Chapter 1

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

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

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

The density of lead is 11.3 \(\mathrm{g} / \mathrm{cm}^{3} .\) 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 \((\mathrm{L})\) by using only the conversions \(1 \mathrm{L}=1000 \mathrm{cm}^{3}\) and \(1 \mathrm{in.}=2.54 \mathrm{cm} .\)

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

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

While driving in an exotic foreign land you see a speed limit sign on a highway 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 \(=5280 \mathrm{ft}\) and \(1 \mathrm{h}=3600 \mathrm{s}\) to convert 60 \(\mathrm{mph}\) to units of \(\mathrm{ff} / \mathrm{s} .(\mathrm{b})\) The acceleration of a freely falling object is 32 \(\mathrm{ff} / \mathrm{s}^{2} .\) Use \(1 \mathrm{ft}=30.48 \mathrm{cm}\) to express this acceleration in units of \(\mathrm{m} / \mathrm{s}^{2} .\) (c) The density of water is 1.0 \(\mathrm{g} / \mathrm{cm}^{3} .\) Convert this density to units of \(\mathrm{kg} / \mathrm{m}^{3} .\)

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

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.)

With a wooden ruler you measure the length of a rectangular piece of sheet metal to be 12 \(\mathrm{mm}\) . You use micrometer calipers to measure the width of the rectangle and obtain the value 5.98 \(\mathrm{mm}\). Give your answers to the following questions to the correct number of significant figures. (a) What is the area of the rectangle? (b) What is the ratio of the rectangle's width to its length? (c) What is the perimeter of the rectangle? (d) What is the difference between the length and width? (e) What is the ratio of the length to the width?

A rectangular piece of aluminum is \(5.10 \pm 0.01 \mathrm{cm}\) long and \(1.90 \pm 0.01 \mathrm{cm}\) 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; see Challenge Problem \(1.98 .\))

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 \(8.50 \pm 0.02 \mathrm{cm}\) and a thickness of \(0.050 \pm 0.005 \mathrm{cm} .\) (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.

A rather ordinary middle-aged man is in the hospital for a routine check-up. The nurse writes the quantity 200 on his medical chart but forgets to inchude the units. Which of the following quantities could the 200 plausibly represent? (a) his mass in kilograms; (b) his height in meters; (c) his height in centimeters; (d) his height in millimeters; (e) his age in months.

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 \(\mathrm{cm}^{3}\) 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?

Hearing rattles from a snake, you make two rapid displacements of magnitude 1.8 \(\mathrm{m}\) and 2.4 \(\mathrm{m}\) . In sketches (roughly to scale), show how your two displacements might add up to give a resultant of magnitude (a) \(4.2 \mathrm{m} ;(\mathrm{b}) 0.6 \mathrm{m} ;(\mathrm{c}) 3.0 \mathrm{m} .\)

A spelumker is surveying a cave. She follows a passage 180 \(\mathrm{m}\) straight west, then 210 \(\mathrm{m}\) in a direction \(45^{\circ}\) east of south, and then 280 \(\mathrm{m}\) at \(30^{\circ}\) east of north. After a fourth unmeasured 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 1.73 for a different approach to this problem.)

Use a scale drawing to find the \(x\) - and \(y\) -components of the following vectors. For each vector the numbers given are the magnitude of the vector and the angle, measured in the sense from the \(+x\) -axis toward the \(+y\) -axis, that it makes with the \(+x\) -axis: (a) magnitude \(9.30 \mathrm{m},\) angle \(60.0^{\circ} ;\)(b) magnitude \(22.0 \mathrm{km},\) angle \(135^{\circ} ;\) (c) magnitude \(6.35 \mathrm{cm},\) angle \(307^{\circ} .\)

Let the angle \(\theta\) be the angle that the vector \(\vec{A}\) makes with the \(+x\) -axis, measured counterelockwise from that axis. Find the angle \(\theta\) for a vector that has the following components: (a) \(A_{x}=2.00 \mathrm{m},\) \(A_{y}=-1.00 \mathrm{m}\) (b) \(A_{x}=2.00 \mathrm{m}, A_{y}=1.00 \mathrm{m}\) (c) \(A_{x}=-2.00 \mathrm{m}\), \(A_{y}=1.00 \mathrm{m}\) (d) \(A_{x}=-200 \mathrm{m}, A_{y}=-1.00 \mathrm{m}\).

A rocket fires two engines simultaneously. One produces a thrust of \(725 \mathrm{~N}\) directly forward, while the other gives a \(513-\mathrm{N}\) thrust at \(32.4^{\circ}\) above the forward direction. Find the magnitude and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.

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

A disoriented physics professor drives 3.25 \(\mathrm{km}\) north, then 4.75 \(\mathrm{km}\) west, and then 1.50 \(\mathrm{km}\) 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 using the method of components.

Vector \(\vec{A}\) has components \(A_{x}=1.30 \mathrm{cm}, A_{y}=2.25 \mathrm{cm} ;\) vector \(\vec{B}\) has components \(B_{x}=4.10 \mathrm{cm}, B_{y}=-3.75 \mathrm{cm} .\) Find \((\mathrm{a})\) the components of the vector sum \(\overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}} ;\) (b) the magnitude and direction of \(\overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}} ;\) (c) the components of the vector difference \(\overrightarrow{\boldsymbol{B}}-\overrightarrow{\boldsymbol{A}}\) (d) the magnitude and direction of \(\overrightarrow{\boldsymbol{B}}-\overrightarrow{\boldsymbol{A}}\)

Vector \(\vec{A}\) is 2.80 \(\mathrm{cm}\) long and is \(60.0^{\circ}\) above the \(x\) -axis in the first quadrant. Vector \(\vec{B}\) is 1.90 \(\mathrm{cm}^{2}\) long and is \(60.0^{\circ}\) below the \(x\) -axis in the fourth quadrant (Fig. 1.35\() .\) Use components to find the magnitude and direction of (a) \(\vec{A}+\vec{B}\) (b) \(\vec{A}-\vec{B} ;\) (c) \(\vec{B}-\vec{A}\) In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.

Ariver flows from south to north at 5.0 \(\mathrm{km} / \mathrm{h}\) . On this river, a boat is heading east to west perpendicular to the current at 7.0 \(\mathrm{km} / \mathrm{h}\) . As viewed by an eagle hovering at rest over the shore, how fast and in what direction is this boat traveling?

Two ropes in a vertical plane exert equal magnitude forces on a hanging weight but pull with an angle of \(86.0^{\circ}\) between them. What pull does each one exert if their resultant pull is 372 N directly upward?

Given two vectors \(\vec{A}=\) \(4.00 \hat{\imath}+3.00 \hat{\jmath}\) and \(\vec{B}=5.00 \hat{\imath}-\) \(2.00 \hat{\jmath}\) (a) find the magnitude of cach vector; (b) write an expression for the vector difference \(\vec{A}-\vec{B}\) using unit vectors; (c) find the magnitude and direction of the vector difference \(\vec{A}-\vec{B}\). (d) In a vector diagram show \(\vec{A}, \vec{B},\) and \(\vec{A}-\vec{B},\) and also show that your diagram agrees qualitatively with your answer in part (c).

(a) Is the vector \((\hat{\imath}+\hat{j}+\hat{k})\) a unit vector? Justify your answer. (b) Can a unit vector have any components with magnitude greater than unity? Can it have any negative components? In each case justify your answer. (c) If \(\overrightarrow{\boldsymbol{A}}=a(3.0 \hat{\imath}+4.0 \hat{\mathbf{y}}),\) where \(\boldsymbol{a}\) is a constant, determine the value of \(a\) that makes \(\vec{A}\) a unit vector.

(a) Use vector components to prove that two vectors commute for both addition and the scalar product. (b) Prove that two vectors anticommute for the vector product; that is, prove that \(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}=-\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{A}}\).

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

By making simple sketches of the appropriate vector products, show that \((a) \vec{A} \cdot \vec{B}\) can be interpreted as the product of the magnitude of \(\overrightarrow{\boldsymbol{A}}\) times the component of \(\overrightarrow{\boldsymbol{B}}\) along \(\overrightarrow{\boldsymbol{A}}\), or the magnitude of \(\vec{B}\) times the component of \(\vec{A}\) along \(\overrightarrow{\boldsymbol{B}}\) (b) \(|\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}|\) can be interpreted as the product of the magnitude of \(\overrightarrow{\boldsymbol{A}}\) times the component of \(\overrightarrow{\boldsymbol{B}}\) perpendicular to \(\overrightarrow{\boldsymbol{A}},\) or the magnitude of \(\overrightarrow{\boldsymbol{B}}\) times the component \(\overrightarrow{\boldsymbol{A}}\) perpendicular to \(\overrightarrow{\boldsymbol{B}}\).

The Hydrogen Maser. 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 sin \(100,000\) 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 \(\mathrm{h} 7\) (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?

Iron has a property such that a \(1.00-\mathrm{m}^{3}\) volume has a mass of \(7.86 \times 10^{3} \mathrm{kg}\) (density equals \(7.86 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} ) .\) You want to manufacture iron into cubes and spheres. Find (a) the length of the side of a cube of iron that has a mass of 200.0 \(\mathrm{g}\) and (b) the radius of a solid sphere of iron that has a mass of 200.0 \(\mathrm{g}\) .

Two workers pull horizontally on a heavy box, bnt one pulls twice as hard as the other. The larger pull is directed at \(25.0^{\circ}\) west of north, and the resultant of these two pulls is 350.0 \(\mathrm{N}\) directly northward. Use vector components to find the magnitude of each of these pulls and the direction of the smaller pull.

Emergency Landing. A plane leaves the airport in Galisteo and flies 170 \(\mathrm{km}\) at \(68^{\circ}\) east of north and then changes direction to fly 230 \(\mathrm{km}\) at \(48^{\circ}\) 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?

As noted in Exercise \(1.33,\) a spelunker is surveying a cave. She follows a passage 180 \(\mathrm{m}\) straight west, then 210 \(\mathrm{m}\) in a direction \(45^{\circ}\) east of south, and then 280 \(\mathrm{m}\) at \(30^{\circ}\) east of north. After a fourth unmeasured 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.

Getting Back. An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 40 steps northeast, then 80 steps \(60^{\circ}\) north of west, then 50 steps due south. Assume his steps all have equal length. (a) Sketch, roughly to scale, the three vectors and their resultant, (b) Save the explorer from becoming hopelessly lost in the jungle by giving him the displacement, calculated using the method of components, that will return him to his hut.

A ship leaves the island of Guam and sails 285 \(\mathrm{km}\) at \(40.0^{\circ}\) north of west. In which direction must it now head and how far must it sail so that its resultant displacement will be 115 \(\mathrm{km}\) directly east of Guam?

Bones and Muscles A patient in therapy has a forearm that weighs 20.5 \(\mathrm{N}\) and that lifts a \(112.0-\mathrm{N}\) weight. These two forces have direction vertically downward. The only other significant forces on his forearm come from the biceps muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produces a pull of 232 \(\mathrm{N}\) when the forearm is raised \(43^{\circ}\) 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 be 132.5 \(\mathrm{N}\) , upward.)

You are hungry and decide to go to your favorite neighbor- hood fast-food restaurant. You leave your apartment and take the elevator 10 flights down (each flight is 3.0 \(\mathrm{m}\) ) and then go 15 \(\mathrm{m}\) south to the apartment exit. You then proceed 0.2 \(\mathrm{km}\) east, turn north, and go 0.1 \(\mathrm{km}\) to the entrance of the restaurant. (a) Determine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, being sure to make clear 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 \(\mathrm{m}\) directly south, then turn and walk 1.25 \(\mathrm{km}\) at \(30.0^{\circ}\) west of north, and finally walk 1.00 \(\mathrm{km}\) at \(40.0^{\circ}\) 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, check it with a graphical solution drawn roughly to scale.

You are camping with two friends, 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 \(\mathrm{m}\) from yours, in the direction \(23.0^{\circ}\) south of east. Karl's tent is 32.0 \(\mathrm{m}\) from yours, in the direction \(37.0^{\circ}\) north of east. What is the distance between Karl's tent and Joe's tent?

Given two vectors \(\overrightarrow{\boldsymbol{A}}=-2.00 \hat{\mathfrak{i}}+3.00 \hat{\mathbf{j}}+4.00 \hat{\boldsymbol{k}}\) and \(\overrightarrow{\boldsymbol{B}}=\) \(3.00 \hat{t}+1.00 \hat{\jmath}-3.00 \hat{k},\) do the following. (a) Find the magnitude of each vector. (b) Write an expression for the vector difference \(\vec{A}-\vec{B},\) using unit vectors. (c) Find the magnitude of the vector difference \(\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}} .\) Is this the same as the magnitude of \(\overrightarrow{\boldsymbol{B}}-\overrightarrow{\boldsymbol{A}} ?\) Explain.

Bond Angle in Methane. In the methane molecule, \(\mathbf{C H}_{4}\), each hydrogen atom is at a corner of a regular terrahedron with the carbon atom at the center. In coordinates where one of the \(C-H\) bonds is in the direction of \(\hat{\imath}+\hat{\jmath}+\hat{k},\) an adjacent \(\mathbf{C}-\mathbf{H}\) bond is in the \(\hat{\imath}-\hat{\jmath}-\hat{k}\) direction. Calculate the angle between these two bonds.

The two vectors \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) are drawn from a common point, and \(\overrightarrow{\boldsymbol{C}}=\overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}},\) (a) Show that if \(\boldsymbol{C}^{2}=\boldsymbol{A}^{2}+\boldsymbol{B}^{2},\) the angle between the vectors \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) is \(90^{\circ} .\) (b) Show that if \(C^{2}< A^{2}+B^{2},\) the angle between the vectors \(\vec{A}\) and \(\vec{B}\) is greater than \(90^{\circ}\) (c) Show that if \(C^{2}>A^{2}+B^{2},\) the angle between the vectors \(\vec{A}\) and \(\vec{B}\) is between \(0^{\circ}\) and \(90^{\circ} .\)

When two vectors \(\vec{A}\) and \(\vec{B}\) are drawn from a common point, the angle between them is \(\phi\) . (a) Using vector techniques, show that the magnitude of their vector sum is given by $$\sqrt{A^{2}+B^{2}+2 A B \cos \phi}$$ (b) If \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) have the same magnitude, for which value of \(\boldsymbol{\phi}\) will their vector sum have the same magnitude as \(\overrightarrow{\boldsymbol{A}}\) or \(\overrightarrow{\boldsymbol{B}}\) ?

You are given vectors \(\overrightarrow{\boldsymbol{A}}=5.0 \hat{\mathfrak{x}}-6.5 \hat{\mathbf{j}}\) and \(\vec{B}=-3.5 \hat{\imath}+\) 7.0\(\hat{\mathrm{J}} .\) A third vector \(\overrightarrow{\boldsymbol{C}}\) lies in the \(x y\) -plane. Vector \(\overrightarrow{\boldsymbol{C}}\) is perpendicular to vector \(\overrightarrow{\boldsymbol{A}},\) and the scalar product of \(\overrightarrow{\boldsymbol{C}}\) with \(\overrightarrow{\boldsymbol{B}}\) is \(15.0 .\) From this information, find the components of vector \(\overrightarrow{\boldsymbol{C}}\).

Two vectors \(\vec{A}\) and \(\vec{B}\) have magaitude \(A=3.00\) and \(B=3.00 .\) Their vector product is \(\vec{A} \times \vec{B}=-5.00 k+2.00 \hat{i}\). What is the angle between \(\vec{A}\) and \(\vec{B} ?\)

Later in our sudy of physics we will encounter quantities represented by \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) , (a) Prove that for any three vectors \(\vec{A}, \vec{B},\) and \(\overrightarrow{\boldsymbol{C}}, \overrightarrow{\boldsymbol{A}} \cdot(\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{C}})=(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) (b) Calculate \((\vec{A} \times \vec{B}) \cdot \vec{C}\) for the three vectors \(\vec{A}\) with magnitude \(A=5.00\) and angle \(\theta_{A}=26.0^{\circ}\) measured in the sense from the \(+x\)-axis toward the \(+y\) -axis, \(\overrightarrow{\boldsymbol{B}}\) with \(B=4.00\) and \(\theta_{B}=63.0^{\circ},\) and \(\overrightarrow{\boldsymbol{C}}\) with magnitude 6.00 and in the \(+z\) -direction. Vectors \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) are in the \(x y\) -plane.