\) 6.00\(\hat{\jmath}\) and \(\overrightarrow{B}\) = 7.00\(\hat{\imath}\) \(+\) 2.00\(\hat{\jmath}\). Let counterclockwise angles be positive. (a) What angle does \(\overrightarrow{A}\) make with the \(+\)x-axis? (b) What angle does \(\overrightarrow{B}\) make with the \(+\)x-axis? (c) Vector \(\overrightarrow{C}\) is the sum of \(\overrightarrow{A}\) and \(\overrightarrow{B}\), so \(\overrightarrow{C}\) = \(\overrightarrow{A}\) \(+\) \(\overrightarrow{B}\). What angle does \(\overrightarrow{C}\) make with the \(+\)x-axis?

Given two vectors \(\overrightarrow{A}\) = \(-\)2.00\(\hat{\imath}\) \(+\) 3.00\(\hat{\jmath}\) \(+\) 4.00\(\hat{k}\) and \(\overrightarrow{B}\) = 3.00\(\hat{\imath}\) \(+\) 1.00\(\hat{\jmath}\) \(-\) 3.00\(\hat{k}\), (a) find the magnitude of each vector; (b) use unit vectors to write an expression for the vector difference \(\overrightarrow{A}\) \(-\) \(\overrightarrow{A}\); and (c) find the magnitude of the vector difference \(\overrightarrow{A}\) \(-\) \(\overrightarrow{A}\). Is this the same as the magnitude of \(\overrightarrow{A}\) \(-\) \(\overrightarrow{A}\)? Explain.

Find the angle between each of these pairs of vectors: (a) \(\overrightarrow{A}\) = \(-\)2.00\(\hat{\imath}\) \(+\) 6.00\(\hat{\jmath}\) and \(\overrightarrow{B}\) = 2.00\(\hat{\imath}\) \(-\) 3.00\(\hat{\jmath}\) (b) \(\overrightarrow{A}\) = 3.00\(\hat{\imath}\) \(+\) 5.00\(\hat{\jmath}\) and \(\overrightarrow{B}\) = 10.00\(\hat{\imath}\) \(+\) 6.00\(\hat{\jmath}\) (c) \(\overrightarrow{A}\) = \(-\)4.00\(\hat{\imath}\) \(+\) 2.00\(\hat{\jmath}\) and \(\overrightarrow{B}\) = 7.00\(\hat{\imath}\) \(+\) 14.00\(\hat{\jmath}\)

An acre has a length of one furlong (\\(\frac{1}{8}\\) mi) and a width one- tenth of its length. (a) How many acres are in a square mile? (b) How many square feet are in an acre? See Appendix E. (c) An acre-foot is the volume of water that would cover 1 acre of flat land to a depth of 1 foot. How many gallons are in 1 acre-foot?

A maser is a laser-type device that produces electromagnetic waves with frequencies in the microwave and radio-wave bands of the electromagnetic spectrum. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The frequency of these waves is 1,420,405,751.786 hertz. (A hertz is another name for one cycle per second.) A clock controlled by a hydrogen maser is off by only 1 s in 100,000 years. For the following questions, use only three significant figures. (The large number of significant figures given for the frequency simply illustrates the remarkable accuracy to which it has been measured.) (a) What is the time for one cycle of the radio wave? (b) How many cycles occur in 1 h? (c) How many cycles would have occurred during the age of the earth, which is estimated to be 4.6 \\(\times\\) 10\(^9\) years? (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

The density of air under standard laboratory conditions is 1.29 kg/m\(^3\), and about 20% of that air consists of oxygen. Typically, people breathe about \\(\frac{1}{2}\\) L of air per breath. (a) How many grams of oxygen does a person breathe in a day? (b) If this air is stored uncompressed in a cubical tank, how long is each side of the tank?

A rectangular piece of aluminum is 7.60 \(\pm\) 0.01 cm long and 1.90 \(\pm\) 0.01 cm wide. (a) Find the area of the rectangle and the uncertainty in the area. (b) Verify that the fractional uncertainty in the area is equal to the sum of the fractional uncertainties in the length and in the width. (This is a general result.)

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 \(\pm\) 0.02 cm and a thickness of 0.050 \(\pm\) 0.005 cm. (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

Two ropes in a vertical plane exert equal-magnitude forces on a hanging weight but pull with an angle of 72.0\(^{\circ}\) between them. What pull does each rope exert if their resultant pull is 372 N directly upward?

Two workers pull horizontally on a heavy box, but one pulls twice as hard as the other. The larger pull is directed at 21.0\(^{\circ}\) west of north, and the resultant of these two pulls is 460.0 N directly northward. Use vector components to find the magnitude of each of these pulls and the direction of the smaller pull.

As noted in Exercise 1.26, a spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45\(^{\circ}\) east of south, and then 280 m at 30\(^{\circ}\) east of north. After a fourth displacement, she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

A plane leaves the airport in Galisteo and flies 170 km at 68.0\(^{\circ}\) east of north; then it changes direction to fly 230 km at 36.0\(^{\circ}\) south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in which direction and how far should this crew fly to go directly to this plane?

You leave the airport in College Station and fly 23.0 km in a direction 34.0\(^{\circ}\) south of east. You then fly 46.0 km due north. How far and in what direction must you then fly to reach a private landing strip that is 32.0 km due west of the College Station airport?

As a test of orienteering skills, your physics class holds a contest in a large, open field. Each contestant is told to travel 20.8 m due north from the starting point, then 38.0 m due east, and finally 18.0 m in the direction 33.0\(^{\circ}\) west of south. After the specified displacements, a contestant will find a silver dollar hidden under a rock. The winner is the person who takes the shortest time to reach the location of the silver dollar. Remembering what you learned in class, you run on a straight line from the starting point to the hidden coin. How far and in what direction do you run?

An explorer in Antarctica leaves his shelter during a whiteout. He takes 40 steps northeast, next 80 steps at 60\(^{\circ}\) north of west, and then 50 steps due south. Assume all of his steps are equal in length. (a) Sketch, roughly to scale, the three vectors and their resultant. (b) Save the explorer from becoming hopelessly lost by giving him the displacement, calculated by using the method of components, that will return him to his shelter.

You are lost at night in a large, open field. Your GPS tells you that you are 122.0 m from your truck, in a direction 58.0\(^{\circ}\) east of south. You walk 72.0 m due west along a ditch. How much farther, and in what direction, must you walk to reach your truck?

A ship leaves the island of Guam and sails 285 km at 62.0\(^{\circ}\) north of west. In which direction must it now head and how far must it sail so that its resultant displacement will be 115 km directly east of Guam?

A physical therapy patient has a forearm that weighs 20.5 N and lifts a 112.0-N weight. These two forces are directed vertically downward. The only other significant forces on this forearm come from the biceps muscle (which acts perpendicular to the forearm) and the force at the elbow. If the biceps produces a pull of 232 N when the forearm is raised 43.0\(^{\circ}\) above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (The sum of the elbow force and the biceps force must balance the weight of the arm and the weight it is carrying, so their vector sum must be132.5 N, upward.)

You decide to go to your favorite neighborhood restaurant. You leave your apartment, take the elevator 10 flights down (each flight is 3.0 m), and then walk 15 m south to the apartment exit. You then proceed 0.200 km east, turn north, and walk 0.100 km to the entrance of the restaurant. (a) Determine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, clearly indicating your choice of coordinates. (b) How far did you travel along the path you took from your apartment to the restaurant, and what is the magnitude of the displacement you calculated in part (a)?

While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at 30.0\(^{\circ}\) west of north, and finally walk 1.00 km at 32.0\(^{\circ}\) north of east, where you find the treasure: a biography of Isaac Newton! (a) To return to the old oak tree, in what direction should you head and how far will you walk? Use components to solve this problem. (b) To see whether your calculation in part (a) is reasonable, compare it with a graphical solution drawn roughly to scale.

A fence post is 52.0 m from where you are standing, in a direction 37.0\(^{\circ}\) north of east. A second fence post is due south from you. How far are you from the second post if the distance between the two posts is 68.0 m?

A dog in an open field runs 12.0 m east and then 28.0 m in a direction 50.0\(^{\circ}\) west of north. In what direction and how far must the dog then run to end up 10.0 m south of her original starting point?

Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks 26.0 m in a direction 60.0\(^{\circ}\) west of north. Jane walks 16.0 m in a direction 30.0\(^{\circ}\) south of west. They then stop and turn to face each other. (a) What is the distance between them? (b) In what direction should Ricardo walk to go directly toward Jane?

You are camping with Joe and Karl. Since all three of you like your privacy, you don't pitch your tents close together. Joe's tent is 21.0 m from yours, in the direction 23.0\(^{\circ}\) south of east. Karl's tent is 32.0 m from yours, in the direction 37.0\(^{\circ}\) north of east. What is the distance between Karl's tent and Joe's tent?

In the methane molecule, CH\(_4\), each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates for which one of the C\(-\)H bonds is in the direction of \(\hat{\imath}\) + \(\hat{\jmath}\) + \(\hat{k}\), an adjacent C\(-\)H bond is in the \(\hat{\imath}\) \(-\) \(\hat{\jmath}\) \(-\) \(\hat{k}\) direction. Calculate the angle between these two bonds.