Convert (a) $47.0^{\circ}$ to radians, (b) 12.0 rad to revolutions, and (c) 75.0 $\mathrm{rpm}$ to rad/s.

Salamat A.

Numerade Educator

A bicycle tire is spinning clockwise at 2.50 $\mathrm{rad} / \mathrm{s}$ . During a time

period $\Delta t=1.25 \mathrm{s}$ , the tire is stopped and spun in the opposite (counterclockwise) direction, also at 2.50 $\mathrm{rad} / \mathrm{s}$ . Calculate

(a) the change in the tire's angular velocity $\Delta \omega$ and $(\mathrm{b})$ the

tire's average angular acceleration $\alpha_{\mathrm{av}}$ .

Averell H.

Carnegie Mellon University

The tires on a new compact car have a diameter of 2.0 ft and are warranted for 60 000 miles. (a) Determine the angle (in radians) through which one of these tires will rotate during the warranty period. (b) How many revolutions of the tire are equivalent to your answer in part (a)?

Salamat A.

Numerade Educator

A potter’s wheel moves uniformly from rest to an angular velocity of 1.00 $\mathrm{rev} / \mathrm{s}$ in 30.0 $\mathrm{s} .$ (a) Find its angular acceleration in radians per second per second. (b) Would doubling the angular acceleration during the given period have doubled final angular velocity?

Averell H.

Carnegie Mellon University

A dentist's drill starts from rest. After 3.20 $\mathrm{s}$ of constant angular acceleration, it turns at a rate of $2.51 \times 10^{4} \mathrm{rev} / \mathrm{min}$ . (a) Find the drill's angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.

Salamat A.

Numerade Educator

A centrifuge in a medical laboratory rotates at an angular velocity of 3600 rev/min. When switched off, it rotates through 50.0 revolutions before coming to rest. Find the constant angular acceleration (in rad/s') of the centrifuge.

Averell H.

Carnegie Mellon University

A bicyclist starting at rest produces a constant angular acceleration of 1.60 $\mathrm{rad} / \mathrm{s}^{2}$ for wheels that are 38.0 $\mathrm{cm}$ in radius.

(a) What is the bicycle's linear acceleration? (b) What is the angular speed of the wheels when the bicyclist reaches 11.0 $\mathrm{m} / \mathrm{s} ?$ (c) How many radians have the wheels turned

through in that time? (d) How far has the bicycle traveled?

Salamat A.

Numerade Educator

A bicycle is turned upside down while its owner repairs a flat tire. A friend spins the other wheel

and observes that drops of water fly off tangentially. She measures the heights reached by drops moving vertically (Fig. P7.8). A drop that breaks loose from the tire on one turn rises vertically 54.0 cm above the tangent point. A drop that breaks loose on the next turn rises 51.0 cm above the tangent

point. The radius of the wheel is 0.381 m. (a) Why does the first drop rise higher than the second drop? (b) Neglecting air friction and using only the observed heights and the radius of the wheel, find the wheel’s angular acceleration (assuming it to be constant).

Averell H.

Carnegie Mellon University

The diameters of the main rotor and tail rotor of a single engine helicopter are 7.60 m and 1.02 m, respectively. The respective rotational speeds are 450 rev/min and 4 138 rev/ min. Calculate the speeds of the tips of both rotors. Compare these speeds with the speed of sound, 343 m/s.

Salamat A.

Numerade Educator

The tub of a washer goes into its spin-dry cycle, starting from rest and reaching an angular speed of 5.0 rev/s in 8.0 s. At this point, the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub slows to rest in 12.0 s. Through how many revolutions does the tub turn during the entire 20-s interval? Assume constant angular acceleration while it is starting and stopping.

Averell H.

Carnegie Mellon University

A car initially traveling at 29.0 $\mathrm{m} / \mathrm{s}$ undergoes a constant negative acceleration of magnitude 1.75 $\mathrm{m} / \mathrm{s}^{2}$ after its brakes are applied. (a) How many revolutions does each tire make before the car comes to a stop, assuming the car does not skid and

the tires have radii of 0.330 m? (b) What is the angular speed of the wheels when the car has traveled half the total distance?

Salamat A.

Numerade Educator

A 45.0 -cm diameter disk rotates with a constant angular acceleration of 2.50 $\mathrm{rad} / \mathrm{s}^{2} .$ It starts from rest at $t=0,$ and a line drawn from the center of the disk to a point $P$ on the rim of the disk makes an angle of $57.3^{\circ}$ with the positive $x$ -axis at this time.

At $t=2.30$ s, find (a) the angular speed of the wheel, (b) the linear speed and tangential acceleration of $P,$ and $(c)$ the position of $P$ (in degrees, with respect to the positive $x$ -axis).

Averell H.

Carnegie Mellon University

A rotating wheel requires 3.00 s to rotate 37.0 revolutions.Its angular velocity at the end of the 3.00- s interval is 98.0 rad/s. What is the constant angular acceleration (in rad/s') of the wheel?

Salamat A.

Numerade Educator

An electric motor rotating a workshop grinding wheel at a rate of $1.00 \times 10^{2}$ rev/min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude 2.00 $\mathrm{rad} / \mathrm{s}^{2}$ . (a) How long does it take for the grinding wheel to stop? (b) Through how many radians has the wheel turned during the interval found in part (a)?

Guilherme B.

Numerade Educator

A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in Figure P7.15. The length of the arc $A B C$ is $235 \mathrm{m},$ and the car completes the turn in 36.0 $\mathrm{s}$ .

(a) Determine the car's speed. (b) What is the magnitude and direction of the acceleration when the car is at point $B$ ?

Salamat A.

Numerade Educator

It has been suggested that rotating cylinders about 10 mi long and 5.0 mi in diameter be placed in space and used as colonies. What angular speed must such a cylinder have so that the centripetal acceleration at its surface equals the free-fall acceleration on Earth?

Averell H.

Carnegie Mellon University

(a) What is the tangential acceleration of a bug on the rim of a 10.0-in.-diameter disk if the disk accelerates uniformly from rest to an angular velocity of 78.0 rev/min in 3.00 s? (b) When the disk is at its final speed, what is the tangential velocity of the bug? One second after the bug starts from rest, what are its (c) tangential acceleration, (d) centripetal acceleration, and (e) total acceleration?

Salamat A.

Numerade Educator

An adventurous archeologist (m 5 85.0 kg) tries to cross a river by swinging from a vine. The vine is 10.0 m long, and his speed at the bottom of the swing is 8.00 m/s . The archeologist doesn't know that the vine has a breaking strength of 1000 N . Does he make it across the river without falling in?

Averell H.

Carnegie Mellon University

One end of a cord is fixed and a small 0.500-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 2.00 m, as shown in Figure P7.19. When $\theta=20.0^{\circ},$ the speed of the object is 8.00 $\mathrm{m} / \mathrm{s}$ . At this instant, find (a) the tension in the string, (b) the tangential and radial components of acceleration, and (c) the total acceleration. (d) Is your answer changed if the object is swinging down toward its lowest point instead of swinging up? (e) Explain your answer to part (d).

Salamat A.

Numerade Educator

Human centrifuges are used to train military pilots and astronauts in preparation for high-g maneuvers. A trained, fit person wearing a g - suit can withstand accelerations up to about 9$g\left(88.2 \mathrm{m} / \mathrm{s}^{2}\right)$ without losing consciousness. (a) If a human centrifuge has a radius of 4.50 m, what angular speed results in a centripetal acceleration of 9g? (b) What linear speed would a person in the centrifuge have at this acceleration?

Averell H.

Carnegie Mellon University

A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius 0.800 m around the pole. (a) Determine the force exerted by the horizontal rope on her arms. (b) Compare this force with her weight.

Salamat A.

Numerade Educator

A 40.0-kg child swings in a swing supported by two chains,

each 3.00 m long. The tension in each chain at the lowest

point is 350 N. Find (a) the child’s speed at the lowest point

and (b) the force exerted by the seat on the child at the lowest

point. (Ignore the mass of the seat.)

Averell H.

Carnegie Mellon University

A certain light truck can go around a flat curve having a radius of 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve having a radius of 75.0 m?

Salamat A.

Numerade Educator

A sample of blood is placed in a centrifuge of radius 15.0 $\mathrm{cm} .$ The mass of a red blood cell is $3.0 \times 10^{-16} \mathrm{kg},$ and the magnitude of the force acting on it as it settles out of the plasma is $4.0 \times 10^{-11} \mathrm{N}$ . At how many revolutions per second should the centrifuge be operated?

Averell H.

Carnegie Mellon University

A 50.0-kg child stands at the rim of a merry-go-round of radius 2.00 m, rotating with an angular speed of 3.00 rad/s.

(a) What is the magnitude of the child’s centripetal acceleration? (b) What is the magnitude of the minimum force between her feet and the floor of the carousel that is required to keep her in the circular path? (c) What minimum coefficient of static friction is required? Is the answer you found

reasonable? In other words, is she likely to stay on the merry-go-round?

Salamat A.

Numerade Educator

A space habitat for a long space voyage consists of two cabins each connected by a cable to a central hub as shown in Figure P7.26. The cabins are set spinning around the hub axis, which is connected to the rest of the spacecraft to generate artificial gravity. (a) What forces are acting on an astronaut in one of the cabins? (b) Write Newton’s second law for an astronaut lying on the “floor” of one of the habitats, relating the astronaut’s mass m, his velocity v, his radial distance from the hub r, and the normal force n. (c) What would n have to equal if the 60.0 - kg astronaut is to experience half his normal Earth weight? (d) Calculate the necessary tangential speed of the habitat from Newton’s second law. (e) Calculate the angular speed from the tangential speed. (f) Calculate the period of rotation from the angular speed. (g) If the astronaut stands up, will his head be moving faster, slower, or at the same speed as his feet? Why? Calculate the tangential speed at the top of

his head if he is 1.80 m tall.

Averell H.

Carnegie Mellon University

An air puck of mass $m_{1}=0.25 \mathrm{kg}$ is tied to a string and allowed to revolve in a circle of radius $R=1.0 \mathrm{m}$ on a frictionless horizontal table. The other end of the string passes through a hole in the center of the table, and a mass of $m_{2}=1.0 \mathrm{kg}$ is tied to it (Fig. $\mathrm{P} 7.27 )$ . The suspended mass remains in equilibrium while the puck on the tabletop revolves. (a) What is the tension in the string? (b) What is the horizontal force acting on the puck? (c) What is the speed of the puck?

Salamat A.

Numerade Educator

A snowboarder drops from rest into a halfpipe of radius R and slides down its frictionless surface to the bottom (Fig. P7.28). Show that (a) the snowboarder’s speed at the bottom of the halfpipe is $v=\sqrt{2 g R}$ (Hint: Use conservation of energy), (b) the snowboarder's centripetal acceleration at the bottom is $a_{c}=2 g,$ and $(c)$ the normal force on the snow boarder at the bottom of the halfpipe has magnitude 3$m g$ (Hint: Use Newton's second law of motion).

Averell H.

Carnegie Mellon University

A woman places her briefcase on the backseat of her car. As she drives to work, the car negotiates an unbanked curve in the road that can be regarded as an arc of a circle of radius 62.0 m. While on the curve, the speed of the car is 15.0 m/s at the instant the briefcase starts to slide across the

backseat toward the side of the car. (a) What force causes the centripetal acceleration of the briefcase when it is stationary relative to the car? Under what condition does the briefcase begin to move relative to the car? (b) What is the coefficient of static friction between the briefcase and seat surface?

Salamat A.

Numerade Educator

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

Averell H.

Carnegie Mellon University

A 40.0-kg child takes a ride on a Ferris wheel that rotates four times each minute and has a diameter of 18.0 m. (a) What is the centripetal acceleration of the child? (b) What force (magnitude and direction) does the seat exert on the child at the lowest point of the ride? (c) What force does the seat exert on the child at the highest point of the ride? (d) What force does the seat exert on the child when the child is halfway between the top and bottom?

Salamat A.

Numerade Educator

A roller-coaster vehicle has a mass of 500 kg when fully loaded with passengers (Fig. P7.32). (a) If the vehicle has a speed of 20.0 m/s at point A, what is the force of the track on the vehicle at this point? (b) What is the maximum speed the vehicle can have at point B for gravity to hold it on the track?

Averell H.

Carnegie Mellon University

(a) Find the magnitude of the gravitational force between a planet with mass $7.50 \times 10^{24} \mathrm{kg}$ and its moon, with mass $2.70 \times 10^{22} \mathrm{kg},$ if the average distance between their centers is $2.80 \times 10^{8} \mathrm{m} .$ (b) What is the acceleration of the moon

towards the planet? (c) What is the acceleration of the planet towards the moon?

Salamat A.

Numerade Educator

The International Space Station has a mass of $4.19 \times 10^{5} \mathrm{kg}$ and orbits at a radius of $6.79 \times 10^{6} \mathrm{m}$ from the center of Earth. Find (a) the gravitational force exerted by Earth on the space station, (b) the space station's gravitational potential energy, and (c) the weight of an $80.0-\mathrm{kg}$ astronaut living inside the station.

Averell H.

Carnegie Mellon University

A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the table as follows: a 2.0-kg object at the origin of the coordinate system, a 3.0-kg object at (0, 2.0), and a 4.0-kg object at (4.0, 0). Find the resultant gravitational force exerted by the other two

objects on the object at the origin.

Salamat A.

Numerade Educator

After the Sun exhausts its nuclear fuel, its ultimate fate may be to collapse to a white dwarf state. In this state, it would have approximately the same mass as it has now, but its radius would be equal to the radius of Earth. Calculate (a) the average density of the white dwarf, (b) the surface free-fall acceleration, and (c) the gravitational potential energy associated with a 1.00-kg object at the surface of the white dwarf.

Averell H.

Carnegie Mellon University

Objects with masses of 200. kg and 500. kg are separated by 0.400 m. (a) Find the net gravitational force exerted by these objects on a 50.0-kg object placed midway between them. (b) At what position (other than infinitely remote ones) can the 50.0-kg object be placed so as to experience a net force of zero?

Salamat A.

Numerade Educator

Use the data of Table 7.3 to find the point between Earth and the Sun at which an object can be placed so that the net gravitational force exerted by Earth and the Sun on that object is zero.

Averell H.

Carnegie Mellon University

A projectile is fired straight upward from the Earth’s surface at the South Pole with an initial speed equal to one third the escape speed. (a) Ignoring air resistance, determine how far from the center of the Earth the projectile travels before stop ping momentarily. (b) What is the altitude of the projectile at this instant?

Salamat A.

Numerade Educator

Two objects attract each other with a gravitational force of magnitude $1.00 \times 10^{-8} \mathrm{N}$ when separated by 20.0 $\mathrm{cm} .$ If the total mass of the objects is $5.00 \mathrm{kg},$ what is the mass of each?

Averell H.

Carnegie Mellon University

A satellite is in a circular orbit around the Earth at an altitude of $2.80 \times 10^{6} \mathrm{m} .$ Find (a) the period of the orbit, (b) the speed of the satellite, and (c) the acceleration of the satellite.

Hint: Modify Equation 7.23 so it is suitable for objects orbiting the Earth rather than the Sun.

Salamat A.

Numerade Educator

An artificial satellite circling the Earth completes each orbit in 110 minutes. (a) Find the altitude of the satellite. (b) What is the value of g at the location of this satellite?

Averell H.

Carnegie Mellon University

A satellite of Mars, called Phoebus, has an orbital radius of $9.4 \times 10^{6} \mathrm{m}$ and a period of $2.8 \times 10^{4}$ s. Assuming the orbit is circular, determine the mass of Mars.

Salamat A.

Numerade Educator

A 600-kg satellite is in a circular orbit about Earth at a height above Earth equal to Earth’s mean radius. Find (a) the satellite’s orbital speed, (b) the period of its revolution, and (c) the gravitational force acting on it.

Averell H.

Carnegie Mellon University

A comet has a period of 76.3 years and moves in an elliptical orbit in which its perihelion (closest approach to the Sun) is 0.610 AU. Find (a) the semimajor axis of the comet and (b) an estimate of the comet’s maximum distance from the Sun, both in astronomical units.

Salamat A.

Numerade Educator

A synchronous satellite, which always remains above the same point on a planet’s equator, is put in circular orbit around Jupiter to study that planet’s famous red spot. Jupiter rotates once every 9.84 h. Use the data of Table 7.3 to find the altitude of the satellite.

Averell H.

Carnegie Mellon University

(a) One of the moons of Jupiter, named Io, has an orbital radius of $4.22 \times 10^{8} \mathrm{m}$ and a period of 1.77 days. Assuming the orbit is circular, calculate the mass of Jupiter. (b) The largest moon of Jupiter, named Ganymede, has an orbital radius of $1.07 \times 10^{9} \mathrm{m}$ and a period of 7.16 days. Calculate the mass of Jupiter from this data. (c) Are your results to parts (a) and

(b) consistent? Explain.

Salamat A.

Numerade Educator

Neutron stars are extremely dense objects that are formed from the remnants of supernova explosions. Many rotate very rapidly. Suppose the mass of a certain spherical neutron star is twice the mass of the Sun and its radius is 10.0 km. Determine the greatest possible angular speed the neutron star can have so that the matter at its surface on the equator is just held in orbit by the gravitational force.

Averell H.

Carnegie Mellon University

One method of pitching a softball is called the “windmill” delivery method, in which the pitcher’s arm rotates through approximately $360^{\circ}$ in a vertical plane before the 198 gram ball is released at the lowest point of the circular motion. An experienced pitcher can throw a ball with a speed of

98.0 mi/h. Assume the angular acceleration is uniform throughout the pitching motion and take the distance between the softball and the shoulder joint to be 74.2 cm. (a) Determine the angular speed of the arm in rev/s at the instant of release. (b) Find the value of the angular acceleration in $\mathrm{rev} / \mathrm{s}^{2}$ and the radial and tangential acceleration of the ball just before it is released. (c) Determine the force exerted on the ball by the pitcher’s hand (both radial and tangential components) just before it is released.

Salamat A.

Numerade Educator

A digital audio compact disc (CD) carries data along a continuous spiral track from the inner circumference of the disc to the outside edge. Each bit occupies 0.6 mm of the track. A CD player turns the disc to carry the track counterclockwise above a lens at a constant speed of 1.30 m/s. Find the required angular speed (a) at the beginning of the recording, where the spiral has a radius of 2.30 cm, and (b) at the end of the recording, where the spiral has a radius of 5.80 cm. (c) A full-length

recording lasts for 74 min, 33 s. Find the average angular acceleration of the disc. (d) Assuming the acceleration is constant, find the total angular displacement of the disc as it plays. (e) Find the total length of the track.

Averell H.

Carnegie Mellon University

An athlete swings a 5.00-kg ball horizontally on the end of a rope. The ball moves in a circle of radius 0.800 m at an angular speed of 0.500 rev/s. What are (a) the tangential speed of the ball and (b) its centripetal acceleration? (c) If the maximum tension the rope can withstand before breaking is 100. N,

what is the maximum tangential speed the ball can have?

Salamat A.

Numerade Educator

The dung beetle is known as one of the strongest animals for its size, often forming balls of dung up to 10 times their own mass and rolling them to locations where they can be buried and stored as food. A typical dung ball formed by the species K. nigroaeneus has a radius of 2.00 cm and is rolled by the beetle at 6.25 cm/s. (a) What is the rolling ball’s angular speed? (b) How many full rotations are required if the beetle rolls the ball a distance of 1.00 m?

Averell H.

Carnegie Mellon University

The Solar Maximum Mission Satellite was placed in a circular orbit about 150 mi above Earth. Determine (a) the orbital speed of the satellite and (b) the time required for one complete revolution.

Salamat A.

Numerade Educator

A 0.400-kg pendulum bob passes through the lowest part of

its path at a speed of 3.00 m/s. (a) What is the tension in

the pendulum cable at this point if the pendulum is 80.0 cm

long? (b) When the pendulum reaches its highest point, what

angle does the cable make with the vertical? (c) What is the

tension in the pendulum cable when the pendulum reaches

its highest point?

Averell H.

Carnegie Mellon University

A car moves at speed v across a bridge made in the shape of a circular arc of radius r. (a) Find an expression for the normal force acting on the car when it is at the top of the arc. (b) At what minimum speed will the normal force become zero (causing the occupants of the car to seem weightless) if

$r=30.0 \mathrm{m} ?$

Salamat A.

Numerade Educator

Keratinocytes are the most common cells in the skin’s outer layer. As these approximately circular cells migrate across a wound during the healing process, they roll in a way that reduces the frictional forces impeding their motion. (a) Given a cell body diameter of $1.00 \times 10^{-5} \mathrm{m}(10 \mu \mathrm{m})$ what minimum angular speed would be required to produce the observed linear speed of $1.67 \times 10^{-7} \mathrm{m} / \mathrm{s}(10 \mu \mathrm{m} / \mathrm{min}) ?$

(b) How many complete revolutions would be required for the cell to roll a distance of $5.00 \times 10^{-3} \mathrm{m}$ ? (Because of slipping as the cells roll, averages of observed angular speeds and the number of complete revolutions are about three times these minimum values.)

Averell H.

Carnegie Mellon University

Because of Earth’s rotation about its axis, a point on the equator has a centripetal acceleration of 0.0340 $\mathrm{m} / \mathrm{s}^{2}$ , whereas a point at the poles has no centripetal acceleration.

(a) Show that, at the equator, the gravitational force on an object (the object's true weight) must exceed the object's apparent weight. (b) What are the apparent weights of a 75.0 -kg person at the equator and at the poles? (Assume Earth is a uniform sphere and take $g=9.800 \mathrm{m} / \mathrm{s}^{2} . )$

Salamat A.

Numerade Educator

A roller coaster travels in a circular path. (a) Identify the forces on a passenger at the top of the circular loop that cause centripetal acceleration. Show the direction of all forces in a sketch. (b) Identify the forces on the passenger at the bot- tom of the loop that produce centripetal acceleration. Show these in a sketch. (c) Based on your answers to parts (a) and (b), at what point, top or bottom, should the seat exert the greatest force on the passenger? (d) Assume the speed of the roller coaster is 4.00 m/s at the top of the loop of radius 8.00 m. Find the force exerted by the seat on a 70.0-kg passenger at the top of the loop. Then, assume the speed remains the same at the bottom of the loop and find the force exerted by the seat on the passenger at this point. Are your answers consistent with your choice of answers for parts (a) and (b)?

Averell H.

Carnegie Mellon University

In Robert Heinlein’s The Moon Is a Harsh Mistress, the colonial inhabitants of the Moon threaten to launch rocks down onto Earth if they are not given independence (or at least representation). Assuming a gun could launch a rock of mass m at twice the lunar escape speed, calculate the speed of the rock as it enters Earth’s atmosphere.

Salamat A.

Numerade Educator

A model airplane of mass 0.750 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 60.0-m control wire as shown in Figure P7.60a. The forces exerted on the airplane are shown in Figure P7.60b; the tension in the control wire, $\theta=20.0^{\circ}$ inward from the vertical. Compute

the tension in the wire, assuming the wire makes a constant angle of $\theta=20.0^{\circ}$ with the horizontal.

Averell H.

Carnegie Mellon University

In a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis, as shown in Figure P7.61. So that the clothes will dry uniformly, they are made to tumble. The rate of rotation of the smooth-walled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of $\theta=68.0^{\circ}$ above the horizontal. If the radius of the tub is $r=0.330 \mathrm{m},$ what rate of revolution is needed in revolutions per second?

Salamat A.

Numerade Educator

Casting of molten metal is important in many industrial processes. Centrifugal casting is used for manufacturing pipes, bearings, and many other structures. A cylindrical enclosure is rotated rapidly and steadily about a horizontal axis, as in Figure P7.62. Molten metal is poured into the rotating cylinder and then cooled, forming the finished product. Turning the cylinder at a high rotation rate forces the solidifying metal strongly to the outside. Any bubbles are displaced toward the axis so that unwanted voids will not be present in the casting.

Suppose a copper sleeve of inner radius 2.10 cm and outer radius 2.20 cm is to be cast. To eliminate bubbles and give high structural integrity, the centripetal acceleration of each bit of metal should be 100g. What rate of rotation is required? State the answer in revolutions per minute.

Averell H.

Carnegie Mellon University

A skier starts at rest at the top of a large hemispherical hill (Fig. P7.63). Neglecting friction, show that the skier will leave the hill and become airborne at a distance $h=R / 3$ below the top of the hill. Hint: At this point, the normal force goes to zero.

Salamat A.

Numerade Educator

A stuntman whose mass is 70 kg swings from the end of a 4.0-m-long rope along the arc of a vertical circle. Assuming he starts from rest when the rope is horizontal, find the tensions in the rope that are required to make him follow his circular path (a) at the beginning of his motion, (b) at a height of 1.5 m above the bottom of the circular arc, and (c) at the bot- tom of the arc.

Averell H.

Carnegie Mellon University

Suppose a 1 800-kg car passes over a bump in a roadway that follows the arc of a circle of

radius 20.4 m, as in Figure P7.65. (a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at 8.94 m/s? (b) What is the maximum speed the car can have without losing contact with the road as it passes this highest point?

Salamat A.

Numerade Educator

The pilot of an airplane executes a constant-speed loop-the-loop maneuver in a vertical circle as in Figure 7.13b. The speed of the airplane is $2.00 \times 10^{2} \mathrm{m} / \mathrm{s}$ , and the radius of the circle is $3.20 \times 10^{3} \mathrm{m} .$ (a) What is the pilot's apparent weight

at the lowest point of the circle if his true weight is 712 $\mathrm{N} ?$ (b) What is his apparent weight at the highest point of the circle? (c) Describe how the pilot could experience weightlessness if both the radius and the speed can be varied. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body. Under what conditions does this occur? (d) What speed would have resulted in the pilot experiencing weightlessness at the top of the loop?

Vishal G.

Numerade Educator

A minimum-energy orbit to an outer planet consists of putting a spacecraft on an elliptical trajectory with the departure planet corresponding to the perihelion of the ellipse, or closest point to the Sun, and the arrival planet corresponding to the aphelion of the ellipse, or farthest point from the Sun.

(a) Use Kepler’s third law to calculate how long it would take to go from Earth to Mars on such an orbit. (Answer in years.)

(b) Can such an orbit be undertaken at any time? Explain.

Salamat A.

Numerade Educator

A coin rests 15.0 $\mathrm{cm}$ from the center of a turntable. The coefficient of static friction between the coin and turntable surface is $0.350 .$ The turntable starts from rest at $t=0$ and

rotates with a constant angular acceleration of 0.730 $\mathrm{rad} / \mathrm{s}^{2}$

(a) Once the turntable starts to rotate, what force causes the centripetal acceleration when the coin is stationary relative to the turntable? Under what condition does the coin begin to move relative to the turntable? (b) After what period of time will the coin start to slip on the turntable?

Keshav S.

Numerade Educator

A 4.00-kg object is attached to a vertical rod by two strings as shown in Figure P7.69. The object rotates in a horizontal circle at constant speed 6.00 m/s. Find the tension in (a) the upper string and (b) the lower string.

Salamat A.

Numerade Educator

A 0.275-kg object is swung in a vertical circular path on a string 0.850 m long as in Figure P7.70. (a) What are the forces acting on the ball at any point along this path? (b) Draw free-body dia-

grams for the ball when it is at the bottom of the circle and when it is at the top. (c) If its speed is 5.20 m/s at the top of the circle, what is the tension in the string there? (d) If the string breaks when its tension exceeds 22.5 N, what is the maximum speed the object can have at the bottom before the string breaks?

Averell H.

Carnegie Mellon University

(a) A luggage carousel at an airport has the form of a section of a large cone, steadily rotating about its vertical axis. Its metallic surface slopes downward toward the outside, making an angle of 20.0° with the horizontal. A 30.0-kg piece of luggage is placed on the carousel, 7.46 m from the axis of

rotation. The travel bag goes around once in 38.0 s. Calculate the force of static friction between the bag and the carousel.

(b) The drive motor is shifted to turn the carousel at a higher constant rate of rotation, and the piece of luggage is bumped to a position 7.94 m from the axis of rotation. The bag is on the verge of slipping as it goes around once every 34.0 s. Calculate the coefficient of static friction between the bag and

the carousel.

Salamat A.

Numerade Educator

The maximum lift force on a bat is proportional to the square of its flying speed v. For the hoary bat (Lasiurus cinereus), the magnitude of the lift force is given by

$$F_{L} \leq\left(0.018 \mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{m}^{2}\right) v^{2}$$

The bat can fly in a horizontal circle by "banking" its wings at an angle $\theta,$ as shown in Figure $\mathrm{P7} .72$ . In this situation, the magnitude of the vertical component of the lift force must

equal the bat's weight. The horizontal component of the force provides the centripetal acceleration. (a) What is the minimum speed that the bat can have if its mass is 0.031 kg?

(b) If the maximum speed of the bat is 10 m/s, what is the maximum banking angle that allows the bat to stay in a horizontal plane? (c) What is the radius of the circle of its flight when the bat flies at its maximum speed? (d) Can the batturn with a smaller radius by flying more slowly?

Averell H.

Carnegie Mellon University

In a popular amusement park ride, a rotating cylinder of radius 3.00 m is set in rotation at an angular speed of 5.00 rad/s, as in Figure P7.73. The floor then drops away, leaving the riders suspended

against the wall in a vertical position. What minimum coefficient of friction between a rider’s clothing and the wall is needed to keep the rider from slipping? Hint: Recall that the magnitude of the maximum force of static friction is equal to msn, where n is the normal force—in this case, the force causing the centripetal acceleration.

Salamat A.

Numerade Educator

A massless spring of constant $k=78.4 \mathrm{N} / \mathrm{m}$ is fixed on the left side of a level track. A block of mass $m=0.50 \mathrm{kg}$ is pressed against the spring and compresses it a distance $d$ , as in Figure $\mathrm{P} 7.74$ . The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius $R=1.5 \mathrm{m}$ . The entire track and the loop-the-loop are frictionless, except for the section of track between points $A$ and $B$ . Given that the coefficient of kinetic friction between the block and the track along $A B$ is $\mu_{k}=0.30$ and that the length of $A B$ is $2.5 \mathrm{m},$ determine the minimum compression $d$ of the spring that enables the block to just make it through the loop-the-loop at point $C$ . Hint: The force exerted by the track on the block will be zero if the block barely makes it through the loop-the-loop.

Averell H.

Carnegie Mellon University

A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane, with the cord making a 30° angle with the vertical. (See Fig. P7.75.) (a) Determine the ball’s speed. (b) If, instead, the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical? (c) If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at which the ball can move?

Salamat A.

Numerade Educator