WHAT WE HAVE LEARNED
**EXAM STUDY GUIDE**

The conversion between Cartesian coordinates,

*x*and*y*, and polar coordinates,*r*and*θ*, is given byThe conversion between polar coordinates and Cartesian coordinates is given by

The linear displacement,

*s*, is related to the angular displacement,*θ*, by*s*=*r**θ*, where*r*is the radius of the circular path and*θ*is measured in radians.The magnitude of the instantaneous angular velocity,

*ω*, is given by .The magnitude of the angular velocity is related to the magnitude of the linear velocity,

*v*, by*v*=*rω*.The magnitude of the instantaneous angular acceleration,

*α*, is given byThe magnitude of the angular acceleration is related to the magnitude of the tangential acceleration,

*a*_{t}, by*a*_{t}=*rα*.The magnitude of the centripetal acceleration,

*a*_{c}, required to keep an object moving in a circle with constant angular velocity is given by .The magnitude of the total acceleration of an object in circular motion is .

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NEW SYMBOLS AND EQUATIONS

, magnitude of the average angular velocity

, magnitude of the instantaneous angular velocity

, angular frequency

, period

, magnitude of the average angular acceleration

, magnitude of the instantaneous angular acceleration

, magnitude of the centripetal acceleration

ANSWERS TO SELF-TEST OPPORTUNITIES

- 9.1
Differential arc length is

*Rdθ*for circle with radius*R*; integral arc length around the circle is the circumference*C* - 9.2
The differential area is shown in the sketch. The differential area is

*dA*= 2*πdr*Area of circle is - 9.3
*ω*^{2}*r*= (2*π*/yr)^{2}(1 au) = 5.9 · 10^{−4}m/s^{2}=*g*/1700. - 9.4
Slightly heavier, but barely noticeable

- 9.5
At

*r*_{1}= 25 mm, the magnitude of the tangential acceleration is*a*_{t}=*αr*_{1}= 1.56 · 10^{−4}m/s^{2}, and that of the centripetal acceleration is*a*_{c}=*vω*(*r*_{1}) = 59.1 m/s^{2}, larger by more than four orders of magnitude. At*r*_{2}= 58 mm, the accelerations are*a*_{t}=*αr*_{2}= 3.61 · 10^{−4}m/s^{2}and*a*_{c}=*vω*(*r*_{2}) = 25.5 m/s^{2} - 9.6
Substitute equation 9.23 into equation 9.22 and find

*T*=*mg*/cos*φ*. - 9.7
If the radius is doubled, the speed at the top of the loop has to increase by a factor of . Thus, the required speed is .

- 9.8
Now we subtract

*θ*_{0}from both sides of the equation and then multiply by*α*:

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PROBLEM-SOLVING PRACTICE

Problem-Solving Guidelines: Circular Motion

Motion in a circle always requires centripetal force and centripetal acceleration. However, remember that centripetal force is not a new kind of force but is simply the net force causing the motion; it consists of the sum of whatever forces are acting on the moving object. This net force equals the mass times the centripetal acceleration; do not make the common mistake of counting mass times acceleration as a force to be added to the net force on one side of the equation of motion.

Be sure you note whether the situation involves angles in degrees or in radians. The radian is not a unit that necessarily has to be carried through a calculation, but check that your result makes sense in terms of the angular units.

The equations of motion with constant angular acceleration have the same form as the equations of motion with constant linear acceleration. However, neither set of equations applies if the acceleration is not constant.

SOLVED PROBLEM9.3
Carnival Ride

PROBLEM

One of the rides found at carnivals is a rotating cylinder, as shown in Figure 9.25. The riders step inside the vertical cylinder and stand with their backs against the curved wall. The cylinder spins very rapidly, and at some angular velocity, the floor is pulled away. The thrill-seekers now hang like flies on the wall. (The cylinder in Figure 9.25, is lifted up and tilted after the ride reaches it operating angular velocity, but we will not deal with this additional complication.) If the radius of the cylinder is *r* = 2.10 m, the rotation axis of the cylinder remains vertical, and the coefficient of static friction between the people and the wall is *μ*_{s} = 0.390, what is the minimum angular velocity, *ω*, at which the floor can be withdrawn?

SOLUTION

THINK

When the floor drops away, the magnitude of the force of static friction between a rider and the wall of the rotating cylinder must equal the magnitude of the force of gravity acting on the rider. The static friction between the rider and the wall depends on the normal force being exerted on the rider and the coefficient of static friction. As the cylinder spins faster, the normal force (which acts as the centripetal force) being exerted on the rider increases. At a certain angular velocity, the maximum magnitude of the force of static friction will equal the magnitude of the force of gravity. That angular velocity is the minimum angular velocity at which the floor can be withdrawn.

SKETCH

A top view of the rotating cylinder is shown in Figure 9.26a. The free-body diagram for one of the riders is shown in Figure 9.26b, where the rotation axis is assumed to be the y-axis. In this sketch, is the force of static friction, is the normal force exerted on the rider of mass *m* by the wall of the cylinder, and is the force of gravity acting on the rider.

RESEARCH

At the minimum angular velocity required to keep the rider from falling, the magnitude of the force of static friction between the rider and the wall is equal to the magnitude of the force of gravity acting on the rider. To analyze these forces, we start with the freebody diagram shown in Figure 9.26b. In the free-body diagram, the *x*-direction is along the radius of the cylinder, and the *y*-direction is vertical. In the *x*-direction, the normal force exerted by the wall on the rider provides the centripetal force that makes the rider move in a circle:

(i)

In the *y*-direction, the thrill-seeking rider sticks to the wall only if the upward force of static friction between the rider and wall balances the downward force of gravity. The force of gravity on the rider is his or her weight, so we can write

(ii)

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We know that the centripetal force is given by

(iii)

and the force of static friction is given by

(iv)

SIMPLIFY

We can combine equations (ii) and (iv) to obtain

(v)

Substituting for *F*_{c} from equation (i) into (iii), we find

(vi)

Combining equations (v) and (vi), we have

which we can solve for *ω*:

Thus, the minimum value of the angular velocity is given by

Note that the mass of the rider canceled out. This is crucial since people of different masses want to ride at the same time!

CALCULATE

Putting in the numerical values, we find

ROUND

Expressing our result to three significant figures gives

DOUBLE-CHECK

To double-check, let's express our result for the angular velocity in revolutions per minute (rpm):

An angular velocity of 33 rpm for the rotating cylinder seems reasonable, because this means that it completes almost one full turn every 2 s. If you have ever been on one of these rides or watched one, you know that the answer is in the right ballpark.

Note that the coefficient of friction, *μ*_{s}, between the clothing of the rider and the wall is not identical in all cases. Our formula, , indicates that a smaller coefficient of friction necessitates a larger angular velocity. Designers of this kind of ride need to make sure that they allow for the smallest coefficient of friction that can be expected to occur. Obviously, they want to have a somewhat sticky contact surface on the wall of the ride, just to make sure!

One last point to check: indicates that the minimum angular velocity required will decrease as a function of the radius of the cylinder. The example of the poker chips on the spinning table in Section 9.5 established that the centripetal force increases with radial distance, which is consistent with this result.

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SOLVED PROBLEM9.4
Flywheel

PROBLEM

The flywheel of a steam engine starts to rotate from rest with a constant angular acceleration of *α* = 1.43 rad/s^{2}. The flywheel undergoes this constant angular acceleration for *t* = 25.9 s and then continues to rotate at a constant angular velocity, *ω*. After the flywheel has been rotating for 59.5 s, what is the total angle through which it has rotated since it started?

SOLUTION

THINK

Here we are trying to determine the total angular displacement, *θ*. For the time interval when the flywheel is undergoing angular acceleration, we can use equation 9.26(i) with *θ*_{0} = 0 and *ω*_{0} = 0. When the flywheel is rotating at a constant angular velocity, we use equation 9.26(i) with *θ*_{0} = 0 and *α* = 0. To get the total angular displacement, we add these two angular displacements.

RESEARCH

Let's call the time during which the flywheel is undergoing angular acceleration *t*_{a} and the total time the flywheel is rotating *t*_{b}. Thus, the flywheel rotates at a constant angular velocity for a time interval equal to *t*_{b}−*t*_{a}. The angular displacement, *θ*_{a}, that occurs while the flywheel is undergoing angular acceleration is given by

(i)

The angular displacement, *θ*_{b}, that occurs while the flywheel is rotating at the constant angular velocity, *ω*, is given by

(ii)

The angular velocity, *ω*, reached by the flywheel after undergoing the angular acceleration *α* for a time *t*_{a} is given by

(iii)

The total angular displacement is given by

(iv)

SIMPLIFY

We can combine equations (ii) and (iii) to obtain the angular displacement while the flywheel is rotating at a constant angular velocity:

(v)

We can combine equations (v), (iv), and (i) to get the total angular displacement of the flywheel:

CALCULATE

Putting in the numerical values gives us

ROUND

Expressing our result to three significant figures, we have

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DOUBLE-CHECK

It is comforting that our answer has the right unit, the rad. Our formula, gives a value that increases linearly with the value of the angular acceleration. It is also always larger than zero, as expected, because *t*_{b}>*t*_{a}.

To perform a further check, let's calculate the angular displacement in two steps. The first step is to calculate the angular displacement while the flywheel is accelerating

The angular velocity of the flywheel after the angular acceleration ends is

Next, we calculate the angular displacement while the flywheel is rotating at constant velocity:

The total angular displacement is then

which agrees with our answer.

MULTIPLE-CHOICE QUESTIONS

**9.1**An object is moving in a circular path. If the centripetal force is suddenly removed, how will the object move?It will move radially outward.

It will move radially inward.

It will move vertically downward.

It will move in the direction in which its velocity vector points at the instant the centripetal force vanishes.

**9.2**The angular acceleration for an object undergoing circular motion is plotted versus time in the figure. If the object started from rest at*t*= 0 s, the net angular displacement of the object at*t*=*t*_{f}is in the clockwise direction.

is in the counterclockwise direction.

is zero.

cannot be determined.

**9.3**The latitude of Lubbock, Texas (known as the Hub City of the South Plains) is 33° N. What is its rotational speed, assuming the radius of the Earth at the Equator to be 6380 km?464 m/s

389 m/s

253 m/s

0.464 m/s

0.389 m/s

**9.4**A rock attached to a string moves clockwise in uniform circular motion. In which direction from point*A*is the rock thrown off when the string is cut?**9.5**A Ferris wheel rotates slowly about a horizontal axis. Passengers sit on seats that remain horizontal on the Ferris wheel as it rotates. Which type of force provides the centripetal acceleration on the passengers when they are at the top of the Ferris wheel?centrifugal

normal

gravity

tension

**9.6**In a conical pendulum, a bob moves in a horizontal circle, as shown in the figure. The period of the pendulum (the time it takes for the bob to perform a complete revolution) is.

.

.

.

.

Page 306**9.7**A ball attached to the end of a string is swung around in a circular path of radius*r*. If the radius is doubled and the linear speed is kept constant, the centripetal accelerationremains the same.

increases by a factor of 2.

decreases by a factor of 2.

increases by a factor of 4.

decreases by a factor of 4.

**9.8**The angular speed of the hour hand of a clock (in radians per second) isThe correct value is not shown.

**9.9**You put three identical coins on a turntable at different distances from the center and then turn the motor on. As the turntable speeds up, the outermost coin slides off first, followed by the one at the middle distance, and, finally, when the turntable is going the fastest, the innermost one. Why is this?For greater distances from the center the centripetal acceleration is higher, and so the force of friction becomes unable to hold the coin in place.

The weight of the coin causes the turntable to flex downward, so the coin nearest the edge falls off first.

Because of the way the turntable is made, the coefficient of static friction decreases with distance from the center.

For smaller distances from the center, the centripetal acceleration is higher.

**9.10**A point on a Blu-ray disc is a distance*R*/4 from the axis of rotation. How far from the axis of rotation is a second point that has, at any instant, a linear velocity twice that of the first point?*R*/16*R*/8*R*/2*R*

**9.11**The figure shows a rider stuck to the wall in the Barrel of Fun at a carnival. Which diagram correctly shows the forces acting on the rider?**9.12**A string is tied to a rock, and the rock is twirled around in a circle at a constant speed. If gravity is ignored and the period of the circular motion is doubled, the tension in the string isreduced to of its original value.

reduced to of its original value.

unchanged.

increased to 2 times its original value.

increased to 4 times its original value.

QUESTIONS

**9.13**A ceiling fan is rotating in clockwise direction (as viewed from below) but it is slowing down. What are the directions of*ω*and*α*?**9.14**A hook above a stage is rated to support 150. lb. A 3-lb rope is attached to the hook, and a 147-lb actor is going to attempt to swing across the stage on the rope. Will the hook hold the actor up during the swing?**9.15**A popular carnival ride consists of seats attached to a central disk through cables, as shown in the figure. The passengers travel in uniform circular motion. The mass of one of the passengers (including the chair he is sitting on) is 65 kg; the mass of an empty chair on the opposite side of the central disk is 5 kg. If*θ*_{1}and*θ*_{2}are the angles that the cables attached to the two chairs make with respect to vertical, how do these two angles compare qualitatively? Is*θ*_{2}larger than, smaller than, or equal to*θ*_{1}?**9.16**A person rides on a Ferris wheel of radius*R*, which is rotating at a constant angular velocity*ω*. Compare the normal force of the seat pushing up on the person at point*A*to that at point*B*in the figure. Which force is greater, or are they the same?**9.17**Bicycle tires range in size from about 25 cm in diameter to about 70 cm in diameter. Why is it impractical to make tires much smaller than 25 cm in diameter? (You will learn why bicycle tires can't be too large in Chapter 10.)**9.18**A CD starts from rest and speeds up to the operating angular frequency of the CD player. Compare the angular velocity and acceleration of a point on the edge of the CD to those of a point halfway between the center and the edge of the CD. Do the same for the linear velocity and acceleration.**9.19**A car is traveling around an unbanked curve at a maximum speed. Which force(s) is(are) responsible for keeping it on the road?**9.20**Two masses hang from two strings of equal length that are attached to the ceiling of a car. One mass is over the driver's seat; the other is over the passenger's seat. As the car makes a sharp turn, both masses swing away from the center of the turn. In their resulting positions, will they be farther apart, closer together, or the same distance apart as they were when the car wasn't turning?**9.21**A point mass*m*starts sliding from a height*h*along the frictionless surface shown in the figure. What is the minimum value of*h*in order for the mass to complete the loop of radius*R*?**9.22**In a conical pendulum, the bob attached to the string (which can be considered massless) moves in a horizontal circle at constant speed. The string sweeps out a cone as the bob rotates. What forces are acting on the bob?**9.23**Is it possible to swing a mass attached to a string in a perfectly horizontal circle (with the mass and the string parallel to the ground)?**9.24**A small ice block of mass*m*starts from rest from the top of an inverted bowl in the shape of a hemisphere, as shown in the figure. The hemisphere is fixed to the ground, and the block slides without friction along the surface of the hemisphere. Find the normal force exerted by the block on the sphere when the line between the block and the center of the sphere makes an angle*θ*with the horizontal. Discuss the result.**9.25**Suppose you are riding on a roller coaster, which moves through a vertical circular loop. Show that your apparent weight at the bottom of the loop is six times your weight when you experience weightlessness at the top, independent of the size of the loop. Assume that friction is negligible.**9.26**The following event actually occurred on the Sunshine Skyway Bridge near St. Petersburg, Florida, in 1997. Five daredevils tied a 55-m-long cable to the center of the bridge. They hoped to swing back and forth under the bridge at the end of this cable. The five people (total weight =*W*) attached themselves to the end of the cable, at the same level and 55 m away from where it was attached to the bridge and dropped straight down from the bridge, following the dashed circular path indicated in the figure. Unfortunately, the daredevils were not well versed in the laws of physics, and the cable broke (at the point it was linked to their seats) at the bottom of their swing. Determine how strong the cable (and all the links where the seats and the bridge are attached to it) would have to be in order to support the five people at the bottom of the swing. Express your result in terms of their total weight,*W*.

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PROBLEMS

A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One • and two •• indicate increasing level of problem difficulty.

Section 9.2

**9.27**What is the angle in radians that the Earth sweeps out in its orbit during winter?**9.28**Assuming that the Earth is spherical and recalling that latitudes range from 0° at the Equator to 90° N at the North Pole, how far apart, measured on the Earth's surface, are Dubuque, Iowa (42.50° N latitude) and Guatemala City (14.62° N latitude)? The two cities lie on approximately the same longitude. Do*not*neglect the curvature of the Earth in determining this distance.**•****9.29**Refer to the information given in Problem 9.28. If one could burrow through the Earth and dig a straight-line tunnel from Dubuque to Guatemala City, how long would the tunnel be? From the point of view of the digger, at what angle below the horizontal would the tunnel be directed?

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Section 9.3

**9.30**A typical Major League fastball is thrown at approximately 88 mph and with a spin rate of 110 rpm. If the distance between the pitcher's point of release and the catcher's glove is exactly 60.5 ft, how many full turns does the ball make between release and catch? Neglect any effect of gravity or air resistance on the ball's flight.**9.31**A vinyl record plays at 33.3 rpm. Assume it takes 5.00 s for it to reach this full speed, starting from rest.What is its angular acceleration during the 5.00 s?

How many revolutions does the record make before reaching its final angular speed?

**9.32**At a county fair, a boy takes his teddy bear on the giant Ferris wheel. Unfortunately, at the top of the ride, he accidentally drops his stuffed buddy. The wheel has a diameter of 12.0 m, the bottom of the wheel is 2.0 m above the ground and its rim is moving at a speed of 1.0 m/s. How far from the base of the Ferris wheel will the teddy bear land?**•****9.33**Having developed a taste for experimentation, the boy in Problem 9.32 invites two friends to bring their teddy bears on the same Ferris wheel. The boys are seated in positions 45° from each other. When the wheel brings the second boy to the maximum height, they all drop their stuffed animals. How far apart will the three teddy bears land?**•****9.34**Mars orbits the Sun at a mean distance of 228 million km, in a period of 687 days. The Earth orbits at a mean distance of 149.6 million km, in a period of 365.26 days.Suppose Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. Exactly 365.26 days later, when the Earth has completed one orbit, what is the angle between the Earth-Sun line and the Mars-Sun line?

The initial situation in part (a) is a closest approach of Mars to Earth. What is the time, in days, between two closest approaches? Assume constant speed and circular orbits for both Mars and Earth.

Another way of expressing the answer to part (b) is in terms of the angle between the lines drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle?

**••9.35**Consider a large simple pendulum that is located at a latitude of 55.0° N and is swinging in a north-south direction with points*A*and*B*being the northernmost and the southernmost points of the swing, respectively. A stationary (with respect to the fixed stars) observer is looking directly down on the pendulum at the moment shown in the figure. The Earth is rotating once every 23 h and 56 min.What are the directions (in terms of N, E, W, and S) and the magnitudes of the velocities of the surface of the Earth at points

*A*and*B*as seen by the observer?*Note*: You will need to calculate answers to at least seven significant figures to see a difference.What is the angular speed with which the 20.0-m diameter circle under the pendulum appears to rotate?

What is the period of this rotation?

What would happen to a pendulum swinging at the Equator?

Section 9.4

**9.36**What is the centripetal acceleration of the Moon? The period of the Moon's orbit about the Earth is 27.3 days, measured with respect to the fixed stars. The radius of the Moon's orbit is*R*_{M}= 3.85 · 10^{8}m.**9.37**You are holding the axle of a bicycle wheel with radius 35.0 cm and mass 1.0 kg. You get the wheel spinning at a rate of 75.0 rpm and then stop it by pressing the tire against the pavement. You notice that it takes 1.20 s for the wheel to come to a complete stop. What is the angular acceleration of the wheel?**9.38**Life scientists use ultracentrifuges to separate biological components or to remove molecules from suspension. Samples in a symmetric array of containers are spun rapidly about a central axis. The centrifugal acceleration they experience in their moving reference frame acts as “artificial gravity” to effect a rapid separation. If the sample containers are 10.0 cm from the rotation axis, what rotation frequency is required to produce an acceleration of 1.00 · 10^{5}*g*?**9.39**A centrifuge in a medical laboratory rotates at an angular speed of 3600 rpm (revolutions per minute). When switched off, it rotates 60.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.**9.40**A discus thrower (with arm length of 1.2 m) starts from rest and begins to rotate counterclockwise with an angular acceleration of 2.5 rad/s^{2}.How long does it take the discus thrower's speed to get to 4.7 rad/s?

How many revolutions does the thrower make to reach the speed of 4.7 rad/s?

What is the linear speed of the discus at 4.7 rad/s?

What is the linear acceleration of the discus thrower at this point?

What is the magnitude of the centripetal acceleration of the discus thrown?

What is the magnitude of the discus's total acceleration?

Page 309**•****9.41**In a department store toy display, a small disk (disk 1) of radius 0.100 m is driven by a motor and turns a larger disk (disk 2) of radius 0.500 m. Disk 2, in turn, drives disk 3, whose radius is 1.00 m. The three disks are in contact, and there is no slipping. Disk 3 is observed to sweep through one complete revolution every 30.0 s.What is the angular speed of disk 3?

What is the ratio of the tangential velocities of the rims of the three disks?

What is the angular speed of disks 1 and 2?

If the motor malfunctions, resulting in an angular acceleration of 0.100 rad/s

^{2}for disk 1, what are disks 2 and 3's angular accelerations?

**•****9.42**A particle is moving clockwise in a circle of radius 1.00 m. At a certain instant, the magnitude of its acceleration is , and the acceleration vector has an angle of*θ*= 50.0° with the position vector, as shown in the figure. At this instant, find the speed, , of this particle.**•9.43**In a tape recorder, the magnetic tape moves at a constant linear speed of 5.6 cm/s. To maintain this constant linear speed, the angular speed of the driving spool (the take-up spool) has to change accordingly.What is the angular speed of the take-up spool when it is empty, with radius

*r*_{1}= 0.80 cm?What is the angular speed when the spool is full, with radius

*r*_{2}= 2.20 cm?If the total length of the tape is 100.80 m, what is the average angular acceleration of the take-up spool while the tape is being played?

**••****9.44**A ring is fitted loosely (with no friction) around a long, smooth rod of length*L*= 0.50 m. The rod is fixed at one end, and the other end is spun in a horizontal circle at a constant angular velocity of*ω*= 4.0 rad/s. The ring has zero radial velocity at its initial position, a distance of*r*_{0}= 0.30 m from the fixed end. Determine the radial velocity of the ring as it reaches the moving end of the rod.**••****9.45**A flywheel with a diameter of 1.00 m is initially at rest. Its angular acceleration versus time is graphed in the figure.What is the angular separation between the initial position of a fixed point on the rim of the flywheel and the point's position 8.00 s after the wheel starts rotating?

The point starts its motion at

*θ*= 0. Calculate and sketch the linear position, velocity vector, and acceleration vector 8 s after the wheel starts rotating.

Section 9.5

**9.46**Calculate the centripetal force exerted on a vehicle of mass*m*= 1500. kg that is moving at a speed of 15.0 m/s around a curve of radius*R*= 400. m. Which force plays the role of the centripetal force in this case?**9.47**What is the apparent weight of a rider on the roller coaster of Solved Problem 9.1 at the*bottom*of the loop?**9.48**Two skaters, A and B, of equal mass are moving in clockwise uniform circular motion on the ice. Their motions have equal periods, but the radius of skater A's circle is half that of skater B's circle.What is the ratio of the speeds of the skaters?

What is the ratio of the magnitudes of the forces acting on each skater?

**•****9.49**A small block of mass*m*is in contact with the inner wall of a large hollow cylinder. Assume the coefficient of static friction between the object and the wall of the cylinder is*μ*. Initially, the cylinder is at rest, and the block is held in place by a peg supporting its weight. The cylinder starts rotating about its center axis, as shown in the figure, with an angular acceleration of*α*. Determine the minimum time interval after the cylinder begins to rotate before the peg can be removed without the block sliding against the wall.**•****9.50**A race car is making U-turn at constant speed. The coefficient of friction between the tires and the track is*μ*_{s}= 1.20. If the radius of the curve is 10.0 m, what is the maximum speed at which the car can turn without sliding? Assume that the car is performing uniform circular motion.**•9.51**A car speeds over the top of a hill. If the radius of curvature of the hill at the top is 9.0 m, how fast can the car be traveling and maintain constant contact with the ground?**•****9.52**A ball of mass*m*= 0.200 kg is attached to a (massless) string of length*L*= 1.00 m and is undergoing circular motion in the horizontal plane, as shown in the figure.Page 310Draw a free-body diagram for the ball.

Which force plays the role of the centripetal force?

What should the speed of the mass be for

*θ*to be 45.0°?What is the tension in the string?

**•****9.53**You are flying to Chicago for a weekend away from the books. In your last physics class, you learned that the airflow over the wings of the plane creates a*lift force,*which acts perpendicular to the wings. When the plane is flying level, the upward lift force exactly balances the downward*weight force*. Since O'Hare is one of the busiest airports in the world, you are not surprised when the captain announces that the flight is in a holding pattern due to the heavy traffic. He informs the passengers that the plane will be flying in a circle of radius 7.00 mi at a speed of 360. mph and an altitude of 2.00 · 10^{4}ft. From the safety information card, you know that the total length of the wingspan of the plane is 275 ft. From this information, estimate the banking angle of the plane relative to the horizontal.**••****9.54**A 20.0-g metal cylinder is placed on a turntable, with its center 80.0 cm from the turntable's center. The coefficient of static friction between the cylinder and the turntable's surface is*μ*_{s}= 0.800. A thin, massless string of length 80.0 cm connects the center of the turntable to the cylinder, and*initially*, the string has zero tension in it. Starting from rest, the turntable very slowly attains higher and higher angular velocities, but the turntable and the cylinder can be considered to have uniform circular motion at any instant. Calculate the tension in the string when the angular velocity of the turntable is 60.0 rpm (rotations per minute).**••9.55**A speedway turn, with radius of curvature*R*, is banked at an angle*θ*above the horizontal.What is the optimal speed at which to take the turn if the track's surface is iced over (that is, if there is very little friction between the tires and the track)?

If the track surface is ice-free and there is a coefficient of friction

*μ*_{s}between the tires and the track, what are the maximum and minimum speeds at which this turn can be taken?Evaluate the results of parts (a) and (b) for

*R*= 400. m,*θ*= 45.0°, and*μ*_{s}= 0.700.

Additional Problems

**9.56**A particular Ferris wheel takes riders in a vertical circle of radius 9.0 m once every 12.0 s.Calculate the speed of the riders, assuming it to be constant.

Draw a free-body diagram for a rider at a time when she is at the bottom of the circle. Calculate the normal force exerted by the seat on the rider at that point in the ride.

Perform the same analysis as in part (b) for a point at the top of the ride.

**9.57**A boy is on a Ferris wheel, which takes him in a vertical circle of radius 9.0 m once every 12.0 s.What is the angular speed of the Ferris wheel?

Suppose the wheel comes to a stop at a uniform rate during one quarter of a revolution. What is the angular acceleration of the wheel during this time?

Calculate the tangential acceleration of the boy during the time interval described in part (b).

**9.58**Consider a 53-cm-long lawn mower blade rotating about its center at 3400 rpm.Calculate the linear speed of the tip of the blade.

If safety regulations require that the blade be stoppable within 3.0 s, what minimum angular acceleration will accomplish this? Assume that the angular acceleration is constant.

**9.59**A car accelerates uniformly from rest and reaches a speed of 22.0 m/s in 9.00 s. The diameter of a tire on this car is 58.0 cm.Find the number of revolutions the tire makes during the car's motion, assuming that no slipping occurs.

What is the final angular speed of a tire in revolutions per second?

**9.60**Gear A, with a mass of 1.00 kg and a radius of 55.0 cm, is in contact with gear B, with a mass of 0.500 kg and a radius of 30.0 cm. The gears do not slip with respect to each other as they rotate. Gear A rotates at 120. rpm and slows to 60.0 rpm in 3.00 s. How many rotations does gear B undergo during this time interval?**9.61**A top spins for 10.0 min, beginning with an angular speed of 10.0 rev/s. Determine its angular acceleration, assuming it is constant, and its total angular displacement.**9.62**A penny is sitting on the edge of an old phonograph disk that is spinning at 33 rpm and has a diameter of 12 inches. What is the minimum coefficient of static friction between the penny and the surface of the disk to ensure that the penny doesn't fly off?**9.63**A vinyl record that is initially turning at rpm slows uniformly to a stop in a time of 15 s. How many rotations are made by the record while stopping?**9.64**Determine the linear and angular speeds and accelerations of a speck of dirt located 2.0 cm from the center of a CD rotating inside a CD player at 250 rpm.**9.65**What is the acceleration of the Earth in its orbit? (Assume the orbit is circular.)**9.66**A day on Mars is 24.6 Earth hours long. A year on Mars is 687 Earth days long. How do the angular velocities of Mars's rotation and orbit compare to the angular velocities of Earth's rotation and orbit?**9.67**A monster truck has tires with a diameter of 1.10 m and is traveling at 35.8 m/s. After the brakes are applied, the truck slows uniformly and is brought to rest after the tires rotate through 40.2 turns.What is the initial angular speed of the tires?

What is the angular acceleration of the tires?

What distance does the truck travel before coming to rest?

**•****9.68**The motor of a fan turns a small wheel of radius*r*_{m}= 2.00 cm. This wheel turns a belt, which is attached to a wheel of radius*r*_{f}= 3.00 cm that is mounted to the axle of the fan blades. Measured from the center of this axle, the tip of the fan blades are at a distance*r*_{b}= 15.0 cm. When the fan is in operation, the motor spins at an angular speed of*ω*= 1200. rpm. What is the tangential speed of the tips of the fan blades?**•****9.69**A car with a mass of 1000. kg goes over a hill at a constant speed of 60.0 m/s. The top of the hill can be approximated as an arc length of a circle with a radius of curvature of 370. m. What force does the car exert on the hill as it passes over the top?**•****9.70**Unlike a ship, an airplane does not use its rudder to turn. It turns by banking its wings: The lift force, perpendicular to the wings, has a horizontal component, which provides the centripetal acceleration for the turn, and a vertical component, which supports the plane's weight. (The rudder counteracts yaw and thus it keeps the plane pointed in the direction it is moving.) The famous spy plane, the SR-71 Blackbird, flying at 4800 km/h, has a turning radius of 290. km. Find its banking angle.**•9.71**A 80.0-kg pilot in an aircraft moving at a constant speed of 500. m/s pulls out of a vertical dive along an arc of a circle of radius 4000. m.Find the centripetal acceleration and the centripetal force acting on the pilot.

What is the pilot's apparent weight at the bottom of the dive?

**•****9.72**A ball having a mass of 1.00 kg is attached to a string 1 m long and is whirled in a vertical circle at a constant speed of 10.0 m/s.Determine the tension in the string when the ball is at the top of the circle.

Determine the tension in the string when the ball is at the bottom of the circle.

Consider the ball at some point other than the top or bottom. What can you say about the tension in the string at this point?

**•****9.73**A car starts from rest and accelerates around a flat curve of radius*R*= 36 m. The tangential component of the car's acceleration remains constant at*a*_{t}= 3.3 m/s^{2}, while the centripetal acceleration increases to keep the car on the curve as long as possible. The coefficient of friction between the tires and the road is*μ*= 0.95. What distance does the car travel around the curve before it begins to skid? (Be sure to include both the tangential and centripetal components of the acceleration.)**•****9.74**A girl on a merry-go-round platform holds a pendulum in her hand. The pendulum is 6.0 m from the rotation axis of the platform. The rotational speed of the platform is 0.020 rev/s. It is found that the pendulum hangs at an angle*θ*to the vertical. Find*θ*.**••9.75**A carousel at a carnival has a diameter of 6.00 m. The ride starts from rest and accelerates at a constant angular acceleration to an angular speed of 0.600 rev/s in 8.00 s.What is the value of the angular acceleration?

What are the centripetal and angular accelerations of a seat on the carousel that is 2.75 m from the rotation axis?

What is the total acceleration, magnitude and direction, 8.00 s after the angular acceleration starts?

**••****9.76**A car of weight*W*= 10.0 kN makes a turn on a track that is banked at an angle of*θ*= 20.0°. Inside the car, hanging from a short string tied to the rear-view mirror, is an ornament. As the car turns, the ornament swings out at an angle of*φ*= 30.0° measured from the vertical inside the car. What is the force of static friction between the car and the road?**••****9.77**A popular carnival ride consists of seats attached to a central disk through cables. The passengers travel in uniform circular motion. As shown in the figure, the radius of the central disk is*R*_{0}= 3.00 m, and the length of the cable is*L*= 3.20 m. The mass of one of the passengers (including the chair he is sitting on) is 65.0 kg.If the angle

*θ*that the cable makes with respect to the vertical is 30.0°, what is the speed,*v*, of this passenger?What is the magnitude of the force exerted by the cable on the chair?

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