Solved: A roller coaster travels in a circular path. (a)

Is it possible to express angular speed in degrees per second? If so, whats the conversion factor from radians per second?

Suppose the radius of the wheel is doubled. Are the answers affected? If so, in what way?

If the propeller had rotated through twice as many revolutions during the process, by what factor would the angular acceleration have changed?

If the angular acceleration were doubled for the same duration, by what factor would the angular displacement change? Why is the answer true in this case but not in general?

What is the angular acceleration of a record player while its playing a song? Can a CD player have the same angular acceleration as a record player? Explain.

What is the angular acceleration of a record player while its playing a song? Can a CD player have the same angular acceleration as a record player? Explain.

If the static friction coeffi cient were increased, would the maximum safe speed be reduced, be increased, or remain the same?

What three physical quantities determine a minimum safe speed on a banked racetrack?

Suppose the car subsequently goes over a rise with the same radius of curvature and at the same speed as part (a). What is the normal force in this case?

Is the gravity force a signifi cant factor in a game of billiards? Explain.

Give two reasons Equation (1) could not be used for every asteroid as it is used in part (a).

As the asteroid approaches Earth, does the gravitational potential energy associated with the asteroidEarth system (a) increase, (b) decrease, (c) remain the same?

Suppose the spacecraft managed to go into an elliptical orbit around Earth, with a nearest point (perigee) and farthest point (apogee). At which point is the kinetic energy of the spacecraft higher, and why?

If the satellite was placed in an orbit three times farther away, about how long would it take to orbit the Earth once? Answer in days, rounding to one digit.

Find the angular speed of Earth around the Sun in radians per second. (a) 2.22  106 rad/s (b) 1.16  107 rad/s (c) 3.17  108 rad/s (d) 4.52  107 rad/s (e) 1.99  107 rad/s

A grindstone increases in angular speed from 4.00 rad/s to 12.00 rad/s in 4.00 s. Through what angle does it turn during that time if the angular acceleration is constant? (a) 8.00 rad (b) 12.0 rad (c) 16.0 rad (d) 32.0 rad (e) 64.0 rad

A cyclist rides a bicycle with a wheel radius of 0.500 m across campus. A piece of plastic on the front rim makes a clicking sound every time it passes through the fork. If the cyclist counts 320 clicks between her apartment and the cafeteria, how far has she traveled? (a) 0.50 km (b) 0.80 km (c) 1.0 km (d) 1.5 km (e) 1.8 km

A 0.400-kg object attached to the end of a string of length 0.500 m is swung in a circular path and in a vertical plane. If a constant angular speed of 8.00 rad/s is maintained, what is the tension in the string when the object is at the top of the circular path? (a) 8.88 N (b) 10.5 N (c) 12.8 N (d) 19.6 N (e) None of these

A merry-go-round rotates with constant angular speed. As a rider moves from the rim of the merry-go-round toward the center, what happens to the magnitude of total centripetal force that must be exerted on him? (a) It increases. (b) It is not zero, but remains the same. (c) It decreases. (d) Its always zero. (e) It increases or decreases, depending on the direction of rotation.

Consider an object on a rotating disk a distance r from its center, held in place on the disk by static friction. Which of the following statements is not true concerning this object? (a) If the angular speed is constant, the object must have constant tangential speed. (b) If the angular speed is constant, the object is not accelerated. (c) The object has a tangential acceleration only if the disk has an angular acceleration. (d) If the disk has an angular acceleration, the object has both a centripetal and a tangential acceleration. (e) The object always has a centripetal acceleration except when the angular speed is zero.

The gravitational force exerted on an astronaut on Earths surface is 650 N down. When she is in the International Space Station, is the gravitational force on her (a) larger, (b) exactly the same, (c) smaller, (d) nearly but not exactly zero, or (e) exactly zero?

An object is located on the surface of a spherical planet of mass M and radius R. The escape speed from the planet does not depend on which of the following? (a) M (b) the density of the planet (c) R (d) the acceleration due to gravity on that planet (e) the mass of the object

A satellite moves in a circular orbit at a constant speed around Earth. Which of the following statements is true? (a) No force acts on the satellite. (b) The satellite moves at constant speed and hence doesnt accelerate. (c) The satellite has an acceleration directed away from Earth. (d) The satellite has an acceleration directed toward Earth. (e) Work is done on the satellite by the force of gravity.

Which of the following statements are true of an object in orbit around Earth? (a) If the orbit is circular, the gravity force is perpendicular to the objects velocity. (b) If the orbit is elliptical, the gravity force is perpendicular to the velocity vector only at the nearest and farthest points. (c) If the orbit is not circular, the speed is greatest when the object is farthest away from Earth. (d) The gravity force on the object always has components both parallel and perpendicular to the objects velocity. (e) All these statements are true.

What is the gravitational acceleration close to the surface of a planet with twice the mass and twice the radius of Earth? Answer as a multiple of g, the gravitational acceleration near Earths surface. (a) 0.25g (b) 0.5g (c) g (d) 2g (e) 4g

A system consists of four particles. How many terms appear in the expression for the total gravitational potential energy of the system? (a) 4 (b) 6 (c) 10 (d) 12 (e) None of these

Halleys comet has a period of approximately 76 years and moves in an elliptical orbit in which its distance from the Sun at closest approach is a small fraction of its maximum distance. Estimate the comets maximum distance from the Sun in astronomical units AU (the distance from Earth to the Sun). (a) 3 AU (b) 6 AU (c) 10 AU (d) 18 AU (e) 36 AU

In a race like the Indianapolis 500, a driver circles the track counterclockwise and feels his head pulled toward one shoulder. To relieve his neck muscles from having to hold his head erect, the driver fastens a strap to one wall of the car and the other to his helmet. The length of the strap is adjusted to keep his head vertical. (a) Which shoulder does his head tend to lean toward? (b) What force or forces produce the centripetal acceleration when there is no strap? (c) What force or forces do so when there is a strap?

Two schoolmates, Romeo and Juliet, catch each others eye across a crowded dance fl oor at a school dance. Find the order of magnitude of the gravitational attraction that Juliet exerts on Romeo and that Romeo exerts on Juliet. State the quantities you take as data and the values you measure or estimate for them.

If a cars wheels are replaced with wheels of greater diam eter, will the reading of the speedometer change? Explain.

At night, you are farther away from the Sun than during the day. Whats more, the force exerted by the Sun on you is downward into Earth at night and upward into the sky during the day. If you had a sensitive enough bathroom scale, would you appear to weigh more at night than during the day?

Correct the following statement: The race car rounds the turn at a constant velocity of 90 miles per hour.

Because of Earths rotation about its axis, you weigh slightly less at the equator than at the poles. Explain.

It has been suggested that rotating cylinders about 10 miles long and 5 miles in diameter be placed in space for colonies. The purpose of their rotation is to simulate gravity for the inhabitants. Explain the concept behind this proposal.

Describe the path of a moving object in the event that the objects acceleration is constant in magnitude at all times and (a) perpendicular to its velocity; (b) parallel to its velocity.

A pail of water can be whirled in a vertical circular path such that no water is spilled. Why does the water remain in the pail, even when the pail is upside down above your head?

Use Keplers second law to convince yourself that Earth must move faster in its orbit during the northernhemisphere winter, when it is closest to the Sun, than during the summer, when it is farthest from the Sun.

Is it possible for a car to move in a circular path in such a way that it has a tangential acceleration but no centripetal acceleration? 12. A satellite in orbit is not truly traveling through a vacuumits moving through very thin air. Does the resulting air friction cause the satellite to slow down?

(a) Find the angular speed of Earths rotation about its axis. (b) How does this rotation affect the shape of Earth?

A wheel has a radius of 4.1 m. How far (path length) does a point on the circumference travel if the wheel is rotated through angles of 30 , 30 rad, and 30 rev, respectively?

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)?

A potters wheel moves uniformly from rest to an angular speed of 1.00 rev/s in 30.0 s. (a) Find its angular acceleration in radians per second per second. (b) Would doubling the angular acceleration during the given period have doubled fi nal angular speed?

A dentists drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51  104 rev/min. (a) Find the drills angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.

A centrifuge in a medical laboratory rotates at an angular speed of 3 600 rev/min. When switched off, it rotates through 50.0 revolutions before coming to rest. Find the constant angular acceleration of the centrifuge.

A machine part rotates at an angular speed of 0.06 rad/s; its speed is then increased to 2.2 rad/s at an angular acceleration of 0.70 rad/s2. (a) Find the angle through which the part rotates before reaching this fi nal speed. (b) In general, if both the initial and fi nal angular speed are doubled at the same angular acceleration, by what factor is the angular displacement changed? Why? Hint: Look at the form of Equation 7.9.

A bicycle is turned upside down while its owner repairs a fl at tire. A friend spins the other wheel and observes that drops of water fl y 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 fi rst drop rise higher than the second drop? (b) Neglecting air friction and using only the observed heights and the radius of the wheel, fi nd the wheels angular acceleration (assuming it to be constant).

The diameters of the main rotor and tail rotor of a singleengine 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.

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.

A car initially traveling at 29.0 m/s undergoes a constant negative acceleration of magnitude 1.75 m/s2 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?

A 45.0-cm diameter disk rotates with a constant angular acceleration of 2.50 rad/s2. 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 with the positive x-axis at this time. At t 2.30 s, fi nd (a) the angular speed of the wheel, (b) the linear velocity and tangential acceleration of P, and (c) the position of P (in degrees, with respect to the positive x-axis).

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 of the wheel?

An electric motor rotating a workshop grinding wheel at a rate of 1.00  102 rev/min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude 2.00 rad/s2. (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)?

Find the centripetal accelerations due to Earths rotation about its axis of a man standing (a) at the equator and (b) at the North Pole. (c) What two forces combine to create these centripetal accelerations?

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?

(a) What is the tangential acceleration of a bug on the rim of a 10-in.-diameter disk if the disk moves from rest to an angular speed of 78 rev/min in 3.0 s? (b) When the disk is at its fi nal speed, what is the tangential velocity of the bug? (c) One second after the bug starts from rest, what are its tangential acceleration, centripetal acceleration, and total acceleration?

The 20-g centrifuge at NASAs Ames Research Center in Mountain View, California, is a cylindrical tube 58 ft long with a radius of 29 ft (Fig. P7.18). If a rider sits in a chair at the end of one arm facing the center, how many revolutions per minute would be required to create a horizontal normal force equal in magnitude to 20.0 times the riders weight?

Part of a roller-coaster ride involves coasting down an incline and entering a loop 8.00 m in diameter. For safety considerations, the roller coasters speed at the top of the loop must be such that the force of the seat on a rider is equal in magnitude to the riders weight. From what height above the bottom of the loop must the roller coaster descend to satisfy this requirement?

A coin rests 15.0 cm from the center of a turntable. The coeffi cient 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 rad/s2. (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?

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.

A race car starts from rest on a circular track of radius 400 m. The cars speed increases at the constant rate of 0.500 m/s2. At the point where the magnitudes of the centripetal and tangential accelerations are equal, determine (a) the speed of the race car, (b) the distance traveled, and (c) the elapsed time.

A certain light truck can go around a fl at 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?

A sample of blood is placed in a centrifuge of radius 15.0 cm. The mass of a red blood cell is 3.0  1016 kg, and the magnitude of the force acting on it as it settles out of the plasma is 4.0  1011 N. At how many revolutions per second should the centrifuge be operated?

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 childs centripetal acceleration? (b) What is the minimum force between her feet and the fl oor of the carousel that is required to keep her in the circular path? (c) What minimum coeffi cient of static friction is required? Is the answer you found reasonable? In other words, is she likely to stay on the merry-goround?

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 artifi cial gravity. (a) What forces are acting on an astronaut in one of the cabins? (b) Write Newtons second law for an astronaut lying on the fl oor of one of the habitats, relating the astronauts 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 Newtons 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.

An air puck of mass 0.25 kg is tied to a string and allowed to revolve in a circle of radius 1.0 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 1.0 kg is tied to it (Fig. P7.27). 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?

An air puck of mass m1 is tied to a string and allowed to revolve in a circle of radius R on a horizontal, frictionless table. The other end of the string passes through a small hole in the center of the table, and an object of mass m2 is tied to it (Fig. P7.27). The suspended object remains in equilibrium while the puck on the tabletop revolves. (a) Find a symbolic expression for the tension in the string in terms of m2 and g. (b) Write Newtons second law for the air puck, using the variables m1, v, R, and T. (c) Eliminate the tension T from the expressions found in parts (a) and (b) and fi nd an expression for the speed of the puck in terms of m1, m2, g, and R. (d) Check your answers by substituting the values of Problem 7.27 and comparing the results with the answers for that problem.

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 cars speedometer registers 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 coeffi cient of static friction between the briefcase and seat surface?

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 pails 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?

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?

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 , 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 for gravity to hold it on the track?

The average distance separating Earth and the Moon is 384 000 km. Use the data in Table 7.3 to fi nd the net gravitational force exerted by Earth and the Moon on a 3.00  104-kg spaceship located halfway between them.

A satellite has a mass of 100 kg and is located at 2.00  106 m above the surface of Earth. (a) What is the potential energy associated with the satellite at this location? (b) What is the magnitude of the gravitational force on the satellite?

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.

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.

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 infi nitely remote ones) can the 50.0-kg object be placed so as to experience a net force of zero?

Use the data of Table 7.3 to fi nd 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.

A rocket is fi red straight up through the atmosphere from the South Pole, burning out at an altitude of 250 km when traveling at 6.0 km/s. (a) What maximum distance from Earths surface does it travel before falling back to Earth? (b) Would its maximum distance increase if it were fi red from a launch site on the equator? Why?

Two objects attract each other with a gravitational force of magnitude 1.00  108 N when separated by 20.0 cm. If the total mass of the objects is 5.00 kg, what is the mass of each?

A satellite moves in a circular orbit around Earth at a speed of 5 000 m/s. Determine (a) the satellites altitude above the surface of Earth and (b) the period of the satellites orbit.

Use Keplers third law to determine how many days it takes a spacecraft to travel in an elliptical orbit from its nearest point, 6 670 km from Earths center, to its farthest point, the Moon, 385 000 km from Earths center. Note: The average radius or semimajor axis is the average of the distance from Earths center to the nearest and farthest points on the elliptical orbit.

Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22  105 km. From these data, determine the mass of Jupiter.

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

A satellite of mass 200 kg is launched from a site on Earths equator into an orbit 200 km above the surface of Earth. (a) Assuming a circular orbit, what is the orbital period of this satellite? (b) What is the satellites speed in its orbit? (c) What is the minimum energy necessary to place the satellite in orbit, assuming no air friction?

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

An artifi cial satellite circles Earth in a circular orbit at a location where the acceleration due to gravity is 9.00 m/s2. Determine the orbital period of the satellite.

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.

One method of pitching a softball is called the windmill delivery method, in which the pitchers arm rotates through approximately 360 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 rev/s2 and the radial and tangential acceleration of the ball just before it is released. (c) Determine the force exerted on the ball by the pitchers hand (both radial and tangential components) just before it is released.

A digital audio compact disc 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, fi nd the total angular displacement of the disc as it plays. (e) Find the total length of the track.

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?

A car rounds a banked curve where the radius of curvature of the road is R, the banking angle is u, and the coeffi cient of static friction is m. (a) Determine the range of speeds the car can have without slipping up or down the road. (b) What is the range of speeds possible if R 100 m, u 10 , and m 0.10 (slippery conditions)?

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.

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?

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 m?

Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.

Because of Earths rotation about its axis, a point on the equator has a centripetal acceleration of 0.034 0 m/s2, whereas a point at the poles has no centripetal acceleration. (a) Show that, at the equator, the gravitational force on an object (the objects true weight) must exceed the objects 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 m/s2.)

A small block of mass m 0.50 kg is fi red with an initial speed of v0 4.0 m/s along a horizontal section of frictionless track, as shown in the top portion of Figure P7.58. The block then moves along the frictionless, semicircular, vertical tracks of radius R 1.5 m. (a) Determine the force exerted by the track on the block at points and . (b) The bottom of the track consists of a section (L 0.40 m) with friction. Determine the coeffi cient of kinetic friction between the block and that portion of the bottom track if the block just makes it to point on the fi rst trip. (Hint: If the block just makes it to point

In Robert Heinleins 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 Earths atmosphere.

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 bottom 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 fi nd 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)?

Assume you are agile enough to run across a horizontal surface at 8.50 m/s, independently of the value of the gravitational fi eld. What would be (a) the radius and (b) the mass of an airless spherical asteroid of uniform density 1.10  103 kg/m3 on which you could launch yourself into orbit by running? (c) What would be your period?

Figure P7.62 shows the elliptical orbit of a spacecraft around Earth. Take the origin of your coordinate system to be at the center of Earth. (a) On a copy of the fi gure (enlarged if necessary), draw vectors representing (i) the position of the spacecraft when it is at and ; (ii) the velocity of the spacecraft when it is at and ; (iii) the acceleration of the spacecraft when it is at and . Make sure that each type of vector can be distinguished. Provide a legend that shows how each type is represented. (b) Have you drawn the velocity vector at longer than, shorter than, or the same length as the one at ? Explain. Have you drawn the acceleration vector at longer than, shorter than, or the same length as the one at ? Explain. (Problem 62 is courtesy of E. F. Redish. For more problems of this type, visit www.physics.umd.edu/perg/.)

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.)

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 P7.64. Molten metal is poured into the rotating cylinder and then cooled, forming the fi nished 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.

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?

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, fi nd 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 bottom of the arc.

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 Keplers 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.

The pilot of an airplane executes a constant-speed loop-the-loop maneuver in a vertical circle as in Figure 7.15b. The speed of the airplane is 2.00 x 102 m/s, and the radius of the circle is 3.20  103 m. (a) What is the pilots apparent weight at the lowest point of the circle if his true weight is 712 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?

A piece of mud is initially at point A on the rim of a bicycle wheel of radius R rotating counterclockwise about a horizontal axis at a constant angular speed v (Fig. P7.8). The mud dislodges from point A when the wheel diameter through A is horizontal. The mud then rises vertically and returns to point A. (a) Find a symbolic expression in terms of R, v, and g for the total time the mud is in the air and returns to point A. (b) If the wheel makes one complete revolution in the time it takes the mud to return to point A, fi nd an expression for the angular speed of the bicycle wheel v in terms of p, g, and R.

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 diagrams for the ball when it is at the bottom of the circle and when it is at the top. (c) If its speed is 5.20 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?

A 4.00-kg object is attached to a vertical rod by two strings as shown in Figure P7.71. 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.

The maximum lift force on a bat is proportional to the square of its fl ying speed v. For the hoary bat (Lasiurus cinereus), the magnitude of the lift force is given by FL (0.018 N s2/m2)v 2 The bat can fl y in a horizontal circle by banking its wings at an angle u, as shown in Figure P7.72. In this situation, the magnitude of the vertical component of the lift force must equal the bats 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 fl ight when the bat fl ies at its maximum speed? (d) Can the bat turn with a smaller radius by fl ying more slowly?

(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 coeffi cient of static friction between the bag and the carousel.

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. P7.74.) (a) Determine the balls 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?

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 P7.75. The fl oor then drops away, leaving the riders suspended against the wall in a vertical position. What minimum coeffi cient of friction between a riders 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 mn, where n is the normal forcein this case, the force causing the centripetal acceleration.)

A massless spring of constant k 78.4 N/m is fi xed on the left side of a level track. A block of mass m 0.50 kg is pressed against the spring and compresses it a distance d, as in Figure P7.76. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R 1.5 m. The entire track and the loop-the-loop are frictionless, except for the section of track betweenpoints A and B. Given that the coeffi cient of kinetic friction between the block and the track along AB is mk 0.30, and that the length of AB is 2.5 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.)