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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

01:57

Problem 1

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 $0.529 \times$ $10^{-10} \mathrm{~m}$ and (b) the centripetal acceleration of the electron.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
03:49

Problem 2

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.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
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Problem 3

A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in Figure $\mathrm{P} 6.3$. The length of the arc $A B C$ is $235 \mathrm{~m},$ 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 $\begin{array}{lll}\text { of } 35.0^{\circ} \text { ? } & \text { Express your }\end{array}$ answer in terms of the unit vectors i and $\mathrm{j}$. Determine (b) the car's average speed and (c) its average acceleration during the $36.0-\mathrm{s}$ interval.

Mkhitar Hobosyan
Mkhitar Hobosyan
Numerade Educator
04:15

Problem 4

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

Corinne Rywalt
Corinne Rywalt
Numerade Educator
02:57

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 $0.480 \mathrm{~m}$. What magnitude of magnetic force is required to maintain the deuteron in a circular path?

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
06:18

Problem 6

Why is the following situation impossible? The object of mass $m=4.00 \mathrm{~kg}$ in Figure $\mathrm{P} 6.6$ 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?

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
02:42

Problem 7

You are working during your summer break as an amusement park ride operator. The ride you are controlling 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. $\mathrm{P} 6.7$ ). The coefficient of static friction between a person of mass $m$ and the wall is $\mu_{s}$, and the radius of the cylinder is $R$. You are rotating the ride with an angular speed $\omega$ suggested by your supervisor. (a) Suppose a very heavy person enters the ride. Do you need to increase the angular speed so that this person will not slide down the wall? (b) Suppose someone enters the ride wearing a very slippery satin workout outfit. In this case, do you need to increase the angular speed so that this person will not slide down the wall?

Ajay Singhal
Ajay Singhal
Numerade Educator
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Problem 8

A driver is suing the state highway department after an accident on a curved freeway. The driver lost control and crashed into a tree located a short distance from the outside edge of the curved roadway. The driver is claiming that the radius of curvature of the unbanked roadway was too small for the speed limit, causing him to slide outward on the curve and hit the tree. You have been hired as an expert witness for the defense, and have been requested to use your knowledge of physics to testify that the radius of curvature of the roadway is appropriate for the speed limit. State regulations show that the radius of curvature of an unbanked roadway on which the speed limit is $65 \mathrm{mi} / \mathrm{h}$ must be at least $150 \mathrm{~m}$. You build an accelerometer, which is a plumb bob with a protractor that you attach to the roof of your car. An associate riding in your car with you observes that the plumb bob hangs at an angle of $15.0^{\circ}$ from the vertical when the car is driven at a safer speed of $23.0 \mathrm{~m} / \mathrm{s}$ on the curve in question. What is your testimony regarding the radius of the curve?

Gregory Devenport
Gregory Devenport
Numerade Educator
02:02

Problem 9

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
Luis Mendoza
Numerade Educator
03:51

Problem 10

A 40.0 -kg child swings in a swing supported by two chains, each $3.00 \mathrm{~m}$ long. The tension in each chain at the lowest point is $350 \mathrm{~N}$. 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.)

Vishal Gupta
Vishal Gupta
Numerade Educator
02:28

Problem 11

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.)

Vishal Gupta
Vishal Gupta
Numerade Educator
13:01

Problem 12

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 $2.00 \mathrm{~m}$ as shown in Figure $\mathrm{P} 6.12 .$ When $\theta=20.0^{\circ}$, the speed of the object is $8.00 \mathrm{~m} / \mathrm{s}$. At this instant, find (a) the tension in the string, (b) the tangential and radial components of acceleration, and (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
Katie Mcalpine
Numerade Educator
06:22

Problem 13

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. $\mathrm{P} 6.13) .$ 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 $40.0 \mathrm{~m}$ high. Suppose the speed at the top of the loop is $13.0 \mathrm{~m} / \mathrm{s}$ and the corresponding centripetal acceleration of the riders is $2 g$. (a) What is the radius of the arc of the teardrop 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 $20.0 \mathrm{~m}$. If the cars have the same speed, $13.0 \mathrm{~m} / \mathrm{s}$ 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.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:33

Problem 14

An object of mass $m=$ $5.00 \mathrm{~kg},$ attached to a spring scale, rests on a frictionless, horizontal surface as shown in Fig-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 $18.0 \mathrm{~N}$ 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.

Dominador Tan
Dominador Tan
Numerade Educator
04:35

Problem 15

A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of $591 \mathrm{~N}$. As the elevator later stops, the scale reading is $391 \mathrm{~N}$. 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.

Vishal Gupta
Vishal Gupta
Numerade Educator
06:24

Problem 16

A student, along with her backpack on the floor next to her, is 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
Katie Mcalpine
Numerade Educator
01:49

Problem 17

A small container of water is placed on a turntable inside a microwave oven, at a radius of $12.0 \mathrm{~cm}$ 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
Luis Mendoza
Numerade Educator
01:38

Problem 18

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
Luis Mendoza
Numerade Educator
06:02

Problem 19

A window washer pulls a rubber squeegee down a very tall vertical window. The squeegee has mass $160 \mathrm{~g}$ 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 $4.00 \mathrm{~N}$. (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).

Pawan Yadav
Pawan Yadav
Numerade Educator
05:31

Problem 20

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 $5.00 \mathrm{~s}$ 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
Katie Mcalpine
Numerade Educator
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Problem 21

A small, spherical bead of mass $3.00 \mathrm{~g}$ 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 $0.632 v_{p}$ and $(c)$ the value of the resistive force when the bead reaches terminal speed.

Mkhitar Hobosyan
Mkhitar Hobosyan
Numerade Educator
02:27

Problem 22

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 $v(t)=v_{i} /\left(1+k t v_{i}\right)$.

Luis Mendoza
Luis Mendoza
Numerade Educator
02:50

Problem 23

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
Katie Mcalpine
Numerade Educator
02:01

Problem 24

A car travels clockwise at constant speed around a circular section of a horizontal road as shown in the aerial view of Figure $\mathrm{P} 6.24 .$ Find the directions of its velocity and acceleration at (a) position A and (b) position B.

Supratim Pal
Supratim Pal
Numerade Educator
02:19

Problem 25

A string under a tension of $50.0 \mathrm{~N}$ is used to whirl a rock in a horizontal circle of radius $2.50 \mathrm{~m}$ at a speed of $20.4 \mathrm{~m} / \mathrm{s}$ on a frictionless surface as shown in Figure $\mathrm{P} 6.25 .$ As the string is pulled in, the speed of the rock increases. When the string on the table is $1.00 \mathrm{~m}$ long and the speed of the rock is $51.0 \mathrm{~m} / \mathrm{s}$, the string breaks. What is the breaking strength, in newtons, of the string?

Supratim Pal
Supratim Pal
Numerade Educator
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Problem 26

Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed bump (called the "Holly hump" ${ }^{\prime \prime}$ and had it installed. Suppose a $1800-\mathrm{kg}$ car passes over a hump in a roadway that follows the arc of a circle of radius $20.4 \mathrm{~m}$ as shown in Figure $\mathrm{P} 6.26$. (a) If the car travels at $30.0 \mathrm{~km} / \mathrm{h},$ 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?

Mkhitar Hobosyan
Mkhitar Hobosyan
Numerade Educator
02:28

Problem 27

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 $\mathrm{P} 6.26$. (a) If the car travels at a speed $v,$ 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?

Vishal Gupta
Vishal Gupta
Numerade Educator
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Problem 28

A child's toy consists of a small wedge that has an acute angle $\theta$ (Fig. $\mathrm{P} 6.28$ ). 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. The wedge is spun by rotating, as an axis, a vertical rod that is firmly attached to the wedge at the bottom end. Show that, when the object sits at rest at a point at distance $L$ up along the wedge, the speed of the object must be $v=(g L \sin \theta)^{1 / 2}$

Mkhitar Hobosyan
Mkhitar Hobosyan
Numerade Educator
06:24

Problem 29

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: $\overrightarrow{\mathbf{R}}=-b \overrightarrow{\mathbf{v}}$. 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?

Guilherme Barros
Guilherme Barros
Numerade Educator
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Problem 30

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 1 of length $\ell=0.500 \mathrm{~m}$. The combination is swung in a vertical circular path on a second string, String $2,$ of length $\ell=$ $0.500 \mathrm{~m}$. During the motion, the two strings are collinear at all times as shown in Figure $\mathrm{P} 6.30 .$ 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?

Mkhitar Hobosyan
Mkhitar Hobosyan
Numerade Educator
02:38

Problem 31

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 $\mathrm{P} 6.31$. (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
Luis Mendoza
Numerade Educator
02:01

Problem 32

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 P6.32. 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 $3.75 \mathrm{~m} / \mathrm{s}$.

Penny Riley
Penny Riley
Numerade Educator
02:28

Problem 33

The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle. The speed of the airplane is $300 \mathrm{mi} / \mathrm{h}$ at the top of the loop and $450 \mathrm{mi} / \mathrm{h}$ at the bottom, and the radius of the circle is $1200 \mathrm{ft}$. (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.

Penny Riley
Penny Riley
Numerade Educator
04:24

Problem 34

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 $25.0-\mathrm{g}$ 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 interval 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.

Sam Stansfield
Sam Stansfield
Numerade Educator
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Problem 35

While learning to drive, you are in a $1200-\mathrm{kg}$ car moving at $20.0 \mathrm{~m} / \mathrm{s}$ 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 $7000 \mathrm{~N}$ 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.

Victor Salazar
Victor Salazar
Numerade Educator
04:55

Problem 36

A truck is moving with constant acceleration $a$ up a hill that makes an angle $\phi$ with the horizontal as in Figure P6.36. 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 $a$ ?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:46

Problem 37

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
Luis Mendoza
Numerade Educator
04:42

Problem 38

A puck of mass $m_{1}$ is tied to a string and allowed to revolve in a circle of radius $R$ on a friction- $-$ less, horizontal table. The other end of the string passes through a small hole in the center of the table, and an object of mass $m_{2}$ is tied to it (Fig. $\mathrm{P} 6.38$ ). 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 $m_{2}$ is instead decreased by removing a part of the hanging load.

Vishal Gupta
Vishal Gupta
Numerade Educator
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Problem 39

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 $v=v_{i}+a t$ 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 $m .$ Describe an example of such a motion or explain why it is unrealistic for (b) the possibility of $k$ positive and (c) the possibility of $k$ negative.

Mkhitar Hobosyan
Mkhitar Hobosyan
Numerade Educator
01:01

Problem 40

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 $v_{r}$ by finding the slope of the straight portion of the curve. Use a leastsquares fit to determine this slope.
$$\begin{array}{crrrrr}
\hline t(\mathrm{~s}) & d(\mathrm{ft}) & t(\mathrm{~s}) & d(\mathrm{ft}) & t(\mathrm{~s}) & d(\mathrm{ft}) \\
\hline 0 & 0 & 7 & 652 & 14 & 1831 \\
1 & 16 & 8 & 808 & 15 & 2005 \\
2 & 62 & 9 & 971 & 16 & 2179 \\
3 & 138 & 10 & 1138 & 17 & 2353 \\
4 & 242 & 11 & 1309 & 18 & 2527 \\
5 & 366 & 12 & 1483 & 19 & 2701 \\
6 & 504 & 13 & 1657 & 20 & 2875 \\
\hline\end{array}$$

Dominador Tan
Dominador Tan
Numerade Educator
09:31

Problem 41

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.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:08

Problem 42

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
05:14

Problem 43

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
Sheh Lit Chang
University of Washington
04:04

Problem 44

A model airplane of mass $0.750 \mathrm{~kg}$ flies with a speed of $35.0 \mathrm{~m} / \mathrm{s}$ in a horizontal circle at the end of a $60.0-\mathrm{m}$ -long control wire as shown in Figure $\mathrm{P} 6.44 \mathrm{a}$. The forces exerted on the airplane are shown in Figure $\mathrm{P} 6.44 \mathrm{~b}$ : 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.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
03:11

Problem 45

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
Luis Mendoza
Numerade Educator
01:26

Problem 46

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: $\overrightarrow{\mathbf{F}}_{\text {net }}=-m k v^{2}$ 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.

Penny Riley
Penny Riley
Numerade Educator
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Problem 47

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 $\mathrm{P} 6.47 .$ 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 southward 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?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 48

A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius $15.0 \mathrm{~cm}$ as shown in Figure $\mathrm{P} 6.48$. The circle is always in a vertical plane and rotates steadily about its vertical diameter with a period of $0.450 \mathrm{~s}$. 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.

Victor Salazar
Victor Salazar
Numerade Educator
01:10

Problem 49

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.

Mayukh Banik
Mayukh Banik
Numerade Educator
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Problem 50

You have a great job working at a major league baseball stadium for the summer! At this stadium, the speed of every pitch is measured using a radar gun aimed at the pitcher by an operator behind home plate. The operator has so much experience with this job that he has perfected a technique by which he can make each measurement at the exact instant at which the ball leaves the pitcher's hand. Your supervisor ask you to construct an algorithm that will provide the speed of the ball as it crosses home plate, $18.3 \mathrm{~m}$ from the pitcher, based on the measured speed $v_{i}$ of the ball as it leaves the pitcher's hand. The speed at home plate will be lower due to the resistive force of the air on the baseball. The vertical motion of the ball is small, so, to a good approximation, we can consider only the horizontal motion of the ball. You begin to develop your algorithm by applying the particle under a net force to the baseball in the horizontal direction. A pitch is measured to have a speed of $40.2 \mathrm{~m} / \mathrm{s}$ as it leaves the pitcher's hand. You need to tell your supervisor how fast it was traveling as it crossed home plate. (Hint: Use the chain rule to express acceleration in terms of a derivative with respect to $x,$ and then solve a differential equation for $v$ to find an expression for the speed of the baseball as a function of its position. The function will involve an exponential. Also make use of Table $6.1 .)$

Victor Salazar
Victor Salazar
Numerade Educator