Mechanics

A useful and easy-to-remember approximate value for the number of seconds in a year is 𝛑 × 10⁷. Determine the percent error in this approximate value. (There are 365.24 days in one year.)

Express each approximation of π 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

How many gallons of gasoline are used in the United States in one day? Assume that there are two cars for every three people, that each car is driven an average of 10,000 miles per year, and that the average car gets 20 miles per gallon.

How many times does a typical person blink her eyes in a lifetime?

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm³ 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?

In Wagner’s opera Das Rheingold, the goddess Freia is ransomed for a pile of gold just tall enough and wide enough to hide her from sight. Estimate the monetary value of this pile. The density of gold is 19.3 g/cm³, and take its value to be about $10 per gram.

How many times does a human heart beat during a person’s lifetime? How many gallons of blood does it pump? (Estimate that the heart pumps 50 cm³ of blood with each beat.)

A postal employee drives a delivery truck along the route shown in Fig. E1.25. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram. (See also Exercise 1.32 for a different approach.)

Compute the x- and y-components of the vectors $\overrightarrow{A},\overrightarrow{B},\overrightarrow{C} \mathbf{and} \overrightarrow{D}$ in Fig. E1.24.

Let $\mathbf{\theta }$be the angle that the vector $\overrightarrow{\mathbf{A}}$ makes with the +x-axis, measured counterclockwise from that axis. Find angle $\mathbf{\theta }$ for a vector that has these components: (a) ${\mathbf{\theta A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$; (b) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$; (c) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$; (d) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{,}\mathbf{\hspace{0.33em}}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{\hspace{0.33em}}\mathbf{m}$

Vector A has y-component Ay = +9.60 m. A makes an angle of 32.0 counterclockwise from the +y-axis. (a) What is the x-component of A ? (b) What is the magnitude of A ?

Vector A  is in the direction 34.0 clockwise from the -yaxis. The x-component of A is Ax = -16.0 m. (a) What is the y-component of A ? (b) What is the magnitude of A ?

A postal employee drives a delivery truck over the route shown in Fig.below. Use the method of components to determine the magnitude and direction of her resultant displacement. 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.

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) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{8}\mathbf{.}\mathbf{60}\mathbf{ }\mathbf{cm}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{cm}$; (b) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{9}\mathbf{.}\mathbf{70}\mathbf{ }\mathbf{m}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{45}\mathbf{ }\mathbf{m}$ (c) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{75}\mathbf{ }\mathbf{km}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{70}\mathbf{ }\mathbf{km}$  

Find the magnitude and direction of the vector represented by the following pairs of components: (a)   ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{8}\mathbf{.}\mathbf{60}\mathbf{ }\mathbf{cm}\mathbf{,}\mathbf{ }{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{cm}$;   (b)   ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{9}\mathbf{.}\mathbf{70}\mathbf{ }\mathbf{m}\mathbf{,}\mathbf{ }{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{45}\mathbf{ }\mathbf{m}$  (c)  ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{75}\mathbf{ }\mathbf{km}\mathbf{,}\mathbf{ }{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{70}\mathbf{ }\mathbf{cm}$

Vector $\overrightarrow{\mathbf{A}}$ S is 2.80 cm long and is $\mathbf{60}\mathbf{°}$ above the x-axis in the first quadrant. Vector $\overrightarrow{\mathbf{B}}$ is 1.90 cm long and is $\mathbf{60}\mathbf{°}$ below the x-axis in the fourth quadrant (Fig. E1.35). Use components to find the magnitude and direction of (a) $\overrightarrow{\mathbf{A}}\mathbf{+}\overrightarrow{\mathbf{B}}$; (b) $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$; (c) $\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}$. In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative 

In each case, find the x- and y-components of vector $\overrightarrow{\mathbf{A}}$: (a) $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{3}\hat{\mathbf{j}}$; (b) $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{11}\mathbf{.}\mathbf{2}\hat{\mathbf{j}}\mathbf{-}\mathbf{9}\mathbf{.}\mathbf{91}\hat{\mathbf{i}}$; (c) $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{-}\mathbf{15}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{22}\mathbf{.}\mathbf{4}\hat{\mathbf{j}}$; (d) $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\overrightarrow{\mathbf{B}}$, where $\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{4}\hat{\mathbf{i}}\mathbf{-}\mathbf{6}\hat{\mathbf{j}}$.

Given two vectors  $\overrightarrow{A}=4.00\widehat{i}+7.00\widehat{j}$ and $\overrightarrow{B}=5.00\widehat{i}-7.00\widehat{j}$ , (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).

(a) Write each vector in Fig. E1.39 in terms of the unit vectors  $\widehat{\mathbf{i}}$  and $\widehat{\mathbf{j}}$ . (b) Use unit vectors to express vector $\overrightarrow{\mathbf{C}}$ , where $\overrightarrow{\mathbf{C}}\mathbf{=}\mathbf{3}\mathbf{.00}\overrightarrow{\mathbf{A}}\mathbf{-}\mathbf{4}\mathbf{.00}\overrightarrow{\mathbf{B}}$  (c) Find the magnitude and direction $\overrightarrow{\mathbf{C}}$ .

You are given two vectors $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{.00}\widehat{\mathbf{i}}\mathbf{+}\mathbf{6}\mathbf{.00}\widehat{\mathbf{j}}$  and $\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{7}\mathbf{.00}\widehat{\mathbf{i}}\mathbf{+}\mathbf{2}\mathbf{.00}\widehat{\mathbf{j}}$ . Let counter- clockwise angles be positive. (a) What angle does $\overrightarrow{A}$ make with the +x-axis? (b) What angle does $\overrightarrow{\mathbf{B}}$  make with the +x-axis? (c) Vector $\overrightarrow{\mathbf{C}}$ is the sum of  $\overrightarrow{\mathit{A}}$ and $\overrightarrow{\mathbf{B}}$ , so  $\overrightarrow{\mathbf{C}}\mathbf{=}\overrightarrow{\mathbf{A}}\mathbf{+}\overrightarrow{\mathbf{B}}$ What angle does  $\overrightarrow{\mathbf{C}}$ make with the +x-axis?

Given two vectors  $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.00}\widehat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.00}\widehat{\mathbf{j}}\mathbf{+}\mathbf{4}\mathbf{.00}\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{3}\mathbf{.00}\widehat{\mathbf{i}}\mathbf{+}\mathbf{1}\mathbf{.00}\widehat{\mathbf{j}}\mathbf{-}\mathbf{3}\mathbf{.00}\widehat{\mathbf{k}}$, (a) find the magnitude of each vector; (b) use unit vectors to write an expression for the vector difference $\overrightarrow{\mathbf{A}}\mathbf{-}\overrightarrow{\mathbf{B}}$ ; and (c) find the magnitude of the vector difference $\overrightarrow{\mathbf{A}}\mathbf{-}\overrightarrow{\mathbf{B}}$ . Is this the same as the magnitude of  $\overrightarrow{\mathbf{B}}\mathbf{-}\overrightarrow{\mathbf{A}}$? Explain.

(I) (a) Find the scalar product of the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ given in Exercise 1.38. (b) Find the angle between these two vectors.

(I) (a) Find the scalar product of the vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ given in Exercise 1.38. (b) Find the angle between these two vectors.

 For the vectors $\overrightarrow{A}$,$\overrightarrow{B}$ and $\overrightarrow{C}$ in Fig. E1.24, find the scalar

products . (a) $\overrightarrow{A}.\overrightarrow{B}$; (b) $\overrightarrow{B}.\overrightarrow{C}$; (c) $\overrightarrow{A}.\overrightarrow{C}$

For the vectors $\overrightarrow{\mathbf{A}}\mathbf{,}\overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{C}}$ in Fig. E1.24, find the scalar products $\left(a\right)\overrightarrow{\mathbf{A}}\mathbf{.}\overrightarrow{\mathbf{B}}\mathbf{;}\left(b\right)\overrightarrow{\mathbf{B}}\mathbf{.}\overrightarrow{\mathbf{C}}\mathbf{;}\left(a\right)\overrightarrow{\mathbf{C}}\mathbf{.}\overrightarrow{\mathbf{A}}\mathbf{.}$

For the two vectors in Fig. E1.35, find the magnitude and

direction of (a) the vector product $\overrightarrow{\mathbf{A}}\mathbf{ }\mathbf{x}\mathbf{ }\overrightarrow{\mathbf{B}}$; (b) the vector product $\overrightarrow{\mathbf{B}}\mathbf{ }\mathbf{x}\mathbf{ }\overrightarrow{\mathbf{A}}$.

For the two vectors $\overrightarrow{\mathbf{A}}$ an $\overrightarrow{\mathbf{D}}$ in Fig. E1.24, find the magnitude and direction of (a) the vector product $\overrightarrow{\mathbf{A}}\mathbf{\times }\overrightarrow{\mathbf{D}}$ ; (b) the vector product $\overrightarrow{\mathbf{D}}\mathbf{\times }\overrightarrow{\mathbf{A}}$

For the two vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ in Fig. E1.39, find (a) the scalar product ; (b) the magnitude and direction of the vector product.

Recall that density is mass divided by volume, and consult Appendix B as needed. (a) Calculate the average density of the earth in $g/c{m}^3$, assuming our planet is a perfect sphere. (b) In about 5 billion years, at the end of its lifetime, our sun will end up as a white dwarf that has about the same mass as it does now but is reduced to about 15,000 km in diameter. What will be its density at that stage? (c) A neutron star is the remnant of certain supernovae (explosions of giant stars). Typically, neutron stars are about 20 km in diameter and have about the same mass as our sun. What is a typical neutron star density in $g/c{m}^3$?

An acre has a length of one furlong $\left(1/8 \mathrm{mi} \right)$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?

In January 2006 astronomers reported the discovery of a planet, comparable in size to the earth, orbiting another star and having a mass about 5.5 times the earth’s mass. It is believed to consist of a mixture of rock and ice, similar to Neptune. If this planet has the same density as Neptune $1.76 \mathrm{g}/{\mathrm{cm}}^3$, what is its radius expressed (a) in kilometres and (b) as a multiple of earth’s radius? Consult Appendix F for astronomical data.

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 $1 \mathrm{s}$ 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 $1\mathrm{hr}$ ? (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?

A rectangular piece of aluminum is $\mathbf{7}\mathbf{.}\mathbf{60}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{01}\mathbf{ }\mathbf{cm}$ long and $\mathbf{1}\mathbf{.}\mathbf{90}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{01}\mathbf{ }\mathbf{cm}$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.

Estimate the number of atoms in your body. (Hint: Based on what you know about biology and chemistry, what are the most common types of atom in your body? What is the mass of each type of atom? Appendix $D$ gives the atomic masses of different elements, measured in atomic mass units; you can find the value of an atomic mass unit, or 1 in Appendix E.)

Two ropes in a vertical plane exert equal-magnitude forces on a hanging weight but pull with an angle of 72.0° 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 $21.0°$ west of north, and the resultant of these two pulls is $460.0 \mathrm{N}$ directly northward. Use vector components to find the magnitude of each of these pulls and the direction of the smaller pull.

A plane leaves the airport in Galisteo and flies 170 km at east of north; then it changes direction to fly 230 km at $\mathbf{36}\mathbf{°}$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?

A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.63.The pulls $\overrightarrow{\mathbf{A}}\mathbf{ }\mathbf{and}\mathbf{ }\overrightarrow{\mathbf{B}}$

 have equal magnitudes and must combine to produce an outward traction force of 12.8 N on the patient’s arm.

How large should these pulls be?

A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, next 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of the starting point (Fig. P1.64). Find the magnitude and direction of the third leg of the journey. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

You leave the airport in College Station and fly 23.0 km in a direction $\mathbf{34}\mathbf{°}$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?

On a training flight, a student pilot flies from Lincoln, Nebraska, to Clarinda, Iowa, next to St. Joseph, Missouri, and then to Manhattan, Kansas

(Fig. P1.66). The directions are shown relative to north: $\mathbf{0}\mathbf{°}$is north, $\mathbf{90}\mathbf{°}$ is east, $\mathbf{180}\mathbf{°}$is south, and $\mathbf{270}\mathbf{°}$is west. Use the method of components to find (a) the distance she has to fly from Manhattan to get back to Lincoln, and (b) the direction (relative to north) she must fly to get there. Illustrate your solutions with a vector diagram.

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^° 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^° 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 cm from your truck, in a direction 58.0^° 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^° 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 produce a pull of 232 N when the forearm is raised 43^° 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 N, upward.)

You decide to go to your favourite neighbourhood 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)?

 A fence post is $52.0m$ from where you are standing, in a direction $37.0°$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 $68.0m$ ?