The earth rotates once per day about its axis, which is

Synchronous or stationary communications satellites are put into an orbit whose radius is r 4.23  107 m. The orbit is in the plane of the equator, and two adjacent satellites have an angular separation of 2.00, as Figure 8.4 illustrates. Find the arc length s (see the drawing) that separates the satellites. Rea

The diameter of the sun is about 400 times greater than that of the moon. By coincidence, the sun is also about 400 times farther from the earth than is the moon. For an observer on earth, compare the angles subtended by the sun and the moon. (a) The angle subtended by the sun is much greater than that subtended by the moon. (b) The angle subtended by the sun is much smaller than that subtended by the moon. (c) The angles subtended by the sun and the moon are approximately equal.

In the drawing, the flat triangular sheet ABC is lying in the plane of the paper. This sheet is going to rotate about first one axis and then another axis. Both of these axes lie in the plane of the paper and pass through point A. For each of the axes the points B and C move on separate circular paths that have the same radii. Identify these two axes.

Three objects are visible in the night sky. They have the following diameters (in multiples of d) and subtend the following angles (in multiples of 0) at the eye of the observer. Object A has a diameter of 4d and subtends an angle of 20. Object B has a diameter of 3d and subtends an angle of 0/2. Object C has a diameter of d/2 and subtends an angle of 0 /8. Rank them in descending order (greatest first) according to their distance from the observer. C A

A gymnast on a high bar swings through two revolutions in a time of 1.90 s, as Figure 8.6 suggests. Find the average angular velocity (in rad/s) of the gymnast.

A jet awaiting clearance for takeoff is momentarily stopped on the runway. As seen from the front of one engine, the fan blades are rotating with an angular velocity of 110 rad/s, where the negative sign indicates a clockwise rotation (see Figure 8.7). As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.

A pair of scissors is being used to cut a piece of paper in half. Does each blade of the scissors have the same angular velocity (both magnitude and direction) at a given instant?

An electric clock is hanging on a wall. As you are watching the second hand rotate, the clocks battery stops functioning, and the second hand comes to a halt over a brief period of time. Which one of the following statements correctly describes the angular velocity  and angular acceleration  of the second hand as it slows down? (a)  and  are both negative. (b)  is positive and  is negative. (c)  is negative and  is positive. (d)  and  are both positive

The blades of an electric blender are whirling with an angular velocity of 375 rad/s while the puree button is pushed in, as Figure 8.8 shows. When the blend button is pressed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of 44.0 rad. The angular acceleration has a constant value of 1740 rad/s2 . Find the final angular velocity of the blades. Re

The blades of a ceiling fan start from rest and, after two revolutions, have an angular speed of 0.50 rev/s. The angular acceleration of the blades is constant. What is the angular speed after eight revolutions?

Equation 8.7 is being used to solve a problem in rotational kinematics. Which one of the following sets of values for the variables 0, , and t cannot be substituted directly into this equation to calculate a value for ? (a) 0 1.0 rad/s,  1.8 rad/s2 , and t 3.8 s (b) 0 0.16 rev/s,  1.8 rad/s2 , and t 3.8 s (c) 0 0.16 rev/s,  0.29 rev/s2 , and t 3.8 s ( 0t

A helicopter blade has an angular speed of  6.50 rev/s and has an angular acceleration of  1.30 rev/s2 . For points 1 and 2 on the blade in Figure 8.11, find the magnitudes of (a) the tangential speeds and (b) the tangential accelerations. R

A thin rod rotates at a constant angular speed. In case A the axis of rotation is perpendicular to the rod at its center. In case B the axis is perpendicular to the rod at one end. In which case, if either, are there points on the rod that have the same tangential speeds?

It is possible to build a clock in which the tips of the hour hand and the second hand move with the same tangential speed. This is normally never done, however. Why? (a) The length of the hour hand would be 3600 times greater than the length of the second hand. (b) The hour hand and the second hand would have the same length. (c) The length of the hour hand would be 3600 times smaller than the length of the second hand.

The earth rotates once per day about its axis, which is perpendicular to the plane of the equator and passes through the north geographic pole. Where on the earths surface should you stand in order to have the smallest possible tangential speed?

A building is located on the earths equator. As the earth rotates about its axis, which floor of the building has the greatest tangential speed? (a) The first floor (b) The tenth floor (c) The twentieth floor

The blade of a lawn mower is rotating at an angular speed of 17 rev/s. The tangential speed of the outer edge of the blade is 32 m/s. What is the radius of the blade?

Discus throwers often warm up by throwing the discus with a twisting motion of their bodies. Figure 8.13a illustrates a top view of such a warm-up throw. Starting from rest, the thrower accelerates the discus to a final angular velocity of 15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves on a circular arc of radius 0.810 m. Find the magnitude a of the total acceleration of the discus just before the discus is released.

A car is up on a hydraulic lift at a garage. The wheels are free to rotate, and the drive wheels are rotating with a constant angular velocity. Which one of the following statements is true? (a) A point on the rim has no tangential and no centripetal acceleration. (b) A point on the rim has both a nonzero tangential acceleration and a nonzero centripetal acceleration. (c) A point on the rim has a nonzero tangential acceleration but no centripetal acceleration. (d) A point on the rim has no tangential acceleration but does have a nonzero centripetal acceleration.

Section 5.6 discusses how the uniform circular motion of a space station can be used to create artificial gravity. This can be done by adjusting the angular speed of the space station, so the centripetal acceleration at an astronauts feet equals g, the magnitude of the acceleration due to the earths gravity (see Figure 5.18). If such an adjustment is made, will the acceleration at the astronauts head due to the artificial gravity be (a) greater than, (b) equal to, or (c) less than g?

A bicycle is turned upside down, the front wheel is spinning (see the drawing), and there is an angular acceleration. At the instant shown, there are six points on the wheel that have arrows associated with them. Which one of the following quantities could the arrows not represent? (a) Tangential velocity (b) Centripetal acceleration (c) Tangential acceleration

A rotating object starts from rest and has a constant angular acceleration. Three seconds later the centripetal acceleration of a point on the object has a magnitude of 2.0 m/s2 . What is the magnitude of the centripetal acceleration of this point six seconds after the motion begins?

An automobile, starting from rest, has a linear acceleration to the right whose magnitude is 0.800 m/s2 (see Figure 8.16). During the next 20.0 s, the tires roll and do not slip. The radius of each wheel is 0.330 m. At the end of this time, what is the angle through which each wheel has rotated?

The speedometer of a truck is set to read the linear speed of the truck, but uses a device that actually measures the angular speed of the rolling tires that came with the truck. However, the owner replaces the tires with larger-diameter versions. Does the reading on the speedometer after the replacement give a speed that is (a) less than, (b) equal to, or (c) greater than the true linear speed of the truck?

Rolling motion is an example that involves rotation about an axis that is not fixed. Give three other examples of rotational motion about an axis that is not fixed.

A rider on a mountain bike is traveling to the left in Figure 8.18. Each wheel has an angular velocity of 21.7 rad/s, where, as usual, the plus sign indicates that the wheel is rotating in the counterclockwise direction. (a) To pass another cyclist, the rider pumps harder, and the angular velocity of the wheels increases from 21.7 to 28.5 rad/s in a time of 3.50 s. (b) After passing the cyclist, the rider begins to coast, and the angular velocity of the wheels decreases from 28.5 to 15.3 rad/s in a time of 10.7 s. In both instances, determine the magnitude and direction of the angular acceleration (assumed constant) of the wheels.

Is the angular acceleration positive or negative when the rider is passing the other cyclist and the angular speed of the wheels is increasing?

Is the angular acceleration positive or negative when the rider is coasting and the angular speed of the wheels is decreasing?

Suppose you are driving a car in a counterclockwise direction on a circular road whose radius is r 390 m (see Figure 8.19). You look at the speedometer and it reads a steady 32 m/s (about 72 mi/h). (a) What is the angular speed of the car? (b) Determine the acceleration (magnitude and direction) of the car. (c) To avoid a rear-end collision with a vehicle ahead, you apply the brakes and reduce your angular speed to 4.9  102 rad/s in a time of 4.0 s. What is the tangential acceleration (magnitude and direction) of the car?

Does an object traveling at a constant tangential speed (for example, vT 32 m/s) along a circular path have an acceleration?

Is there a tangential acceleration when the angular speed of an object changes (e.g., when the cars angular speed decreases to 4.9 102 rad/s)?

The moon is 3.85 108 m from the earth and has a diameter of 3.48 106 m. You have a pea (diameter 0.50 cm) and a dime (diameter 1.8 cm). You close one eye and hold each object at arms length (71 cm) between your open eye and the moon. Which objects, if an any, completely cover your view of the moon? Assume that the moon and both objects are sufficiently far from your eye that the given diameters are equal to arc lengths when calculating angles. (a) Both (b) Neither (c) Pea (d) Dime

The radius of the circle traced out by the second hand on a clock is 6.00 cm. In a time t the tip of the second hand moves through an arc length of 24.0 cm. Determine the value of t in seconds

A rotating object has an angular acceleration of 0 rad/s2 . Which one or more of the following three statements is consistent with a zero angular acceleration? A. The angular velocity is 0 rad/s at all times. B. The angular velocity is 10 rad/s at all times. C. The angular displacement  has the same value at all times. (a) A, B, and C (b) A and B, but not C (c) A only (d) B only (e) C only Sect

A rotating wheel has a constant angular acceleration. It has an angular velocity of 5.0 rad/s at time t 0 s, and 3.0 s later has an angular velocity of 9.0 rad/s. What is the angular displacement of the wheel during the 3.0-s interval? (a) 15 rad (b) 21 rad (c) 27 rad (d) There is not enough information given to determine the angular displacement.

A rotating object starts from rest at t 0 s and has a constant angular acceleration. At a time of t 7.0 s the object has an angular velocity of 16 rad/s. What is its angular velocity at a time of t 14 s? Sect

A merry-go-round at a playground is a circular platform that is mounted parallel to the ground and can rotate about an axis that is perpendicular to the platform at its center. The angular speed of the merry-go-round is constant, and a child at a distance of 1.4 m from the axis has a tangential speed of 2.2 m/s. What is the tangential speed of another child, who is located at a distance of 2.1 m from the axis? (a) 1.5 m/s (b) 3.3 m/s (c) 2.2 m/s (d) 5.0 m/s (e) 0.98 m/s

A small fan has blades that have a radius of 0.0600 m. When the fan is turned on, the tips of the blades have a tangential acceleration of 22.0 m/s2 as the fan comes up to speed. What is the angular acceleration of the blades?

A wheel rotates with a constant angular speed . Which one of the following is true concerning the angular acceleration of the wheel, the tangential acceleration a T of a point on the rim of the wheel, and the centripetal acceleration ac of a point on the rim? (a) 0 rad/s2 , a T 0 m/s2 , and ac 0 m/s2 (b) 0 rad/s2 , a T 0 m/s2 , and ac 0 m/s2 (c) 0 rad/s2 , a T 0 m/s2 , and ac 0 m/s2 (d) 0 rad/s2 , a T 0 m/s2 , and ac 0 m/s2 (e) 0 rad/s2 , a T 0 m/s2 , and ac 0 m/s2 14. A platform

A platform is rotating with an angular speed of 3.00 rad/s and an angular acceleration of 11.0 rad/s2 . At a point on the platform that is 1.25 m from the axis of rotation, what is the magnitude of the total acceleration a?

The radius of each wheel on a bicycle is 0.400 m. The bicycle travels a distance of 3.0 km. Assuming that the wheels do not slip, how many revolutions does each wheel make? (a) 1.2 103 revolutions (b) 2.4 102 revolutions (c) 6.0 103 revolutions (d) 8.4 104 revolutions (e) Since the time of travel is not given, there is not enough information for a solution.

A pitcher throws a curveball that reaches the catcher in 0.60 s. The ball curves because it is spinning at an average angular velocity of 330 rev/min (assumed constant) on its way to the catchers mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?

The table that follows lists four pairs of initial and final angles of a wheel on a moving car. The elapsed time for each pair of angles is 2.0 s. For each of the four pairs, determine the average angular velocity (magnitude and direction as given by the algebraic sign of your answer).

The earth spins on its axis once a day and orbits the sun once a year . Determine the average angular velocity (in rad/s) of the earth as it (a) spins on its axis and (b) orbits the sun. In each case, take the positive direction for the angular displacement to be the direction of the earths motion.

Our sun rotates in a circular orbit about the center of the Milky Way galaxy. The radius of the orbit is 2.2 1020 m, and the angular speed of the sun is 1.1 1015 rad/s. How long (in years) does it take for the sun to make one revolution around the center?

In Europe, surveyors often measure angles in grads. There are 100 grads in one-quarter of a circle. How many grads are in one radian?

The initial angular velocity and the angular acceleration of four rotating objects at the same instant in time are listed in the table that follows. For each of the objects (a), (b), (c), and (d), determine the final angular speed after an elapsed time of 2.0 s.

The table that follows lists four pairs of initial and final angular velocities for a rotating fan blade. The elapsed time for each of the four pairs of angular velocities is 4.0 s. For each of the four pairs, find the average angular acceleration (magnitude and direction as given by the algebraic sign of your answer). Initial angular Final angular velocity 0 velocity (a) 2.0 rad/s 5.0 rad/s (b) 5.0 rad/s 2.0 rad/s (c) 7.0 rad/s 3.0 rad/s (d) 4.0 rad/s 4.0 rad/s 8. Con

Conceptual Example 2 provides some relevant background for this problem. A jet is circling an airport control tower at a distance of 18.0 km. An observer in the tower watches the jet cross in front of the moon. As seen from the tower, the moon subtends an angle of 9.04 103 radians. Find the distance traveled (in meters) by the jet as the observer watches the nose of the jet cross from one side of the moon to the other.

A Ferris wheel rotates at an angular velocity of 0.24 rad/s. Starting from rest, it reaches its operating speed with an average angular acceleration of 0.030 rad/s2 . How long does it take the wheel to come up to operating speed?

A floor polisher has a rotating disk that has a 15-cm radius. The disk rotates at a constant angular velocity of 1.4 rev/s and is covered with a soft material that does the polishing. An operator holds the polisher in one place for 45 s, in order to buff an especially scuffed area of the floor. How far (in meters) does a spot on the outer edge of the disk move during this time?

The sun appears to move across the sky, because the earth spins on its axis. To a person standing on the earth, the sun subtends an angle of sun 9.28 103 rad (see Conceptual Example 2). How much time (in seconds) does it take for the sun to move a distance equal to its own diameter?

A propeller is rotating about an axis perpendicular to its center, as the drawing shows. The axis is parallel to the ground. An arrow is fired at the propeller, travels parallel to the axis, and passes through one of the open spaces between the propeller blades. The angular open spaces between the three propeller blades are each /3 rad (60.0). The vertical drop of the arrow may be ignored. There is a maximum value for the angular speed of the propeller, beyond which the arrow cannot pass through an open space without being struck by one of the blades. Find this maximum value when the arrow has the lengths L and speeds v shown in the following table.

Two people start at the same place and walk around a circular lake in opposite directions. One walks with an angular speed of 1.7 103 rad/s, while the other has an angular speed of 3.4 103 rad/s. How long will it be before they meet?

A space station consists of two donut-shaped living chambers, A and B, that have the radii shown in the drawing. As the station rotates, an astronaut in chamber A is moved 2.40 102 m along a circular arc. How far along a circular arc is an astronaut in chamber B moved during the same time?

The drawing shows a device that can be used to measure the speed of a bullet. The device consists of two rotating disks, separated by a distance of d 0.850 m, and rotating with an angular speed of 95.0 rad/s. The bullet first passes through the left disk and then through the right disk. It is found that the angular displacement between the two bullet holes is  0.240 rad. From these data, determine the speed of the bullet. *

An automatic dryer spins wet clothes at an angular speed of 5.2 rad/s. Starting from rest, the dryer reaches its operating speed with an average angular acceleration of 4.0 rad/s2 . How long does it take the dryer to come up to speed?

A stroboscope is a light that flashes on and off at a constant rate. It can be used to illuminate a rotating object, and if the flashing rate is adjusted properly, the object can be made to appear stationary. (a) What is the shortest time between flashes of light that will make a three-bladed propeller appear stationary when it is rotating with an angular speed of 16.7 rev/s? (b) What is the next shortest time?

Review Conceptual Example 2 before attempting to work this problem. The moon has a diameter of 3.48 106 m and is a distance of 3.85 108 m from the earth. The sun has a diameter of 1.39 109 m and is 1.50 1011 m from the earth. (a) Determine (in radians) the angles subtended by the moon and the sun, as measured by a person standing on the earth. (b) Based on your answers to part (a), decide whether a total eclipse of the sun is really total. Give your reasoning. (c) Determine the ratio, expressed as a percentage, of the apparent circular area of the moon to the apparent circular area of the sun.

The drawing shows a golf ball passing through a windmill at a miniature golf course. The windmill has 8 blades and rotates at an angular speed of 1.25 rad/s. The opening between successive blades is equal to the width of a blade. A golf ball (diameter 4.50 102 m) has just reached the edge of one of the rotating blades (see the drawing). Ignoring the thickness of the blades, find the minimum linear speed with which the ball moves along the ground, such that the ball will not be hit by the next blade.

A figure skater is spinning with an angular velocity of 15 rad/s. She then comes to a stop over a brief period of time. During this time, her angular displacement is 5.1 rad. Determine (a) her average angular acceleration and (b) the time during which she comes to rest.

A gymnast is performing a floor routine. In a tumbling run she spins through the air, increasing her angular velocity from 3.00 to 5.00 rev/s while rotating through one-half of a revolution. How much time does this maneuver take?

The angular speed of the rotor in a centrifuge increases from 420 to 1420 rad/s in a time of 5.00 s. (a) Obtain the angle through which the rotor turns. (b) What is the magnitude of the angular acceleration?

A wind turbine is initially spinning at a constant angular speed. As the winds strength gradually increases, the turbine experiences a constant angular acceleration of 0.140 rad/s2 . After making 2870 revolutions, its angular speed is 137 rad/s. (a) What is the initial angular velocity of the turbine? (b) How much time elapses while the turbine is speeding up?

A car is traveling along a road, and its engine is turning over with an angular velocity of 220 rad/s. The driver steps on the accelerator, and in a time of 10.0 s the angular velocity increases to 280 rad/s. (a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of 220 rad/s during the entire 10.0-s interval? (b) What would have been the angular displacement if the angular velocity had been equal to its final value of 280 rad/s during the entire 10.0-s interval? (c) Determine the actual value of the angular displacement during the 10.0-s interval. 25.

The wheels of a bicycle have an angular velocity of 20.0 rad/s. Then, the brakes are applied. In coming to rest, each wheel makes an angular displacement of 15.92 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the angular acceleration (in rad/s2 ) of each wheel? *

A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.05 104 rad/s to an angular speed of 3.14 104 rad/s. In the process, the bit turns through 1.88 104 rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of 7.85 104 rad/s, starting from rest?

A motorcyclist is traveling along a road and accelerates for 4.50 s to pass another cyclist. The angular acceleration of each wheel is 6.70 rad/s2 , and, just after passing, the angular velocity of each wheel is 74.5 rad/s, where the plus signs indicate counterclockwise directions. What is the angular displacement of each wheel during this time? *

A top is a toy that is made to spin on its pointed end by pulling on a string wrapped around the body of the top. The string has a length of 64 cm and is wound around the top at a spot where its radius is 2.0 cm. The thickness of the string is negligible. The top is initially at rest. Someone pulls the free end of the string, thereby unwinding it and giving the top an angular acceleration of 12 rad/s2 . What is the final angular velocity of the top when the string is completely unwound?

The drive propeller of a ship starts from rest and accelerates at 2.90 103 rad/s2 for 2.10 103 s. For the next 1.40 103 s the propeller rotates at a constant angular speed. Then it decelerates at 2.30 103 rad/s2 until it slows (without reversing direction) to an angular speed of 4.00 rad/s. Find the total angular displacement of the propeller.

The drawing shows a graph of the angular velocity of a rotating wheel as a function of time. Although not shown in the graph, the angular velocity continues to increase at the same rate until t 8.0 s. What is the angular displacement of the wheel from 0 to 8.0 s?

At the local swimming hole, a favorite trick is to run horizontally off a cliff that is 8.3 m above the water. One diver runs off the edge of the cliff, tucks into a ball, and rotates on the way down with an average angular speed of 1.6 rev/s. Ignore air resistance and determine the number of revolutions she makes while on the way down.

A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of 4.00 rad/s2 . Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of 25.0 rad/s. While this change occurs, the angular displacement of the wheel is zero. (Note the similarity to that of a ball being thrown vertically upward, coming to a momentary halt, and then falling downward to its initial position.) Find the time required for the change in the angular velocity to occur.

A child, hunting for his favorite wooden horse, is running on the ground around the edge of a stationary merry-go-round. The angular speed of the child has a constant value of 0.250 rad/s. At the instant the child spots the horse, one-quarter of a turn away, the merry-go-round begins to move (in the direction the child is running) with a constant angular acceleration of 0.0100 rad/s2 . What is the shortest time it takes for the child to catch up with the horse?

A fan blade is rotating with a constant angular acceleration of 12.0 rad/s2 . At what point on the blade, as measured from the axis of rotation, does the magnitude of the tangential acceleration equal that of the acceleration due to gravity? 3

Some bacteria are propelled by biological motors that spin hairlike flagella. A typical bacterial motor turning at a constant angular velocity has a radius of 1.5 108 m, and a tangential speed at the rim of 2.3 105 m/s. (a) What is the angular speed (the magnitude of the angular velocity) of this bacterial motor? (b) How long does it take the motor to make one revolution?

An auto race takes place on a circular track. A car completes one lap in a time of 18.9 s, with an average tangential speed of 42.6 m/s. Find (a) the average angular speed and (b) the radius of the track.

A string trimmer is a tool for cutting grass and weeds; it utilizes a length of nylon string that rotates about an axis perpendicular to one end of the string. The string rotates at an angular speed of 47 rev/s, and its tip has a tangential speed of 54 m/s. What is the length of the rotating string?

In 9.5 s a fisherman winds 2.6 m of fishing line onto a reel whose radius is 3.0 cm (assumed to be constant as an approximation). The line is being reeled in at a constant speed. Determine the angular speed of the reel

he take-up reel of a cassette tape has an average radius of 1.4 cm. Find the length of tape (in meters) that passes around the reel in 13 s when the reel rotates at an average angular speed of 3.4 rad/s

The earth has a radius of 6.38  106 m and turns on its axis once every 23.9 h. (a) What is the tangential speed (in m/s) of a person living in Ecuador, a country that lies on the equator? (b) At what latitude (i.e., the angle in the drawing) is the tangential speed one-third that of a person living in Ecuador?

A baseball pitcher throws a baseball horizontally at a linear speed of 42.5 m/s (about 95 mi/h). Before being caught, the baseball travels a horizontal distance of 16.5 m and rotates through an angle of 49.0 rad. The baseball has a radius of 3.67 cm and is rotating about an axis as it travels, much like the earth does. What is the tangential speed of a point on the equator of the baseball?

A person lowers a bucket into a well by turning the hand crank, as the drawing illustrates. The crank handle moves with a constant tangential speed of 1.20 m/s on its circular path. The rope holding the bucket unwinds without slipping on the barrel of the crank. Find the linear speed with which the bucket moves down the well.

A thin rod (length 1.50 m) is oriented vertically, with its bottom end attached to the floor by means of a frictionless hinge. The mass of the rod may be ignored, compared to the mass of an object fixed to the top of the rod. The rod, starting from rest, tips over and rotates downward. (a) What is the angular speed of the rod just before it strikes the floor? (Hint: Consider using the principle of conservation of mechanical energy.) (b) What is the magnitude of the angular acceleration of the rod just before it strikes the floor?

One type of slingshot can be made from a length of rope and a leather pocket for holding the stone. The stone can be thrown by whirling it rapidly in a horizontal circle and releasing it at the right moment. Such a slingshot is used to throw a stone from the edge of a cliff, the point of release being 20.0 m above the base of the cliff. The stone lands on the ground below the cliff at a point X. The horizontal distance of point X from the base of the cliff (directly beneath the point of release) is thirty times the radius of the circle on which the stone is whirled. Determine the angular speed of the stone at the moment of release.

A racing car travels with a constant tangential speed of 75.0 m/s around a circular track of radius 625 m. Find (a) the magnitude of the cars total acceleration and (b) the direction of its total acceleration relative to the radial direction.

Two Formula One racing cars are negotiating a circular turn, and they have the same centripetal acceleration. However, the path of car A has a radius of 48 m, while that of car B is 36 m. Determine the ratio of the angular speed of car A to the angular speed of car B.

The earth orbits the sun once a year (3.16  107 s) in a nearly circular orbit of radius 1.50  1011 m. With respect to the sun, determine (a) the angular speed of the earth, (b) the tangential speed of the earth, and (c) the magnitude and direction of the earths centripetal acceleration.

Review Multiple-Concept Example 7 in this chapter as an aid in solving this problem. In a fast-pitch softball game the pitcher is impressive to watch, as she delivers a pitch by rapidly whirling her arm around so that the ball in her hand moves on a circle. In one instance, the radius of the circle is 0.670 m. At one point on this circle, the ball has an angular acceleration of 64.0 rad/s2 and an angular speed of 16.0 rad/s. (a) Find the magnitude of the total acceleration (centripetal plus tangential) of the ball. (b) Determine the angle of the total acceleration relative to the radial direction.

A rectangular plate is rotating with a constant angular speed about an axis that passes perpendicularly through one corner, as the drawing shows. The centripetal acceleration measured at corner A is n times as great as that measured at corner B. What is the ratio L1/L2 of the lengths of the sides of the rectangle when n 2.00?

Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.380 m and are rotating about a fixed axis with an angular velocity of 1.50 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of 2.00 rad/s2 . After 0.500 s has elapsed since the switch was reset, what is (a) the total acceleration (in m/s2 ) of a point on the tip of a blade and (b) the angle  between the total acceleration and the centripetal acceleration ? (See Figure 8.12b.)

The sun has a mass of 1.99  1030 kg and is moving in a circular orbit about the center of our galaxy, the Milky Way. The radius of the orbit is 2.3  104 light-years (1 light-year 9.5  1015 m), and the angular speed of the sun is 1.1  1015 rad/s. (a) Determine the tangential speed of the sun. (b) What is the magnitude of the net force that acts on the sun to keep it moving around the center of the Milky Way? *

An electric drill starts from rest and rotates with a constant angular acceleration. After the drill has rotated through a certain angle, the magnitude of the centripetal acceleration of a point on the drill is twice the magnitude of the tangential acceleration. What is the angle?

A motorcycle accelerates uniformly from rest and reaches a linear speed of 22.0 m/s in a time of 9.00 s. The radius of each tire is 0.280 m. What is the magnitude of the angular acceleration of each tire?

A car is traveling with a speed of 20.0 m/s along a straight horizontal road. The wheels have a radius of 0.300 m. If the car speeds up with a linear acceleration of 1.50 m/s2 for 8.00 s, find the angular displacement of each wheel during this period.

Suppose you are riding a stationary exercise bicycle, and the electronic meter indicates that the wheel is rotating at 9.1 rad/s. The wheel has a radius of 0.45 m. If you ride the bike for 35 min, how far would you have gone if the bike could move?

Multiple-Concept Example 8 provides useful background for part b of this problem. A motorcycle, which has an initial linear speed of 6.6 m/s, decelerates to a speed of 2.1 m/s in 5.0 s. Each wheel has a radius of 0.65 m and is rotating in a counterclockwise (positive) direction. What are (a) the constant angular acceleration (in rad/s2 ) and (b) the angular displacement (in rad) of each wheel?

A dragster starts from rest and accelerates down a track. Each tire has a radius of 0.320 m and rolls without slipping. At a distance of 384 m, the angular speed of the wheels is 288 rad/s. Determine (a) the linear speed of the dragster and (b) the magnitude of the angular acceleration of its wheels.

Over the course of a multi-stage 4520-km bicycle race, the front wheel of an athletes bicycle makes 2.18 106 revolutions. How many revolutions would the wheel have made during the race if its radius had been 1.2 cm larger?

A bicycle is rolling down a circular portion of a path; this portion of the path has a radius of 9.00 m. As the drawing illustrates, the angular displacement of the bicycle is 0.960 rad. What is the angle (in radians) through which each bicycle wheel (radius 0.400 m) rotates?

The penny-farthing is a bicycle that was popular between 1870 and 1890. As the drawing shows, this type of bicycle has a large front wheel and a small rear wheel. During a ride, the front wheel (radius 1.20 m) makes 276 revolutions. How many revolutions does the rear wheel (radius 0.340 m) make? *

A ball of radius 0.200 m rolls with a constant linear speed of 3.60 m/s along a horizontal table. The ball rolls off the edge and falls a vertical distance of 2.10 m before hitting the floor. What is the angular displacement of the ball while the ball is in the air?

The differential gear of a car axle allows the wheel on the left side of a car to rotate at a different angular speed than the wheel on the right side. A car is driving at a constant speed around a circular track on level ground, completing each lap in 19.5 s. The distance between the tires on the left and right sides of the car is 1.60 m, and the radius of each wheel is 0.350 m. What is the difference between the angular speeds of the wheels on the left and right sides of the car?

The trap-jaw ant can snap its mandibles shut in as little as 1.3 104 s. In order to shut, each mandible rotates through a 90 angle. What is the average angular velocity of one of the mandibles of the trap-jaw ant when the mandibles snap shut?

A 220-kg speedboat is negotiating a circular turn (radius 32 m) around a buoy. During the turn, the engine causes a net tangential force of magnitude 550 N to be applied to the boat. The initial tangential speed of the boat going into the turn is 5.0 m/s. (a) Find the tangential acceleration. (b) After the boat is 2.0 s into the turn, find the centripetal acceleration.

A flywheel has a constant angular deceleration of 2.0 rad/s2 . (a) Find the angle through which the flywheel turns as it comes to rest from an angular speed of 220 rad/s. (b) Find the time for the flywheel to come to rest.

An electric fan is running on HIGH. After the LOW button is pressed, the angular speed of the fan decreases to 83.8 rad/s in 1.75 s. The deceleration is 42.0 rad/s2 . Determine the initial angular speed of the fan.

Refer to Multiple-Concept Example 7 for insight into this problem. During a tennis serve, a racket is given an angular acceleration of magnitude 160 rad/s2 . At the top of the serve, the racket has an angular speed of 14 rad/s. If the distance between the top of the racket and the shoulder is 1.5 m, find the magnitude of the total acceleration of the top of the racket.

The drawing shows a chain-saw blade. The rotating sprocket tip at the end of the guide bar has a radius of 4.0 102 m. The linear speed of a chain link at point A is 5.6 m/s. Find the angular speed of the sprocket tip in rev/s.

In a large centrifuge used for training pilots and astronauts, a small chamber is fixed at the end of a rigid arm that rotates in a horizontal circle. A trainee riding in the chamber of a centrifuge rotating with a constant angular speed of 2.5 rad/s experiences a centripetal acceleration of 3.2 times the acceleration due to gravity. In a second training exercise, the centrifuge speeds up from rest with a constant angular acceleration. When the centrifuge reaches an angular speed of 2.5 rad/s, the trainee experiences a total acceleration equal to 4.8 times the acceleration due to gravity. (a) How long is the arm of the centrifuge? (b) What is the angular acceleration of the centrifuge in the second training exercise?

A compact disc (CD) contains music on a spiral track. Music is put onto a CD with the assumption that, during playback, the music will be detected at a constant tangential speed at any point. Since v T r, a CD rotates at a smaller angular speed for music near the outer edge and a larger angular speed for music near the inner part of the disc. For music A at the outer edge (r 0.0568 m), the angular speed is 3.50 rev/s. Find (a) the constant tangential speed at which music is detected and (b) the angular speed (in rev/s) for music at a distance of 0.0249 m from the center of a CD.

After 10.0 s, a spinning roulette wheel at a casino has slowed down to an angular velocity of 1.88 rad/s. During this time, the wheel has an angular acceleration of 5.04 rad/s2 . Determine the angular displacement of the wheel.

At a county fair there is a betting game that involves a spinning wheel. As the drawing shows, the wheel is set into rotational motion with the beginning of the angular section labeled 1 at the marker at the top of the wheel. The wheel then decelerates and eventually comes to a halt on one of the numbered sections. The wheel in the drawing is divided into twelve sections, each of which is an angle of 30.0. Determine the numbered section on which the wheel comes to a halt when the deceleration of the wheel has a magnitude of 0.200 rev/s2 and the initial angular velocity is (a) 1.20 rev/s and (b) 1.47 rev/s.

A racing car, starting from rest, travels around a circular turn of radius 23.5 m. At a certain instant, the car is still accelerating, and its angular speed is 0.571 rad/s. At this time, the total acceleration (centripetal plus tangential) makes an angle of 35.0 with respect to the radius. (The situation is similar to that in Figure 8.12b.) What is the magnitude of the total acceleration?

A quarterback throws a pass that is a perfect spiral. In other words, the football does not wobble, but spins smoothly about an axis passing through each end of the ball. Suppose the ball spins at 7.7 rev/s. In addition, the ball is thrown with a linear speed of 19 m/s at an angle of 55 with respect to the ground. If the ball is caught at the same height at which it left the quarterbacks hand, how many revolutions has the ball made while in the air?

Take two quarters and lay them on a table. Press down on one quarter so it cannot move. Then, starting at the 12:00 position, roll the other quarter along the edge of the stationary quarter, as the drawing suggests. How many revolutions does the rolling quarter make when it travels once around the circumference of the stationary quarter? Surprisingly, the answer is not one revolution.

An automobile tire has a radius of 0.330 m, and its center moves forward with a linear speed of v 15.0 m/s. (a) Determine the angular speed of the wheel. (b) Relative to the axle, what is the tangential speed of a point located 0.175 m from the axle?