Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 6

Circular Motion and Other Applications of Newton’s Laws - all with Video Answers

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Chapter Questions

Problem 1

A light string can support a stationary hanging load of 25.0 kg before breaking. An object of mass $m=$ 3.00 $\mathrm{kg}$ attached to the string rotates on $\mathrm{a}$ frictionless, horizontal table in a circle of radius $r=0.800 \mathrm{m},$ and the other end of the string is held fixed as in Figure P6.1. What range of speeds can the object have before the string breaks?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 2

A curve in a road forms part of a horizontal circle. As a car goes around it at constant speed 14.0 m/s, the total horizontal force on the driver has magnitude 130 N. What is the total horizontal force on the driver if the speed on the same curve is 18.0 m/s instead?

Katie Mcalpine

Numerade Educator

Problem 3

In the Bohr model of the hydrogen atom, an electron moves in a circular path around a proton. The speed of the electron is approximately $2.20 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Find (a) the force acting on the electron as it revolves in a circular orbit of radius $0.530 \times 10^{-10} \mathrm{m}$ and (b) the centripetal acceleration of the electron.

Keshav Singh

Numerade Educator

Problem 4

Whenever two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the orbit to be circular and 100 $\mathrm{km}$ above the surface of the Moon, where the acceleration due to gravity is 1.52 $\mathrm{m} / \mathrm{s}^{2}$ . The radius of the Moon is $1.70 \times 10^{6} \mathrm{m} .$ Determine (a) the astronaut's orbital speed and (b) the period of the orbit.

Katie Mcalpine

Numerade Educator

Problem 5

In a cyclotron (one type of particle accelerator), a deuteron (of mass 2.00 u) reaches a final speed of 10.0% of the speed of light while moving in a circular path of radius 0.480 m. What magnitude of magnetic force is required to maintain the deuteron in a circular path?

Keshav Singh

Numerade Educator

Problem 6

A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in Figure P6.6. The length of the arc ABC is 235 m, and the car completes the turn in 36.0 s. (a) What is the acceleration when the car is at B located at an angle of $35.0^{\circ} ?$ Express your answer in terms of the unit vectors i and $\hat{\mathbf{j}}$. Determine (b) the car’s average speed and (c) its average acceleration during the 36.0-s interval.

Katie Mcalpine

Numerade Educator

Problem 7

A space station, in the form of a wheel 120 $\mathrm{m}$ in diameter, rotates to provide an "artificial gravity" of 3.00 $\mathrm{m} / \mathrm{s}^{2}$ for persons who walk around on the inner wall of the outer rim. Find the rate of the wheel's rotation in revolutions per minute that will produce this effect.

Luis Mendoza

Numerade Educator

Problem 8

Consider a conical pendulum (Fig. P6.8) with a bob of mass $m=80.0 \mathrm{kg}$ on a string of length $L=10.0 \mathrm{m}$ that makes an angle of $\theta=5.00^{\circ}$ with the vertical. Determine (a) the horizontal and vertical components of the force exerted by the string on the pendulum and (b) the radial acceleration of the bob.

Katie Mcalpine

Numerade Educator

Problem 9

A crate of eggs is located in the middle of the flatbed of a pickup truck as the truck negotiates a curve in the flat road. The curve may be regarded as an arc of a circle of radius 35.0 m. If the coefficient of static friction between crate and truck is 0.600, how fast can the truck be moving without the crate sliding?

Luis Mendoza

Numerade Educator

Problem 10

Why is the following situation impossible? The object of mass $m=4.00 \mathrm{kg}$ in Figure $\mathrm{P} 6.10$ is attached to a vertical rod by two strings of length $\ell=2.00 \mathrm{m} .$ The strings are attached to the rod at points a distance $d=3.00 \mathrm{m}$ apart. The object rotates in a horizontal circle at a constant speed of $v=3.00 \mathrm{m} / \mathrm{s}$ , and the strings remain taut. The rod rotates along with the object so that the strings do not wrap onto the rod. What If? Could this situation be possible on another planet?

Katie Mcalpine

Numerade Educator

Problem 11

A coin placed 30.0 $\mathrm{cm}$ from the center of a rotating, horizontal turntable slips when its speed is 50.0 $\mathrm{cm} / \mathrm{s}$ . (a) What force causes the centripetal acceleration when the coin is stationary relative to the turntable? (b) What is the coefficient of static friction between coin and turntable?

Keshav Singh

Numerade Educator

Problem 12

A pail of water is rotated in a vertical circle of radius 1.00 m. (a) What two external forces act on the water in the pail? (b) Which of the two forces is most important in causing the water to move in a circle? (c) What is the pail’s minimum speed at the top of the circle if no water is to spill out? (d) Assume the pail with the speed found in part (c) were to suddenly disappear at the top of the circle. Describe the subsequent motion of the water. Would it differ from the motion of a projectile?

Katie Mcalpine

Numerade Educator

Problem 13

A hawk flies in a horizontal arc of radius 12.0 $\mathrm{m}$ at constant speed 4.00 $\mathrm{m} / \mathrm{s}$ (a) Find its centripetal acceleration. (b) It continues to fly along the same horizontal arc, but increases its speed at the rate of 1.20 $\mathrm{m} / \mathrm{s}^{2}$ . Find the acceleration (magnitude and direction) in this situation at the moment the hawk's speed is 4.00 $\mathrm{m} / \mathrm{s}$.

Luis Mendoza

Numerade Educator

Problem 14

A 40.0-kg child swings in a swing supported by two chains, each 3.00 m long. The tension in each chain at the lowest point is 350 N. Find (a) the child’s speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)

Averell Hause

Carnegie Mellon University

Problem 15

A child of mass m swings in a swing supported by two chains, each of length R. If the tension in each chain at the lowest point is T, find (a) the child’s speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 16

A roller-coaster car (Fig. P6.16) has a mass of 500 kg when fully loaded with passengers. The path of the coaster from its initial point shown in the figure to point involves only up-and-down motion (as seen by the riders), with no motion to the left or right. (a) If the vehicle has a speed of 20.0 m/s at point , what is the force exerted by the track on the car at this point? (b) What is the maximum speed the vehicle can have at point $\mathbb{B}$ and still remain on the track? Assume the roller-coaster tracks at points $@$ and (B) are parts of vertical circles of radius $r_{1}=10.0 \mathrm{m}$ and $r_{2}=$ $15.0 \mathrm{m},$ respectively.

Katie Mcalpine

Numerade Educator

Problem 17

An adventurous archeologist $(m=85.0 \mathrm{kg})$ tries to cross a river by swinging from a vine. The vine is 10.0 $\mathrm{m}$ long, and his speed at the bottom of the swing is 8.00 $\mathrm{m} / \mathrm{s}$ . The archeologist doesn't know that the vine has a breaking strength of 1000 $\mathrm{N}$ . Does he make it across the river without falling in?

Luis Mendoza

Numerade Educator

Problem 18

One end of a cord is fixed and a small 0.500-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 2.00 m as shown in Figure $\mathrm{P} 6.18$ . When $\theta=20.0^{\circ}$ the speed of the object is 8.00 $\mathrm{m} / \mathrm{s}$ . At this instant, find $(\mathrm{a})$ the tension in the string, (b) the tangential and radial components of acceleration, and $(\mathrm{c})$ the total acceleration. (d) Is your answer changed if the object is swinging down toward its lowest point instead of swinging up? (e) Explain your answer to part (d).

Katie Mcalpine

Numerade Educator

Problem 19

A roller coaster at the Six Flags Great America amusement park in Gurnee, Illinois, incorporates some clever design technology and some basic physics. Each vertical loop, instead of being circular, is shaped like a teardrop (Fig. P6.19). The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure the cars remain on the track. The biggest loop is 40.0 m high. Suppose the speed at the top of the loop is 13.0 m/s and the corresponding centripetal acceleration of the riders is 2g. (a) What is the radius of the arc of the tear- drop at the top? (b) If the total mass of a car plus the riders is M, what force does the rail exert on the car at the top? (c) Suppose the roller coaster had a circular loop of radius 20.0 m. If the cars have the same speed, 13.0 m/s at the top, what is the centripetal acceleration of the riders at the top? (d) Comment on the normal force at the top in the situation described in part (c) and on the advantages of having teardrop-shaped loops.

Luis Mendoza

Numerade Educator

Problem 20

An object of mass $m=$ 5.00 kg, attached to a spring scale, rests on a frictionless, horizontal surface as shown in Figure P6.20. The spring scale, attached to the front end of a boxcar, reads zero when the car is at rest. (a) Determine the acceleration of the car if the spring scale has a constant reading of 18.0 N when the car is in motion. (b) What constant reading will the spring scale show if the car moves with constant velocity? Describe the forces on the object as observed (c) by someone in the car and (d) by someone at rest outside the car.

Katie Mcalpine

Numerade Educator

Problem 21

An object of mass $m=$ 0.500 $\mathrm{kg}$ is suspended from the ceiling of an accelerating truck as shown in Figure $\mathrm{P} 6.21$ Taking $a=3.00 \mathrm{m} / \mathrm{s}^{2}$ Taking $a=3.00 \mathrm{m} / \mathrm{s}^{2}$ find $(\mathrm{a})$ the angle $\theta$ that the string makes with the vertical and $(\mathrm{b})$ the tension $T$ in the string.

Luis Mendoza

Numerade Educator

Problem 22

A student, along with her backpack on the floor next to her, are in an elevator that is accelerating upward with acceleration $a$ . The student gives her backpack a quick kick at $t=0$ , imparting to it speed $v$ and causing it to slide across the elevator floor. At time $t,$ the backpack hits the opposite wall a distance $L$ away from the student. Find the coefficient of kinetic friction $\mu_{k}$ between the backpack and the elevator floor.

Katie Mcalpine

Numerade Educator

Problem 23

A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N. As the elevator later stops, the scale reading is 391 N. Assuming the magnitude of the acceleration is the same during starting and stopping, determine (a) the weight of the person, (b) the person’s mass, and (c) the acceleration of the elevator.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 24

A child lying on her back experiences 55.0 N tension in the muscles on both sides of her neck when she raises her head to look past her toes. Later, sliding feet first down a water slide at terminal speed 5.70 m/s and riding high on the outside wall of a horizontal curve of radius 2.40 m, she raises her head again to look forward past her toes. Find the tension in the muscles on both sides of her neck while she is sliding.

Rashmi Sinha

Numerade Educator

Problem 25

A small container of water is placed on a turntable inside a microwave oven, at a radius of 12.0 cm from the center. The turntable rotates steadily, turning one revolution in each 7.25 s. What angle does the water surface make with the horizontal?

Luis Mendoza

Numerade Educator

Problem 26

A skydiver of mass 80.0 kg jumps from a slow-moving air-craft and reaches a terminal speed of 50.0 m/s. (a) What is her acceleration when her speed is 30.0 m/s? What is the drag force on the skydiver when her speed is (b) 50.0 m/s and (c) 30.0 m/s?

Katie Mcalpine

Numerade Educator

Problem 27

The mass of a sports car is 1200 $\mathrm{kg}$ . The shape of the body is such that the aerodynamic drag coefficient is 0.250 and the frontal area is $2.20 \mathrm{m}^{2} .$ Ignoring all other sources of friction, calculate the initial acceleration the car has if it has been traveling at 100 $\mathrm{km} / \mathrm{h}$ and is now shifted into neutral and allowed to coast.

Luis Mendoza

Numerade Educator

Problem 28

Review. (a) Estimate the terminal speed of a wooden sphere (density 0.830 $\mathrm{g} / \mathrm{cm}^{3} )$ falling through air, taking its radius as 8.00 $\mathrm{cm}$ and its drag coefficient as $0.500 .$ (b) From what height would a freely falling object reach this speed in the absence of air resistance?

Katie Mcalpine

Numerade Educator

Problem 29

Calculate the force required to pull a copper ball of radius 2.00 cm upward through a fluid at the constant speed 9.00 cm/s. Take the drag force to be proportional to the speed, with proportionality constant 0.950 kg/s. Ignore the buoyant force.

Luis Mendoza

Numerade Educator

Problem 30

A small piece of Styrofoam packing material is dropped from a height of 2.00 $\mathrm{m}$ above the ground. Until it reaches terminal speed, the magnitude of its acceleration is given
by $a=g-B v$ . After falling 0.500 $\mathrm{m}$ , the Styrofoam effectively reaches terminal speed and then takes 5.00 s more to reach the ground. (a) What is the value of the constant $B$ ? (b) What is the acceleration at $t=0 ?$ (c) What is the acceleration when the speed is 0.150 $\mathrm{m} / \mathrm{s}$ ?

Katie Mcalpine

Numerade Educator

Problem 31

A small, spherical bead of mass 3.00 g is released from rest at $t=0$ from a point under the surface of a viscous liquid. The terminal speed is observed to be $v_{T}=2.00 \mathrm{cm} / \mathrm{s}$ . Find (a) the value of the constant $b$ that appears in Equation $6.2,$ (b) the time $t$ at which the bead reaches 0.632$v_{T}$, and $(\mathrm{c})$ the value of the resistive force when the bead reaches terminal speed.

Luis Mendoza

Numerade Educator

Problem 32

A window washer pulls a rubber squeegee down a very tall vertical window. The squeegee has mass 160 g and is mounted on the end of a light rod. The coefficient of kinetic friction between the squeegee and the dry glass is 0.900. The window washer presses it against the window with a force having a horizontal component of 4.00 N. (a) If she pulls the squeegee down the window at constant velocity, what vertical force component must she exert? (b) The window washer increases the downward force component by 25.0%, while all other forces remain the same. Find the squeegee’s acceleration in this situation. (c) The squeegee is moved into a wet portion of the window, where its motion is resisted by a fluid drag force R proportional to its velocity according to $R=-20.0 v,$ where $R$ is in newtons and $v$ is in meters per second. Find the terminal velocity that the squeegee approaches, assuming the window washer exerts the same force described in part (b).

Keshav Singh

Numerade Educator

Problem 33

Assume the resistive force acting on a speed skater is proportional to the square of the skater's speed $v$ and is given by $f=-k m v^{2},$ where $k$ is a constant and $m$ is the skater's mass. The skater crosses the finish line of a straight-line race with speed $v_{i}$ and then slows down by coasting on his skates. Show that the skater's speed at any time $t$ after crossing the finish line is $v(t)=v_{i} /\left(1+k t v_{i}\right) .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 34

At major league baseball games, it is commonplace to flash on the scoreboard a speed for each pitch. This speed is determined with a radar gun aimed by an operator positioned behind home plate. The gun uses the Doppler shift of microwaves reflected from the baseball, an effect we will study in Chapter 39. The gun determines the speed at some particular point on the baseball’s path, depending on when the operator pulls the trigger. Because the ball is subject to a drag force due to air proportional to the square of its speed given by $R=k m v^{2},$ it slows as it travels 18.3 $\mathrm{m}$ toward the plate according to the formula $v=v_{i} e^{-k x}$ . Suppose the ball leaves the pitcher's hand at $90.0 \mathrm{mi} / \mathrm{h}=$ 40.2 $\mathrm{m} / \mathrm{s}$ . Ignore its vertical motion. Use the calculation of $R$ for baseballs from Example 6.11 to determine the speed of the pitch when the ball crosses the plate.

Prashant Bana

Numerade Educator

Problem 35

A motorboat cuts its engine when its speed is 10.0 m/s and then coasts to rest. The equation describing the motion of the motorboat during this period is $v=v_{i} e^{-c t},$ where $v$ is the speed at time $t, v_{i}$ is the initial speed at $t=0,$ and $c$ is a constant. At $t=20.0 \mathrm{s}$ , the speed is 5.00 $\mathrm{m} / \mathrm{s}$ . (a) Find the constant $c .$ (b) What is the speed at $t=40.0 \mathrm{s}$ ? (c) Differentiate the expression for $v(t)$ and thus show that the acceleration of the boat is proportional to the speed at any time.

Luis Mendoza

Numerade Educator

Problem 36

You can feel a force of air drag on your hand if you stretch your arm out of the open window of a speeding car. Note: Do not endanger yourself. What is the order of magnitude of this force? In your solution, state the quantities you measure or estimate and their values.

Katie Mcalpine

Numerade Educator

Problem 37

A car travels clockwise at constant speed around a circular section of a horizontal road as shown in the aerial view of Figure P6.37. Find the directions of its velocity and acceleration at (a) position and (b) position.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 38

The mass of a roller-coaster car, including its passengers, is 500 $\mathrm{kg}$ . Its speed at the bottom of the track in Figure $\mathrm{P} 6.16$ is 19 $\mathrm{m} / \mathrm{s}$ . The radius of this section of the track is $r_{1}=25 \mathrm{m} .$ Find the force that a seat in the roller-coaster car exerts on a $50-\mathrm{kg}$ passenger at the lowest point.

Vishal Gupta

Numerade Educator

Problem 39

A string under a tension of 50.0 N is used to whirl a rock in a horizontal circle of radius 2.50 m at a speed of 20.4 m/s on a frictionless surface as shown in Figure P6.39. As the string is pulled in, the speed of the rock increases. When the string is 1.00 m long and the speed of the rock is 51.0 m/s, the string breaks. What is the breaking strength, in newtons, of the string?

Luis Mendoza

Numerade Educator

Problem 40

Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed bump (called the “Holly hump”) and had it installed. Suppose a 1 800-kg car passes over a hump in a roadway that follows the arc of a circle of radius 20.4 m as shown in Figure P6.40. (a) If the car travels at 30.0 km/h, what force does the road exert on the car as the car passes the highest point of the hump? (b) What If? What is the maximum speed the car can have without losing contact with the road as it passes this highest point?

Katie Mcalpine

Numerade Educator

Problem 41

A car of mass $m$ passes over a hump in a road that follows the arc of a circle of radius $R$ as shown in Figure $\mathrm{P} 6.40 .$ (a) If the car travels at a speed $v,$ what force does the road

Luis Mendoza

Numerade Educator

Problem 42

A child’s toy consists of a small wedge that has an acute angle $\theta$ (Fig. P6.42). The sloping side of the wedge is frictionless, and an object of mass $m$ on it remains at constant height if the wedge is spun at a certain constant speed. as an axis, a vertical rod that is as an axis, a vertical rod that is firmly attached to the wedge at the bottom end. Show that, when the boject sits at rest at a point at distance $L$ up along the wedge, the speed of the object must be $v=(g L \sin \theta)^{1 / 2}$

Katie Mcalpine

Numerade Educator

Problem 43

A seaplane of total mass $m$ lands on a lake with initial speed $v_{i}$ i. The only horizontal force on it is a resistive force on its pontoons from the water. The resistive force is proportional to the velocity of the seaplane: $\overline{\mathbf{R}}=-b \overrightarrow{\mathbf{v}}$ . Newton's second law applied to the plane is $-b v \hat{\mathbf{i}}=m(d v / d t) \hat{\mathbf{i}}$ . From the fundamental theorem of calculus, this differential equation implies that the speed changes according to $$\int_{v_{i}}^{v} \frac{d v}{v}=-\frac{b}{m} \int_{0}^{t} d t$$
(a) Carry out the integration to determine the speed of the seaplane as a function of time. (b) Sketch a graph of the speed as a function of time. (c) Does the seaplane come to a complete stop after a finite interval of time? (d) Does the seaplane travel a finite distance in stopping?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 44

An object of mass $m_{1}=$ 4.00 $\mathrm{kg}$ is tied to an object of mass $m_{2}=3.00 \mathrm{kg}$ with String $\mathrm{I}$ of length $\ell=$ $0.500 \mathrm{m} .$ The combination is swung in a vertical cirring, String $2,$ of length $\ell=0.500 \mathrm{m} .$ During the motion, the two strings are collinear at all times as shown in Figure P6.44. At the top of its motion, $m_{2}$ is traveling at $v=4.00 \mathrm{m} / \mathrm{s}$ . (a) What is the tension in String 1 at this instant? (b) What is the tension in String 2 at this instant? (c) Which string will break first if the combination is rotated faster and faster?

Katie Mcalpine

Numerade Educator

Problem 45

A ball of mass $m=0.275 \mathrm{kg}$ swings in a vertical circular path on a string $L=0.850 \mathrm{m}$ long as in Figure $\mathrm{P} 6.45$ (a) What are the forces acting on the ball at any point on the path? (b) Draw force diagrams for the ball when it is at the bottom of the circle and when it is at the top. (c) If its speed is 5.20 $\mathrm{m} / \mathrm{s}$ at the top of the circle, what is the tension in the string there? (d) If the string breaks when its tension exceeds $22.5 \mathrm{N},$ what is the maximum speed the ball can have at the bottom before that happens?

Luis Mendoza

Numerade Educator

Problem 46

Why is the following situation impossible? A mischievous child goes to an amusement park with his family. On one ride, after a severe scolding from his mother, he slips out of his seat and climbs to the top of the ride’s structure, which is shaped like a cone with its axis vertical and its sloped sides making an angle of $\theta=20.0^{\circ}$ with the horizontal as shown in Figure $\mathrm{P} 6.46 .$ This part of the structure rotates about the vertical central axis when the ride operates. The child sits on the sloped surface at a point $d=5.32 \mathrm{m}$ down the sloped side from the center of the cone and pouts. The coefficient of static friction between the boy and the cone is 0.700 . The ride operator does not notice that the child has slipped away from his seat and so continues to operate the ride. As a result, the sitting, pouting boy rotates in a circular path at a speed of 3.75 $\mathrm{m} / \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

(a) A luggage carousel at an airport has the form of a section of a large cone, steadily rotating about its vertical axis. Its metallic surface slopes downward toward the outside, making an angle of $20.0^{\circ}$ with the horizontal. A piece of luggage having mass 30.0 kg is placed on the carousel at a position 7.46 m measured horizontally from the axis of rotation. The travel bag goes around once in 38.0 s. Calculate the force of static friction exerted by the carousel on the bag. (b) The drive motor is shifted to turn the carousel at a higher constant rate of rotation, and the piece of luggage is bumped to another position, 7.94 m from the axis of rotation. Now going around once in every 34.0 s, the bag is on the verge of slipping down the sloped surface. Calculate the coefficient of static friction between the bag and the carousel.

Luis Mendoza

Numerade Educator

Problem 48

a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis as shown in Figure P6.48. So that the clothes will dry uniformly, they are made to tumble. The rate of rotation of the smooth-walled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of $\theta=$ $68.0^{\circ}$ above the horizontal. If the radius of the tub is $r=$ $0.330 \mathrm{m},$ what rate of revolution is needed?

Katie Mcalpine

Numerade Educator

Problem 49

Interpret the graph in Figure 6.16(b), which describes the results for falling coffee filters discussed in Example 6.10. Proceed as follows. (a) Find the slope of the straight line, including its units. (b) From Equation $6.6, R=\frac{1}{2} D \rho A v^{2},$ identify the theoretical slope of a graph of resistive force versus squared speed. (c) Set the experimental and theoretical slopes equal to each other and proceed to calculate the drag coefficient of the filters. Model the cross-sectional area of the filters as that of a circle of radius 10.5 cm and take the density of air to be $1.20 \mathrm{kg} / \mathrm{m}^{3} .$ (d) Arbitrarily choose the eighth data point on the graph and find its vertical separation from the line of best fit. Express this scatter as a percentage. (e) In a short paragraph, state what the graph demonstrates and compare it with the theoretical prediction. You will need to make reference to the quantities plotted on the axes, to the shape of the graph line, to the data points, and to the results of parts (c) and (d).

Luis Mendoza

Numerade Educator

Problem 50

A basin surrounding a drain has the shape of a circular cone opening upward, having everywhere an angle of $35.0^{\circ}$ with the horizontal. A 25.0-g ice cube is set sliding around the cone without friction in a horizontal circle of radius R. (a) Find the speed the ice cube must have as a function of R. (b) Is any piece of data unnecessary for the solution? Suppose R is made two times larger. (c) Will the required speed increase, decrease, or stay constant? If it changes, by what factor? (d) Will the time required for each revolution increase, decrease, or stay constant? If it changes, by what factor? (e) Do the answers to parts (c) and (d) seem contradictory? Explain.

Keshav Singh

Numerade Educator

Problem 51

A truck is moving with constant acceleration $a$ up a hill that makes an angle $\phi$ with the horizontal as in Figure P6.51. A small sphere of mass $m$ is suspended from the ceiling of the truck by a light cord. If the pendulum makes a constant angle $\theta$ with the perpendicular to the ceiling, what is $a ?$

Vishal Gupta

Numerade Educator

Problem 52

The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle. The speed of the airplane is 300 mi/h at the top of the loop and 450 mi/h at the bottom, and the radius of the circle is 1 200 ft. (a) What is the pilot’s apparent weight at the lowest point if his true weight is 160 lb? (b) What is his apparent weight at the highest point? (c) What If? Describe how the pilot could experience weightlessness if both the radius and the speed can be varied. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body.

Katie Mcalpine

Numerade Educator

Problem 53

Because the Earth rotates about its axis, a point on the equator experiences a centripetal acceleration of $0.0337 \mathrm{m} / \mathrm{s}^{2},$ whereas a point at the poles experiences no centripetal acceleration. If a person at the equator has a mass of $75.0 \mathrm{kg},$ calculate (a) the gravitational force (true weight) on the person and (b) the normal force (apparent weight) on the person. (c) Which force is greater? Assume the Earth is a uniform sphere and take $g=9.800 \mathrm{m} / \mathrm{s}^{2} .$

Luis Mendoza

Numerade Educator

Problem 54

A puck of mass $m_{1}$ is tied to a string and allowed to revolve in a circle of radius $R$ on a frictionless, horizontal table. The other end of table. The other end of the string passes through a small hole in the cen-object of mass $m_{2}$ is tied to it (Fig. P $6.54 ) .$ The suspended object remains in equilibrium while the puck on the tabletop revolves. Find symbolic expressions for (a) the tension in the string, (b) the radial force acting on the puck, and (c) the speed of the puck. (d) Qualitatively describe what will happen in the motion of the puck if the value of $m_{2}$ is increased by placing a small additional load on the puck. (e) Qualitatively describe what will happen in the motion of the puck if the value of $m_{2}$ instead decreased by removing a part of the hanging load.

Katie Mcalpine

Numerade Educator

Problem 55

While learning to drive, you are in a 1 200-kg car moving at 20.0 m/s across a large, vacant, level parking lot. Suddenly you realize you are heading straight toward the brick sidewall of a large supermarket and are in danger of running into it. The pavement can exert a maximum horizontal force of 7 000 N on the car. (a) Explain why you should expect the force to have a well-defined maximum value. (b) Suppose you apply the brakes and do not turn the steering wheel. Find the minimum distance you must be from the wall to avoid a collision. (c) If you do not brake but instead maintain constant speed and turn the steering wheel, what is the minimum distance you must be from the wall to avoid a collision? (d) Of the two methods in parts (b) and (c), which is better for avoiding a collision? Or should you use both the brakes and the steering wheel, or neither? Explain. (e) Does the conclusion in part (d) depend on the numerical values given in this problem, or is it true in general? Explain.

Luis Mendoza

Numerade Educator

Problem 56

In Example 6.5, we investigated the forces a child experiences on a Ferris wheel. Assume the data in that example applies to this problem. What force (magnitude and direction) does the seat exert on a 40.0-kg child when the child is halfway between top and bottom?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

Figure P6.57 shows a photo of a swing ride at an amusement park. The structure consists of a horizontal, rotating, circular platform of diameter D from which seats of mass $m$ are sus-
pended at the end of massless chains of length d. When the system rotates at constant speed, the chains swing outward and make an angle $\theta$ with the vertical. Consider such a ride with the following parameters: $D=$ $8.00 \mathrm{m}, d=2.50 \mathrm{m}, m=$$10.0 \mathrm{kg},$ and $\theta=28.0^{\circ} .$ (a) What is the speed of each seat? (b) Draw a diagram of forces acting on a $40.0-\mathrm{kg}$ child riding in a seat and $(\mathrm{c})$ find the tension in the chain.

Luis Mendoza

Numerade Educator

Problem 58

A piece of putty is initially located at point A on the rim of a grinding wheel rotating at constant angular speed about a horizontal axis. The putty is dislodged from point A when the diameter through A is horizontal. It then rises vertically and returns to A at the instant the wheel completes one revolution. From this information, we wish to find the speed $v$ of the putty when it leaves the wheel and the force holding it to the wheel. (a) What analysis model is appropriate for the motion of the putty as it rises and falls? (b) Use this model to find a symbolic expression for the time interval between when the putty leaves point $A$ and when it arrives back at $A,$ in terms of $v$ and g. (c) What is the appropriate analysis model to describe point $A$ on the wheel? (d) Find the period of the motion of point $A$ in terms of the tangential speed $v$ and the radius $R$ of the wheel. (e) Set the time interval from part (b) equal to the period from part (d) and solve for the speed $v$ of the putty as it leaves the wheel. (f) If the mass of the putty is $m$, what is the magnitude of the force that held it to the wheel before it was released?

Sheh Lit Chang

University of Washington

Problem 59

An amusement park ride consists of a large vertical cylinder that spins about its axis fast enough that any person inside is held up against the wall when the floor drops away (Fig. P6.59). The coefficient of static friction between person and wall is $\mu_{s},$ and the radius of the cylinder is $R .$ (a) Show that the maximum period of revolution necessary to keep the person from falling is $T=\left(4 \pi^{2} R \mu_{\delta} / g\right)^{1 / 2} .$ (b) If the rate of revolution of the cylinder is made to be somewhat larger, what happens to the magnitude of each one of the forces acting on the person? What happens in the motion of the person? (c) If the rate of revolution of the cylinder is instead made to be somewhat smaller, what happens to the magnitude of each one of the forces acting on the person? How does the motion of the person change?

Luis Mendoza

Numerade Educator

Problem 60

Members of a skydiving club were given the following data to use in planning their jumps. In the table, $d$ is the distance fallen from rest by a skydiver in a "free-fall stable spread position" versus the time of fall $t$ . (a) Convert the distances in feet into meters. (b) Graph $d$ (in meters) versus $t$ (c) Determine the value of the terminal speed $v_{T}$ by finding the slope of the straight portion of the curve. Use a least-squares fit to determine this slope.

Dominador Tan

Numerade Educator

Problem 61

A car rounds a banked curve as discussed in Example 6.4 and shown in Figure 6.5. The radius of curvature of the road is $R,$ the banking angle is $\theta,$ and the coefficient of static friction is $\mu_{s}$ . (a) Determine the range of speeds the car can have without slipping up or down the road. (b) Find the minimum value for $\mu_{s}$ such that the minimum speed is zero.

Luis Mendoza

Numerade Educator

Problem 62

Galileo thought about whether acceleration should be defined as the rate of change of velocity over time or as the rate of change in velocity over distance. He chose the former, so let’s use the name “vroomosity” for the rate of change of velocity over distance. For motion of a particle on a straight line with constant acceleration, the equation $v=$ $v_{i}+$ at gives its velocity $v$ as a function of time. Similarly, for a particle's linear motion with constant vroomosity $k,$ the equation $v=v_{i}+k x$ gives the velocity as a function of the position $x$ if the particle's speed is $v_{i}$ at $x=0 .$ (a) Find the law describing the total force acting on this object of mass $m$ . (b) Describe an example of such a motion or explain why it is unrealistic. Consider (c) the possibility of $k$ positive and (d) the possibility of $k$ negative.

Katie Mcalpine

Numerade Educator

Problem 63

A model airplane of mass 0.750 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 60.0-m-long control wire as shown in Figure P6.63a. The forces exerted on the airplane are shown in Figure P6.63b: the tension in the control wire, the gravitational force, and aerodynamic lift that acts at $\theta=20.0^{\circ}$ inward from the vertical. Compute the tension in the wire, assuming it makes a constant angle of $\theta=20.0^{\circ}$ with the horizontal.

Luis Mendoza

Numerade Educator

Problem 64

Because of the Earth's rotation, a plumb bob does not hang exactly along a line directed to the center of the Earth. How much does the plumb bob deviate from a radial line at $35.0^{\circ}$ north latitude? Assume the Earth is spherical.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

A 9.00-kg object starting from rest falls through a viscous medium and experiences a resistive force given by Equation 6.2. The object reaches one half its terminal speed in 5.54 s. (a) Determine the terminal speed. (b) At what time is the speed of the object three-fourths the terminal speed? (c) How far has the object traveled in the first 5.54 s of motion?

Luis Mendoza

Numerade Educator

Problem 66

For $t<0,$ an object of mass $m$ experiences no force and moves in the positive $x$ direction with a constant speed $v_{i}$ . Beginning at $t=0,$ when the object passes position $x=0$ , it experiences a net resistive force proportional to the square of its speed: $\overrightarrow{\mathbf{F}}_{\text { net }}= m k v^{2} \hat{\mathbf{i}},$ where $k$ is a constant. The speed of the object after $t=0$ is given by $v=v_{i} /\left(1+k v_{i} t\right)$
(a) Find the position $x$ of the object as a function of time.
(b) Find the object's velocity as a function of position.

Katie Mcalpine

Numerade Educator

Problem 67

A golfer tees off from a location precisely at $\phi_{i}=$ $35.0^{\circ}$ north latitude. He hits the ball due south, with range $285 \mathrm{m} .$ The ball's initial velocity is at $48.0^{\circ}$ above the horizontal. Suppose air resistance is negligible for the golf ball. (a) For how long is the ball in flight? The cup is due south of the golfer's location, and the golfer would have a hole-in-one if the Earth were not rotating. The Earth's rotation makes the tee move in a circle of radius $R_{E} \cos \phi_{i}=\left(6.37 \times 10^{6} \mathrm{m}\right) \cos 35.0^{\circ}$ as shown in Figure $\mathrm{P} 6.67$ . The tee completes one revolution each day. (b) Find the eastward speed of the tee relative to the stars. The hole is also moving east, but it is 285 $\mathrm{m}$ farther south and thus at a slightly lower latitude $\phi_{f} .$ Because the hole moves in a slightly larger circle, its speed must be greater than that of the tee. (c) By how much does the hole’s speed exceed that of the tee? During the time interval the ball is in flight, it moves upward and downward as well as south-ward with the projectile motion you studied in Chapter 4, but it also moves eastward with the speed you found in part (b). The hole moves to the east at a faster speed, however, pulling ahead of the ball with the relative speed you found in part (c). (d) How far to the west of the hole does the ball land?

Ajay Singhal

Numerade Educator

Problem 68

A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius 15.0 cm as shown in Figure P6.68. The circle is always in a vertical plane and rotates steadily about its vertical diameter with a period of 0.450 s. The position of the bead is described by the angle $\theta$ that the radial line, from the center of the loop to the bead, makes with the vertical. (a) At what angle up from the bottom of the circle can the bead stay motionless relative to the turning circle? (b) What If? Repeat the problem, this time taking the period of the circle’s rotation as 0.850 s. (c) Describe how the solution to part (b) is different from the solution to part (a). (d) For any period or loop size, is there always an angle at which the bead can stand still relative to the loop? (e) Are there ever more than two angles? Arnold Arons suggested the idea for this problem.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

The expression $F=a r v+b r^{2} v^{2}$ gives the magnitude of the resistive force (in newtons) exerted on a sphere of radius $r$ (in meters) by a stream of air moving at speed $v$ (in meters per second), where $a$ and $b$ are constants with appropriate SI units. Their numerical values are $a=3.10 \times 10^{-4}$ and $b=0.870 .$ Using this expression, find the terminal speed for water droplets falling under their own weight in air, taking the following values for the drop radii: (a) 10.0$\mu \mathrm{m}$ , (b) $100 \mu \mathrm{m},$ (c) 1.00 $\mathrm{mm}$ . For parts (a) and (c), you can obtain accurate answers without solving a quadratic equation by considering which of the two contributions to the air resistance is dominant and ignoring the lesser contribution.

Luis Mendoza

Numerade Educator