Before attempting this problem, review Examples 7 and 8.

A car travels at a constant speed around a circular track whose radius is 2.6 km. The car goes once around the track in 360 s. What is the magnitude of the centripetal acceleration of the car?

The following table lists data for the speed and radius of three examples of uniform circular motion. Find the magnitude of the centripetal acceleration for each example.

Review Conceptual Example 2 as background for this problem. One kind of slingshot consists of a pocket that holds a pebble and is whirled on a circle of radius r. The pebble is released from the circle at the angle u, so that it will hit the target. The distance to the target from the center of the circle is d. (See the drawing, which is not to scale.) The circular path is parallel to the ground, and the target lies in the plane of the circle. The distance d is ten times the radius r. Ignore the eff ect of gravity in pulling the stone downward after it is released and fi nd the angle u.

Speedboat A negotiates a curve whose radius is 120 m. Speedboat B negotiates a curve whose radius is 240 m. Each boat experiences the same centripetal acceleration. What is the ratio vA/vB of the speeds of the boats?

How long does it take a plane, traveling at a constant speed of 110 m/s, to fl y once around a circle whose radius is 2850 m?

The aorta is a major artery, rising upward from the left ventricle of the heart and curving down to carry blood to the abdomen and lower half of the body. The curved artery can be approximated as a semicircular arch whose diameter is 5.0 cm. If blood fl ows through the aortic arch at a speed of 0.32 m/s, what is the magnitude (in m/s2 ) of the bloods centripetal acceleration?

The blade of a windshield wiper moves through an angle of 90.08 in 0.40 s. The tip of the blade moves on the arc of a circle that has a radius of 0.45 m. What is the magnitude of the centripetal acceleration of the tip of the blade?

There is a clever kitchen gadget for drying lettuce leaves after you wash them. It consists of a cylindrical container mounted so that it can be rotated about its axis by turning a hand crank. The outer wall of the cylinder is perforated with small holes. You put the wet leaves in the container and turn the crank to spin off the water. The radius of the container is 12 cm. When the cylinder is rotating at 2.0 revolutions per second, what is the magnitude of the centripetal acceleration at the outer wall?

Computer-controlled display screens provide drivers in the Indianapolis 500 with a variety of information about how their cars are performing. For instance, as a car is going through a turn, a speed of 221 mi/h (98.8 m/s) and centripetal acceleration of 3.00 g (three times the acceleration due to gravity) are displayed. Determine the radius of the turn (in meters).

A computer is reading data from a rotating CD-ROM. At a point that is 0.030 m from the center of the disc, the centripetal acceleration is 120 m/s2 . What is the centripetal acceleration at a point that is 0.050 m from the center of the disc?

A centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance, to cause the more dense red blood cells to settle through the less dense blood serum and collect at the bottom of the container. Suppose the centripetal acceleration of the sample is 6.25 3 103 times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of 5.00 cm from the axis of rotation?

The earth rotates once per day about an axis passing through the north and south poles, an axis that is perpendicular to the plane of the equator. Assuming the earth is a sphere with a radius of 6.38 3 106 m, determine the speed and centripetal acceleration of a person situated (a) at the equator and (b) at a latitude of 30.08 north of the equator.

Review Example 3, which deals with the bobsled in Figure 5.5. Also review Conceptual Example 4. The mass of the sled and its two riders in Figure 5.5 is 350 kg. Find the magnitude of the centripetal force that acts on the sled during the turn with a radius of (a) 33 m and (b) 24 m.

At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of 3.2 m/s, and an 83-kg person feels a 560-N force pressing against his back. What is the radius of the chamber?

Multiple-Concept Example 7 reviews the concepts that play a role in this problem. Car A uses tires for which the coeffi cient of static friction is 1.1 on a particular unbanked curve. The maximum speed at which the car can negotiate this curve is 25 m/s. Car B uses tires for which the coeffi cient of static friction is 0.85 on the same curve. What is the maximum speed at which car B can negotiate the curve?

A speed skater goes around a turn that has a radius of 31 m. The skater has a speed of 14 m/s and experiences a centripetal force of 460 N. What is the mass of the skater?

For background pertinent to this problem, review Conceptual Example 6. In Figure 5.7 the man hanging upside down is holding a partner who weighs 475 N. Assume that the partner moves on a circle that has a radius of 6.50 m. At a swinging speed of 4.00 m/s, what force must the man apply to his partner in the straight-down position?

A car is safely negotiating an unbanked circular turn at a speed of 21 m/s. The road is dry, and the maximum static frictional force acts on the tires. Suddenly a long wet patch in the road decreases the maximum static frictional force to one-third of its dry-road value. If the car is to continue safely around the curve, to what speed must the driver slow the car?

See Conceptual Example 6 to review the concepts involved in this problem. A 9.5-kg monkey is hanging by one arm from a branch and is swinging on a vertical circle. As an approximation, assume a radial distance of 85 cm between the branch and the point where the monkeys mass is located. As the monkey swings through the lowest point on the circle, it has a speed of 2.8 m/s. Find (a) the magnitude of the centripetal force acting on the monkey and (b) the magnitude of the tension in the monkeys arm.

Multiple-Concept Example 7 deals with the concepts that are important in this problem. A penny is placed at the outer edge of a disk (radius 5 0.150 m) that rotates about an axis perpendicular to the plane of the disk at its center. The period of the rotation is 1.80 s. Find the minimum coeffi cient of friction necessary to allow the penny to rotate along with the disk

The hammer throw is a track-and-fi eld event in which a 7.3-kg ball (the hammer) is whirled around in a circle several times and released. It then moves upward on the familiar curving path of projectile motion and eventually returns to earth some distance away. The world record for this distance is 86.75 m, achieved in 1986 by Yuriy Sedykh. Ignore air resistance and the fact that the ball is released above the ground rather than at ground level. Furthermore, assume that the ball is whirled on a circle that has a radius of 1.8 m and that its velocity at the instant of release is directed 418 above the horizontal. Find the magnitude of the centripetal force acting on the ball just prior to the moment of release

An 830-kg race car can drive around an unbanked turn at a maximum speed of 58 m/s without slipping. The turn has a radius of curvature of 160 m. Air fl owing over the cars wing exerts a downwardpointing force (called the downforce) of 11 000 N on the car. (a) What is the coeffi cient of static friction between the track and the cars tires? (b) What would be the maximum speed if no downforce acted on the car?

A swing ride at a carnival consists of chairs that are swung in a circle by 15.0-m cables attached to a vertical rotating pole, as the drawing shows. Suppose the total mass of a chair and its occupant is 179 kg. (a) Determine the tension in the cable attached to the chair. (b) Find the speed of the chair.

On a banked race track, the smallest circular path on which cars can move has a radius of 112 m, while the largest has a radius of 165 m, as the drawing illustrates. The height of the outer wall is 18 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.

Before attempting this problem, review Examples 7 and 8. Two curves on a highway have the same radii. However, one is unbanked and the other is banked at an angle u. A car can safely travel along the unbanked curve at a maximum speed v0 under conditions when the coeffi cient of static friction between the tires and the road is ms 5 0.81. The banked curve is frictionless, and the car can negotiate it at the same maximum speed v0. Find the angle u of the banked curve.

A woman is riding a Jet Ski at a speed of 26 m/s and notices a seawall straight ahead. The farthest she can lean the craft in order to make a turn is 228. This situation is like that of a car on a curve that is banked at an angle of 228. If she tries to make the turn without slowing down, what is the minimum distance from the seawall that she can begin making her turn and still avoid a crash?

Two banked curves have the same radius. Curve A is banked at an angle of 138, and curve B is banked at an angle of 198. A car can travel around curve A without relying on friction at a speed of 18 m/s. At what speed can this car travel around curve B without relying on friction?

A racetrack has the shape of an inverted cone, as the drawing shows. On this surface the cars race in circles that are parallel to the ground. For a speed of 34.0 m/s, at what value of the distance d should a driver locate his car if he wishes to stay on a circular path without depending on friction?

A jet fl ying at 123 m/s banks to make a horizontal circular turn. The radius of the turn is 3810 m, and the mass of the jet is 2.00 3 105 kg. Calculate the magnitude of the necessary lifting force.

The drawing shows a baggage carousel at an airport. Your suitcase has not slid all the way down the slope and is going around at a constant speed on a circle (r 5 11.0 m) as the carousel turns. The coeffi cient of static friction between the suitcase and the carousel is 0.760, and the angle u in the drawing is 36.08. How much time is required for your suitcase to go around once?

Two satellites are in circular orbits around the earth. The orbit for satellite A is at a height of 360 km above the earths surface, while that for satellite B is at a height of 720 km. Find the orbital speed for each satellite.

A rocket is used to place a synchronous satellite in orbit about the earth. What is the speed of the satellite in orbit?

A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.70 3 104 m/s, and the radius of the orbit is 5.25 3 106 m. A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of 8.60 3 106 m. What is the orbital speed of the second satellite?

Multiple-Concept Example 13 off ers a helpful perspective for this problem. Suppose the surface (radius 5 r) of the space station in Figure 5.18 is rotating at 35.8 m/s. What must be the value of r for the astronauts to weigh one-half of their earth-weight?

A satellite is in a circular orbit about the earth (ME 5 5.98 3 1024 kg). The period of the satellite is 1.20 3 104 s. What is the speed at which the satellite travels?

A satellite circles the earth in an orbit whose radius is twice the earths radius. The earths mass is 5.98 3 1024 kg, and its radius is 6.38 3 106 m. What is the period of the satellite?

A satellite moves on a circular earth orbit that has a radius of 6.7 3 106 m. A model airplane is fl ying on a 15-m guideline in a horizontal circle. The guideline is parallel to the ground. Find the speed of the plane such that the plane and the satellite have the same centripetal acceleration

A satellite has a mass of 5850 kg and is in a circular orbit 4.1 3 105 m above the surface of a planet. The period of the orbit is 2.00 hours. The radius of the planet is 4.15 3 106 m. What would be the true weight of the satellite if it were at rest on the planets surface?

Two newly discovered planets follow circular orbits around a star in a distant part of the galaxy. The orbital speeds of the planets are determined to be 43.3 km/s and 58.6 km/s. The slower planets orbital period is 7.60 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?

To create artifi cial gravity, the space station shown in the drawing is rotating at a rate of 1.00 rpm. The radii of the cylindrically shaped chambers have the ratio rA/rB 5 4.00. Each chamber A simulates an acceleration due to gravity of 10.0 m/s2 . Find values for (a) rA, (b) rB, and (c) the acceleration due to gravity that is simulated in chamber B.

A motorcycle has a constant speed of 25.0 m/s as it passes over the top of a hill whose radius of curvature is 126 m. The mass of the motorcycle and driver is 342 kg. Find the magnitudes of (a) the centripetal force and (b) the normal force that acts on the cycle.

Pilots of high-performance fi ghter planes can be subjected to large centripetal accelerations during high-speed turns. Because of these accelerations, the pilots are subjected to forces that can be much greater than their body weight, leading to an accumulation of blood in the abdomen and legs. As a result, the brain becomes starved for blood, and the pilot can lose consciousness (black out). The pilots wear anti-G suits to help keep the blood from draining out of the brain. To appreciate the forces that a fi ghter pilot must endure, consider the magnitude FN of the normal force that the pilots seat exerts on him at the bottom of a dive. The magnitude of the pilots weight is W. The plane is traveling at 230 m/s on a vertical circle of radius 690 m. Determine the ratio FN/W. For comparison, note that blackout can occur for values of FN/W as small as 2 if the pilot is not wearing an anti-G suit.

For the normal force in Figure 5.19 to have the same magnitude at all points on the vertical track, the stunt driver must adjust the speed to be diff erent at diff erent points. Suppose, for example, that the track has a radius of 3.0 m and that the driver goes past point 1 at the bottom with a speed of 15 m/s. What speed must she have at point 3, so that the normal force at the top has the same magnitude as it did at the bottom?

A special electronic sensor is embedded in the seat of a car that takes riders around a circular loop-the-loop ride at an amusement park. The sensor measures the magnitude of the normal force that the seat exerts on a rider. The loop-the-loop ride is in the vertical plane and its radius is 21 m. Sitting on the seat before the ride starts, a rider is level and stationary, and the electronic sensor reads 770 N. At the top of the loop, the rider is upside down and moving, and the sensor reads 350 N. What is the speed of the rider at the top of the loop?

A 0.20-kg ball on a stick is whirled on a vertical circle at a constant speed. When the ball is at the three oclock position, the stick tension is 16 N. Find the tensions in the stick when the ball is at the twelve oclock and at the six oclock positions.

A stone is tied to a string (length 5 1.10 m) and whirled in a circle at the same constant speed in two diff erent ways. First, the circle is horizontal and the string is nearly parallel to the ground. Next, the circle is vertical. In the vertical case the maximum tension in the string is 15.0% larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.

A motorcycle is traveling up one side of a hill and down the other side. The crest of the hill is a circular arc with a radius of 45.0 m. Determine the maximum speed that the cycle can have while moving over the crest without losing contact with the road

In an automatic clothes dryer, a hollow cylinder moves the clothes on a vertical circle (radius r 5 0.32 m), as the drawing shows. The appliance is designed so that the clothes tumble gently as they dry. This means that when a piece of clothing reaches an angle of u above the horizontal, it loses contact with the wall of the cylinder and falls onto the clothes below. How many revolutions per second should the cylinder make in order that the clothes lose contact with the wall when u 5 70.08?

In a skating stunt known as crack-the-whip, a number of skaters hold hands and form a straight line. They try to skate so that the line rotates about the skater at one end, who acts as the pivot. The skater farthest out has a mass of 80.0 kg and is 6.10 m from the pivot. He is skating at a speed of 6.80 m/s. Determine the magnitude of the centripetal force that acts on him.

In designing rotating space stations to provide for artifi cial- gravity environments, one of the constraints that must be considered is motion sickness. Studies have shown that the negative eff ects of motion sickness begin to appear when the rotational motion is faster than two revolutions per minute. On the other hand, the magnitude of the centripetal acceleration at the astronauts feet should equal the magnitude of the acceleration due to gravity on earth. Thus, to eliminate the diffi - culties with motion sickness, designers must choose the distance between the astronauts feet and the axis about which the space station rotates to be greater than a certain minimum value. What is this minimum value?

The second hand and the minute hand on one type of clock are the same length. Find the ratio (ac, second/ac, minute) of the centripetal accelerations of the tips of the second hand and the minute hand.

A child is twirling a 0.0120-kg plastic ball on a string in a horizontal circle whose radius is 0.100 m. The ball travels once around the circle in 0.500 s. (a) Determine the centripetal force acting on the ball. (b) If the speed is doubled, does the centripetal force double? If not, by what factor does the centripetal force increase?

Two satellites, A and B, are in diff erent circular orbits about the earth. The orbital speed of satellite A is three times that of satellite B. Find the ratio (TA/TB) of the periods of the satellites.

Two cars are traveling at the same speed of 27 m/s on a curve that has a radius of 120 m. Car A has a mass of 1100 kg, and car B has a mass of 1600 kg. Find the magnitude of the centripetal acceleration and the magnitude of the centripetal force for each car

A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to twice her weight as she goes through the dip. If r 5 20.0 m, how fast is the roller coaster traveling at the bottom of the dip?

he National Aeronautics and Space Administration (NASA) studies the physiological eff ects of large accelerations on astronauts. Some of these studies use a machine known as a centrifuge. This machine consists of a long arm, to one end of which is attached a chamber in which the astronaut sits. The other end of the arm is connected to an axis about which the arm and chamber can be rotated. The astronaut moves on a circular path, much like a model airplane fl ying in a circle on a guideline. The chamber is located 15 m from the center of the circle. At what speed must the chamber move so that an astronaut is subjected to 7.5 times the acceleration due to gravity?

Each of the space shuttles main engines is fed liquid hydrogen by a high-pressure pump. Turbine blades inside the pump rotate at 617 rev/s. A point on one of the blades traces out a circle with a radius of 0.020 m as the blade rotates. (a) What is the magnitude of the centripetal acceleration that the blade must sustain at this point? (b) Express this acceleration as a multiple of g 5 9.80 m/s2

The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is 6.7 m, measured from its tip to the center of the circle. Find the ratio of the centripetal acceleration at the end of the blade to that which exists at a point located 3.0 m from the center of the circle.

A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 28 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius 5 150 m), the block swings toward the outside of the curve. Then the string makes an angle u with the vertical. Find u

A rigid massless rod is rotated about one end in a horizontal circle. There is a particle of mass m1 attached to the center of the rod and a particle of mass m2 attached to the outer end of the rod. The inner section of the rod sustains a tension that is three times as great as the tension that the outer section sustains. Find the ratio m1/m2.

Redo Example 5, assuming that there is no upward lift on the plane generated by its wings. Without such lift, the guideline slopes downward due to the weight of the plane. For purposes of signifi cant fi gures, use 0.900 kg for the mass of the plane, 17.0 m for the length of the guideline, and 19.0 and 38.0 m/s for the speeds.

A space laboratory is rotating to create artifi cial gravity, as the fi gure indicates. Its period of rotation is chosen so that the outer ring (rO 5 2150 m) simulates the acceleration due to gravity on earth (9.80 m/s2 ). What should be the radius r1 of the inner ring, so it simulates the acceleration due to gravity on the surface of Mars (3.72 m/s2 )?

At time t 5 0 s, automobile A is traveling at a speed of 18 m/s along a straight road and is picking up speed with an acceleration that has a magnitude of 3.5 m/s2 as in part a of the fi gure. At t 5 0 s, automobile B is traveling at a speed of 18 m/s in uniform circular motion as it negotiates a turn (part b of the fi gure). It has a centripetal acceleration whose magnitude is also 3.5 m/s2 . Concepts: (i) Which automobile has a constant acceleration? (ii) For which automobile do the equations of kinematics apply? Calculations: Determine the speed of each automobile when t 5 2 s.

Ball A is attached to one end of a rigid massless rod, while an identical ball B is attached to the center of the rod, as shown in the fi gure. Each ball has a mass of m 5 0.50 kg, and the length of each half of the rod is L 5 0.40 m. This arrangement is held by the empty end and is whirled around in a horizontal circle at a constant rate, so that each ball is in uniform circular motion. Ball A travels at a constant speed of vA 5 5.0 m/s. Concepts: (i) How many tension forces contribute to the centripetal force that acts on ball A? (ii) How many tension forces contribute to the centripetal force that acts on ball B? (iii) Is the speed of ball A the same as that of ball B? Calculations: Find the tension in each half of the rod.