Derive a formula for the mass of a planet in terms of its radius r, the acceleration due

A child sitting 10 m from the center of a merry-go- round moves with a speed of 25 m/s. Calculate (a) the centripetal acceleration of the child, and (b) the net horizontal force exerted on the child (mass = 25.0 kg).

A jet plane traveling 1890 km/h (525 m/s) pulls out of a dive by moving in an arc of radius 6.00 km. What is the planes acceleration in gs?

Calculate the centripetal acceleration of the Earth in its orbit around the Sun. and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orbit is a circle of radius 50 x 10M m. [Hint: see the Tables inside the front cover of this book.)

A horizontal force of 210 N is exerted on a 2.0-kg discus as it rotates uniformly in a horizontal circle (at arms length) of radius 0.90 m. Calculate the speed of the discus.

Suppose the space shuttle is in orbit 400 km from the Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g. the gravitational acceleration at the Earths surface.

What is the magnitude of the acceleration of a speck of clay on the edge of a potters wheel turning at 45rpm (revolutions per minute) if the wheels diameter is 32 cm?

A ball on the end of a string is revolved at a uniform rate in a vertical circle of radius 72.0 cm. as shown in Fig. 5-33. If its speed is 4.00 m/s and its mass is 0.300 kg. calculate the tension in the string when the ball is (a) at the top of its path, and (b) at the bottom of its path.

A 0.45-kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 3 m on a friction less horizontal surface. If the cord will break when the tension in it exceeds 75 N, what is the maximum speed the ball can have?

What is the maximum speed with which a 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is this result independent of the mass of the car?

How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95 km/h?

A device for training astronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m. If the force felt by the trainee on her back is 7.85 times her own weight, how fast is she rotating? Express your answer in both m/s and rev/s.

A coin is placed 10 cm from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36rpm is reached and the coin slides off. What is the coefficient of static friction between the coin and the turntable?

At what minimum speed must a roller coaster be traveling when upside down at the top of a circle (Fig. 5-34) so that the passengers will not fall out? Assume a radius of curvature of 7.4 m.

A sports car of mass 950 kg (including the driver) crosses the rounded top of a hill (radius = 95 m) at 22 m/s. Determine (a) the normal force exerted by the road on the car. (b) the normal force exerted by the car on the 72-kg driver, and (c) the car speed at which the normal force on the driver equals zero.

How many revolutions per minute would a 15-m-diameter Ferris wheel need to make for the passengers to feel weightless at the topmost point?

A bucket of mass 2.00 kg is whirled in a vertical circle of radius 10m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N. (a) Find the speed of the bucket. (/>) How fast must the bucket move at the top of the circle so that the rope does not go slack?

How fast (in rpm) must a centrifuge rotate if a particle 9.00 cm from the axis of rotation is to experience an acceleration of 115,000 g's?

In a Rotor-ride at a carnival, people are rotated in a cylindrically walled room. (Sec Fig. 5-35.) The room radius is 4.6 m. and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were pressed against the wall. Is there really an outward force pressing them against the wall? If so. what is its source? If not, what is the proper description of their situation (besides scary)? [Hint: First draw the free-body diagram for a person.)

A flat puck (mass M) is rotated in a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m) through a central hole as shown in Fig. 5-36. Show that the speed of the puck is given by

) Redo Example 5-3. precisely this time, by not ignoring the weight of the ball which revolves on a string 0.600 m long. In particular, find the magnitude of Fp. and the angle it makes with the horizontal. [Hint: Set the horizontal component of FT equal to maR; also, since there is no vertical motion, what can you say about the vertical component of Fy?)

If a curve with a radius of 88 m is perfectly banked for a car traveling 75 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 95 km/h?

22. A 1200-kg car rounds a curve of radius 67 m banked at an angle of 12. If the car is traveling at 95 km/h. will a friction force be required? If so, how much and in what direction?

22. Two blocks, of masses m\ and m2, arc connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency / (revolutions per second) on a frictionless horizontal surface at distances rx and r2 from the post. Derive an algebraic expression for the tension in each segment of the cord.

A pilot performs an evasive maneuver by diving vertically at 310m/s. If he can withstand an acceleration of 9.0 gs without blacking out. at what altitude must he begin to pull out of the dive to avoid crashing into the sea?

Determine the tangential and centripetal components of the net force exerted on the car (by the ground) in Example 5-8 when its speed is 15 m/s. The cars mass is 11(X) kg.

A car at the Indianapolis 500 accelerates uniformly from the pit area, going from rest to 320 km/h in a semi-circular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding?

A particle revolves in a horizontal circle of radius 2.90 m. At a particular instant, its acceleration is 05 m/s2, in a direction that makes an angle of 32.0 to its direction of motion. Determine its speed (a) at this moment, and (/>) 2.00 s later, assuming constant tangential acceleration.

Calculate the force of Earth's gravity on a spacecraft 12.800 km (2 Earth radii) above the Earths surface if its mass is 1350 kg.

) At the surface of a certain planet, the gravitational acceleration g has a magnitude of 12.0 m/s2. A 20-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet, and (b) the weight of the brass ball on the Earth and on the planet?

Calculate the acceleration due to gravity on the Moon. The Moons radius is 74 x 106m and its mass is 7.35 x 10 kg.

A hypothetical planet has a radius 5 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface?

A hypothetical planet has a mass 66 times that of Earth, but the same radius. What is g near its surface?

Two objects attract each other gravitationally with a force of 2.5 x 10 10 N when they are 0.25 m apart. Their total mass is 4.0 kg. Find their individual masses.

Calculate the effective value of g. the acceleration of gravity, at () 3200 m, and (b) 3200 km. above the Earth's surface.

What is the distance from the Earths center to a point outside the Earth where the gravitational acceleration due to the Earth is ^ of its value at the Earth's surface?

) A certain neutron star has five times the mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on this monster.

A typical white-dwarf star, which once wras an average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on this star?

You are explaining why astronauts feel weightless while orbiting in the space shuttle. Your friends respond that they thought gravity wras just a lot weaker up there. Convince them and yourself that it isnt so by calculating the acceleration of gravity 250km above the Earths surface in terms of g.

) Four 9.5-kg spheres are located at the corners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force exerted on one sphere by the other three.

Every few hundred years most of the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5- 38). The masses are My = 0.815A/e Mj = 318A4;. Ms = 95.1ME, and their mean distances from the Sun are 108. 150. 778. and 1430 million km. respectively. What fraction of the Suns force on the Earth is this?

Given that the acceleration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars' radius is 3400 km. determine the mass of Mars.

) Determine the mass of the Sun using the known value for the period of the Earth and its distance from the Sun. [Note: Compare your answer to that obtained using Keplers laws. Example 5-16.]

) Calculate the speed of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km.

The space shuttle releases a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs?

At wrhat rate must a cylindrical spaceship rotate if occupants are to experience simulated gravity of 0.60 g? Assume the spaceships diameter is 32 m. and give your answer as the time needed for one revolution. (See Question 2 Fig 5-32.)

) Determine the time it takes for a satellite to orbit the Earth in a circular near-Earth" orbit. A near- Earth' orbit is one at a height above the surface of the Earth which is very small compared to the radius of the Earth. Does your result depend on the mass of the satellite?

At what horizontal velocity would a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbit around the Earth?

During an Apollo lunar landing mission, the command module continued to orbit the Moon at an altitude of about 100 km. How long did it take to go around the Moon once?

The rings of Saturn are composed of chunks of ice that orbit the planet. The inner radius of the rings is 73.0 km. while the outer radius is 170,000 km. Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturns mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 X 1026 kg.

A Ferris wheel 24.0 m in diameter rotates once every 5 s (see Fig. 5-9). What is the ratio of a persons apparent weight to her real weight (a) at the top. and (6) at the bottom?

What is the apparent weight of a 75-kg astronaut 4200 km from the center of the Earths Moon in a space vehicle (a) moving at constant velocity, and (b) accelerating toward the Moon at 2.9 m/s2? State the direction in each case.

Suppose that a binary-star system consists of two stars of equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?

What will a spring scale read for the weight of a 55-kg woman in an elevator that moves (a) upward with constant speed of 6.0 m/s, (6) downward writh constant speed of 6.0 m/s, (c) upward with acceleration of 0.33 g. (d) downward with acceleration 0.33g. and (e) in free fall?

A 17.0-kg monkey hangs from a cord suspended from the ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevators minimum acceleration (magnitude and direction)?

) Show that if a satellite orbits very near the surface of a planet with period T. the density (mass/volume) of the planet is p = m/V = 3tt/GT2. (/>) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min.

) Use Keplers laws and the period of the Moon (27.4 d) to determine the period of an artificial satellite orbiting very near the Earths surface.

The asteroid Icarus, though only a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from the Sun?

Neptune is an average distance of 4.5 x 10^ km from the Sun. Estimate the length of the Neptunian year given that the Earth is 50 X 108 km from the Sun on the average.

Halleys comet orbits the Sun roughly once every 76 years. It comes very close to the surface of the Sun on its closest approach (Fig. 5-39). Estimate the greatest distance of the comet from the Sun. Is it still in" the Solar System? What planet's orbit is nearest when it is out there? [Hint: The mean distance s in Keplers third law is half the sum of the nearest and farthest distance from the Sun.]

Our Sun rotates about the center of the Galaxy (Mg % 4 x 1041 kg) at a distance of about 3 x 104 light- years (l ly = 3 X 108m/s x 3.16 x 107s/y X 1 y). What is the period of our orbital motion about the center of the Galaxy?

Table 5-3 gives the mass, period, and mean distance for the four largest moons of Jupiter (those discovered by Galileo in 1609). (a) Determine the mass of Jupiter using the data for Io. (/>) Determine the mass of Jupiter using data for each of the other three moons. Arc the results consistent?

Determine the mass of the Earth from the known period and distance of the Moon.

Determine the mean distance from Jupiter for each of Jupiters moons, using Keplers third law. Use the distance of Io and the periods given in Table 5-3. Compare to the values in the Table.

The asteroid belt between Mars and Jupiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destroyed). () If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is. how long would it have taken this hypothetical planet to orbit the Sun? (b) Can we use these data to deduce the mass of this planet?

A science-fiction tale describes an artificial "planet in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planets year, in Earth days?

Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. 5-41). If his arms arc capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg. and the vine is 5.5 m long.

How far above the Earths surface will the acceleration of gravity be half what it is on the surface?

On an ice rink, two skaters of equal mass grab hands and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?

Because the Earth rotates once per day. the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didnt rotate. Estimate the magnitude of this effect. What fraction of g is this?

At what distance from the Earth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forces?

You know your mass is 65 kg, but when you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, and in which direction?

A projected space station consists of a circular tube that will rotate about its center (like a tubular bicycle tire) (Fig. 5-42). The circle formed by the tube has a diameter of about 1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (l.Og) is to be felt?

A jet pilot takes his aircraft in a vertical loop (Fig. 5-43). (a) If the jet is moving at a speed of 1300 km/h at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 gs. (b) Calculate the 78-kg pilot's effective weight (the force with which the seat pushes up on him) at the bottom of the circle, and (c) at the top of the circle (assume the same speed).

Derive a formula for the mass of a planet in terms of its radius r, the acceleration due to gravity at its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from the vertical by an angle 6 due to a massive mountain nearby (Fig. 5-44). (a) Find an approximate formula for 6 in terms of the mass of the mountain. mM, the distance to its center. |, and the radius and mass of the Earth. (b) Make a rough estimate of the mass of Mt. Everest, assuming it has the shape of a cone 4000 m high and base of diameter 4000 m. Assume its mass per unit volume is 3000 kg per m3. (c) Estimate the angle 0 of the plumb bob if it is 5 km from the center of Mt. Everest.

A curve of radius 67 m is banked for a design speed of 95 km/h. If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the curve?

How long would a day be if the Earth were rotating so fast that objects at the equator were apparently weightless?

Two equal-mass stars maintain a constant distance apart of 8.0 X 10lom and rotate about a point midway between them at a rate of one revolution every 12.6 yr. () Why don't the two stars crash into one another due to the gravitational force between them? (b) What must be the mass of each star?

A train traveling at a constant speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an angle of 17.5 throughout the curve. What is the speed of the train?

Jupiter is about 320 times as massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people cant survive more than a few gs. Calculate the number of gs a person would experience at the equator of such a planet. Use the following data for Jupiter: mass = 9 X 1027kg, equatorial radius = 7.1 X 104km. rotation period = 9 hr 55 min. Take the centripetal acceleration into account.

Astronomers using the Hubble Space Telescope deduced the presence of an extremely massive core in the distant galaxy M87. so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be 780 km/s at a distance of 60 light-years (5.7 X 101 m) from the core. Deduce the mass of the core, and compare it to the mass of our Sun.

A car maintains a constant speed v as it traverses the hill and valley shown in Fig. 5-45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain, (c) How fast can the car go without losing contact with the road at A?

The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using triangulation' and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational "fixes. The satellites orbit at an altitude of approximately 1000 nautical miles [1 nautical mile = 852 km = 6076 ft|. (a) Determine the speed of each satellite, (b) Determine the period of each satellite.

The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion km. is meant to orbit the asteroid Eros at a height of about 15 km. Eros is roughly 40 km X 6 km X 6 km. Assume Eros has a density (mass/volume) of about 2.3 X 103kg/m\ (a) What will be the period of NEAR as it orbits Eros? (h) If Eros were a sphere with the same mass and density, what would its radius be? (c) What would g be at the surface of this spherical Eros?

You are an astronaut in the space shuttle pursuing a satellite in need of repair. You arc in a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it. (a) How long will it take to overtake the satellite if you reduce your orbital radius by 0 km? (b) By how much must you reduce your orbital radius to catch up in 7.0 hours?

The comet Hale-Bopp has a period of 3000 years. (a) What is its mean distance from the Sun? (/>) At its closest approach, the comet is about 1 A.U. from the Sun (1 A.U. = distance from Earth to the Sun). What is the farthest distance? (c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint. Use Keplers second law and estimate areas by a triangle (as in Fig. 5-29. but smaller distance travelled; see also Hint for Problem 59.)

Estimate what the value of G would need to be if you could actually "feel yourself gravitationally attracted to someone near you. Make reasonable assumptions, like F % IN.

The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46) at a distance of about 30.000 light-years from the center (l ly = 9.5 X 10l5m). If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun (2 X lO30 kg), how many stars would there be in our Galaxy?

Four 0-kg masses are located at the corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 0-kg mass placed at the midpoint of the bottom side of the square.

A satellite of mass 5500 kg orbits the Earth (mass = 6.0 x 1024kg) and has a period of 6200 s. Find (<7) the magnitude of the Earth's gravitational force on the satellite, (h) the altitude of the satellite.

What is the acceleration experienced by the tip of the 5-cm-long sweep second hand on your wrist watch?

While fishing, you gel bored and start to swing a sinker weight around in a circle below you on a 0.25- m piece of fishing line. Hie weight makes a complete circle every 0.50 s. What is the angle that the fishing line makes with the vertical ? [Hint: See Fig. 5-10.)

A circular curve of radius R in a new highway is designed so that a car traveling at speed can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from vmjn to t?max. Derive formulas for vmjn and t\nax as functions of and R.

Amtrak's high speed train, the Acela, utilizes tilt of the cars when negotiating curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the normal force. The passengers experience less friction force against the seat, thus feeling more comfortable. Consider an Acela train that rounds a curve with a radius of 620 m at a speed of 160 km/h (approximately lOOmi/h). (a) Calculate the friction force needed on a train passenger of mass 75 kg if the track is not banked and the train does not tilt. (h) Calculate the friction force on the passenger if the train tilts to its maximum tilt of 8.0 toward the center of the curve.

Problem 1P A child sitting 1.10 m from the center of a merry-go-round moves with a speed of 1.25 m/s. Calculate (a) the centripetal acceleration of the child, and (b) the net horizontal force exerted on the child (mass = 25.0 kg).

Problem 1Q Sometimes people say that water is removed from clothes in a spin dryer by centrifugal force throwing the water outward. What is wrong with this statement?

Problem 27P A particle revolves in a horizontal circle of radius 2.90 m. At a particular instant, its acceleration is 1.05 m/s2, in a direction that makes an angle of 32.0° to its direction of motion. Determine its speed (a) at this moment, and (b) 2.00 s later, assuming constant tangential acceleration.

Problem 28P Calculate the force of Earth’s gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 kg.

Problem 29P At the surface of a certain planet, the gravitational acceleration g has a magnitude of 12.0 m/s2. A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet, and (b) the weight of the brass ball on the Earth and on the planet?

Problem 30P (II) Calculate the acceleration due to gravity on the Moon, which has radius 1.74 X 106 m and mass 7.35 X 1022 kg.

Problem 31P A hypothetical planet has a radius 1.5 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface?

Problem 80GP Jupiter is about 320 times as massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter because people cannot survive more than a few g’s. Calculate the number of g’s a person would experience at Jupiter’s equator, using the following data for Jupiter: mass = 1.9 X 1027 kg, equatorial radius= 7.1 X 104km, rotation Period = 9hr 55 min. Take the centripetal acceleration into account. [See Sections 5–2, 5–6, and 5–7.]

Problem 81GP Astronomers using the Hubble Space Telescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be 780 km/s at a distance of 60 light-years (= 5.7× 1017 m) from the core. Deduce the mass of the core, and compare it to the mass of our Sun.

A car maintains a constant speed \(v\) as it traverses the hill and valley shown in Fig. 5-45. Both the hill and valley have a radius of curvature \(R\). (a) How do the normal forces acting on the car at \(A\), \(B\), and \(C\) compare? (Which is largest? Smallest?) Explain. (b) Where would the drive feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the road at \(A\) Figure 5-45 Equation Transcription: Text Transcription: v R A B C

Problem 83GP The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed around the Earth, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical mile = 1.852 km = 6076ft]. (a) Determine the speed of each satellite. (b) Determine the period of each satellite.

Problem 84GP The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km. Eros is roughly 40 km × 6 km × 6 km. Assume Eros has a density (mass/volume) of about 2.3 × 103 kg/m3. (a) What will be the period of NEAR as it orbits Eros? (b) If Eros were a sphere with the same mass and density, what would its radius be? (c) What would g be at the surface of this spherical Eros?

Problem 2P A jet plane traveling 1890 km/h (525 m/s) pulls out of a dive by moving in an arc of radius 6.00 km. What is the plane’s acceleration in g’s?

Problem 2Q Will the acceleration of a car be the same when a car travels around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at the same speed? Explain.

Problem 32P A hypothetical planet has a mass 1.66 times that of Earth, but the same radius. What is g near its surface?

Problem 33P Two objects attract each other gravitationally with a force of 2.5 × 10–10 N when they are 0.25 m apart. Their total mass is 4.0 kg. Find their individual masses.

Problem 85GP You are an astronaut in the space shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it. (a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km? (b) By how much must you reduce your orbital radius to catch up in 7.0 hours?

The comet Hale-Bopp has a period of 3000 years (a) What is its mean distance from the Sun? (b) At its closest approach, the comet is about 1 A.U. from the Sun (1 A.U. = distance from Earth to the Sun). What is the farthest distance? (c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5-29, but smaller distance traveled; see also Hint for Problem 59.]

Problem 23Q The Earth moves faster in its orbit around the Sun in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons—the main factor is the tilt of the Earth’s axis relative to the plane of its orbit.]

Problem 24P A pilot performs an evasive maneuver by diving vertically at 310 m/s. If he can withstand an acceleration of 9.0 g’s without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?

Problem 24Q The mass of the “planet” Pluto was not known until it was discovered to have a moon. Explain how this enabled an estimate of Pluto’s mass.

Problem 74GP Derive a formula for the mass of a planet in terms of its radius r, the acceleration due to gravity at its surface gP, and the gravitational constant G.

Determine the tangential and centripetal components of the net force exerted on the car (by the ground) in Example 5–8 when its speed is 15 m/s. The car’s mass is 1100 kg.

Problem 77GP How long would a day be if the Earth were rotating so fast that objects at the equator were apparently weightless?

Problem 78GP Two equal-mass stars maintain a constant distance apart of 8.0 × 1010 m and rotate about a point midway between them at a rate of one revolution every 12.6 yr. (a) Why don’t the two stars crash into one another due to the gravitational force between them? (b) What must be the mass of each star?

Problem 79GP A train traveling at a constant speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an angle of 17.5° throughout the curve. What is the speed of the train?

(III) Two blocks, of masses and , are connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency (revolutions per second) on a frictionless horizontal surface at distances and from the post. Derive an algebraic expression for the tension in each segment of the cord.

Problem 3P Calculate the centripetal acceleration of the Earth in its orbit around the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth’s orbit is a circle of radius 1.50 × 1011 m. [Hint: see the Tables inside the front cover of this book.]

Problem 4P A horizontal force of 210 N is exerted on a 2.0-kg discus as it rotates uniformly in a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.

Problem 4Q Describe all the forces acting on a child riding a horse on a merry-go-round. Which of these forces provides the centripetal acceleration of the child?

Problem 5P Suppose the space shuttle is in orbit 400 km from the Earth’s surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g, the gravitational acceleration at the Earth’s surface.

Problem 34P Calculate the effective value of g, the acceleration of gravity, at (a) 3200 m, and (b) 3200 km, above the Earth’s surface.

What is the distance from the Earth’s center to a point outside the Earth where the gravitational acceleration due to the Earth is of its value at the Earth’s surface?

Problem 36P (II) A certain neutron star has five times the mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on this monster.

Problem 37P A typical white-dwarf star, which once was an average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on this star?

Problem 38P You are explaining why astronauts feel weightless while orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface in terms of g.

Problem 87GP Estimate what the value of G would need to be if you could actually “feel” yourself gravitationally attracted to someone near you. Make reasonable assumptions, like F ? 1 N.

The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46) at a distance of about 30,000 light-years from the center . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun , how many stars would there be in our Galaxy? Figure 5-46 Problem 88. Edge -on view of our Galaxy

Problem 89GP Four 1.0-kg masses are located at the corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg mass placed at the midpoint of the bottom side of the square.

Problem 90GP A satellite of mass 5500 kg orbits the Earth (mass = 6.0 × 1024 kg) and has a period of 6200 s. Find (a) the magnitude of the Earth’s gravitational force on the satellite, (b) the altitude of the satellite.

Problem 91GP What is the acceleration experienced by the tip of the 1.5-cm-long sweep second hand on your wrist watch?

Problem 3Q Suppose a car moves at constant speed along a hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Problem 5Q A bucket of water can be whirled in a vertical circle without the water spilling out, even at the top of the circle when the bucket is upside down. Explain.

Problem 6P What is the magnitude of the acceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diameter is 32 cm?

Problem 6Q How many “accelerators” do you have in your car? There are at least three controls in the car which can be used to cause the car to accelerate. What are they? What accelerations do they produce?

(II) A ball on the end of a string is revolved at a uniform rate in a vertical circle of radius 72.0 cm, as shown in Fig. 5-33. If its speed is 4.00 m/s and its mass is 0.300 kg, calculate the tension in the string when the ball is (a) at the top of its path, and (b) at the bottom of its path. Figure 5-33

A child on a sled comes flying over the crest of a small hill, as shown in Fig. 5-31. His sled does not leave the ground (he does not achieve “air”), but he feels the normal force between his chest and the sled decreases as he goes over the hill. Explain this decrease using Newton’s second law. Figure 5-31

Problem 39P Four 9.5-kg Spheres are located at the corners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force exerted on one sphere by the other three.

(II) Every few hundred years most of the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. ). The masses are , and their mean distances from the Sun are , and 1430 million , respectively. What fraction of the Sun's force on the Earth is this?

Problem 41P (II) Given that the acceleration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the mass of Mars.

Determine the mass of the Sun using the known value for the period of the Earth and its distance from the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]

Problem 43P Calculate the speed of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km.

While fishing, you get bored and start to swing a sinker weight around in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5-10.]

Problem 94GP Amtrak’s high speed train, the Acela, utilizes tilt of the cars when negotiating curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the normal force. The passengers experience less friction force against the seat, thus feeling more comfortable. Consider an Acela train that rounds a curve with a radius of 620 m at a speed of 160 km/h (approximately 100 mi/h). (a) Calculate the friction force needed on a train passenger of mass 75 kg if the track is not banked and the train does not tilt. (b) Calculate the friction force on the passenger if the train tilts to its maximum tilt of 8.0° toward the center of the curve.

Problem 93GP A circular curve of radius R in a new highway is designed so that a car traveling at speed v0 can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from vmin to vmax. Derive formulas for vmin and vmax as functions of ?s, v0, and R.

Problem 8P A 0.45-kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3 m on a frictionless horizontal surface. If the cord will break when the tension in it exceeds 75 N, what is the maximum speed the ball can have?

Problem 8Q Why do bicycle riders lean inward when rounding a curve at high speed?

Problem 9P What is the maximum speed with which a 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is this result independent of the mass of the car?

Why do airplanes bank when they turn? How would you compute the banking angle given its speed and radius of the turn?

Problem 10P How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95 km/h?

Problem 44P The space shuttle releases a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs?

(II) At what rate must a cylindrical spaceship rotate if occupants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.)

Problem 46P (II) Determine the time it takes for a satellite to orbit the Earth in a circular near-Earth orbit. A “near-Earth” orbit is at a height above the surface of the Earth that is very small compared to the radius of the Earth. [Hint: You may take the acceleration due to gravity as essentially the same as that on the surface.] Does your result depend on the mass of the satellite?

Problem 47P At what horizontal velocity would a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbit around the Earth?

Problem 48P During an Apollo lunar landing mission, the command module continued to orbit the Moon at an altitude of about 100 km. How long did it take to go around the Moon once?

A plumb bob (a mass hanging on a string) is deflected from the vertical by an angle due to a massive mountain nearby (Fig. 5-44). (a) Find an approximate formula for in terms of the mass of the mountain, , the distance to its center, , and the radius and mass of the Earth. (b) Make a rough estimate of the mass of Mt. Everest, assuming it has the shape of a cone high and base of diameter . Assume its mass per unit volume is per . (c) Estimate the angle of the plumb bob if it is from the center of Mt. Everest.

Problem 10Q A girl is whirling a ball on a string around her head in a horizontal plane. She wants to let go at precisely the right time so that the ball will hit a target on the other side of the yard. When should she let go of the string?

Problem 11P A device for training astronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m. If the force felt by the trainee on her back is 7.85 times her own weight, how fast is she rotating? Express your answer in both m/s and rev/s.

Problem 11Q Does an apple exert a gravitational force on the Earth? If so, how large a force? Consider an apple (a) attached to a tree and (b) falling.

Problem 12P A coin is placed 11.0 cm from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between the coin and the turntable?

Problem 12Q If the Earth’s mass were double what it is, in what ways would the Moon’s orbit be different?

Problem 49P The rings of Saturn are composed of chunks of ice that orbit the planet. The inner radius of the rings is 73,000 km, while the outer radius is 170,000 km. Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn’s mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 × 1026 kg.

Problem 50P A Ferris wheel 24.0 m in diameter rotates once every 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) at the top, and (b) at the bottom? Figure 5–9

Problem 51P What is the apparent weight of a 75-kg astronaut 4200 km from the center of the Earth’s Moon in a space vehicle (a) moving at constant velocity, and (b) accelerating toward the Moon at 2.9 m/s2? State the “direction” in each case.

Problem 52P Suppose that a binary-star system consists of two stars of equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?

Problem 53P What will a spring scale read for the weight of a 55-kg woman in an elevator that moves (a) upward with constant speed of 6.0 m/s, (b) downward with constant speed of 6.0 m/s, (c) upward with acceleration of 0.33 g, (d) downward with acceleration 0.33 g, and (e) in free fall?

(II) At what minimum speed must a roller coaster be traveling when upside down at the top of a circle (Fig. 5-34) so that the passengers will not fall out? Assume a radius of curvature of 7.4 m.

Problem 13Q Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the Earth? Which accelerates more?

Problem 14P A sports car of mass 950 kg (including the driver) crosses the rounded top of a hill (radius = 95 m) at 22 m/s. Determine (a) the normal force exerted by the road on the car, (b) the normal force exerted by the car on the 72-kg driver, and (c) the car speed at which the normal force on the driver equals zero.

Problem 14Q The Sun’s gravitational pull on the Earth is much larger than the Moon’s. Yet the Moon’s is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Problem 15P How many revolutions per minute would a 15-m-diameter Ferris wheel need to make for the passengers to feel “weightless” at the topmost point?

Problem 54P A 17.0-kg monkey hangs from a cord suspended from the ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)?

(III) (a) Show that if a satellite orbits very near the surface of a planet with period T, the density (mass/volume) of the planet is . (b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min.

Problem 56P (I) Use Kepler’s laws and the period of the Moon (27.4 d) to determine the period of an artificial satellite orbiting very near the Earth’s surface.

Problem 58P (I) Neptune is an average distance of 4.5 X 109 km from the Sun. Estimate the length of the Neptunian year using the fact that the Earth is 1.50 X 108 from the Sun on average.

Problem 57P The asteroid Icarus, though only a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from the Sun?

Problem 76GP A curve of radius 67 m is banked for a design speed of 95 km/h. If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the curve?

Problem 15Q Will an object weigh more at the equator or at the poles? What two effects are at work? Do they oppose each other?

Problem 16P A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N. (a) Find the speed of the bucket. (b) How fast must the bucket move at the top of the circle so that the rope does not go slack?

Problem 16Q The gravitational force on the Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn’t the Moon pulled away from the Earth?

Problem 17Q Is the centripetal acceleration of Mars in its orbit around the Sun larger or smaller than the centripetal acceleration of the Earth? Explain.

Problem 17P How fast (in rpm) must a centrifuge rotate if a particle 9.00 cm from the axis of rotation is to experience an acceleration of 115,000 g’s?

(II) Halley’s comet orbits the Sun roughly once every 76 years. It comes very close to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the greatest distance of the comet from the Sun. Is it still “in” the solar system? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.]

Problem 60P (II) Our Sun revolves about the center of our Galaxy (mG ? 4 X 1041 kg) at a distance of about 3X 104 light-years [1 Iy =(3.00 X 108 m/s) . (3.16 X 107 s/yr) . (1.00 yr)]. What is the period of the Sun’s orbital motion about the center of the Galaxy?

(II) Table 5–3 gives the mass, period, and mean distance for the four largest moons of Jupiter (those discovered by Galileo in 1609). (a) Determine the mass of Jupiter using the data for Io; (b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?

Problem 62P (II) Determine the mass of the Earth from the known period and distance of the Moon.

(II) Determine the mean distance from Jupiter for each of Jupiter’s principal moons, using Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the values in the Table.

Problem 26P A car at the Indianapolis 500 accelerates uniformly from the pit area, going from rest to 320 km/h in a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding?

(II) In a "Rotor-ride" at a carnival, people are rotated in a cylindrically walled "room." (See Fig. ) The room radius is , and the rotation frequency is revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.] FIGURE 5-35 Problem 18

Problem 18Q Would it require less speed to launch a satellite (a) toward the east or (b) toward the west? Consider the Earth’s rotation direction.

(II) A flat puck (mass ) is rotated in a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass ) through a central hole as shown in Fig. . Show that the speed of the puck is given by

Problem 19Q When will your apparent weight be the greatest, as measured by a scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at constant speed? In which case would your weight be the least? When would it be the same as when you are on the ground?

(II) Redo Example , precisely this time, by not ignoring the weight of the ball which revolves on a string long. In particular, find the magnitude of , and the angle it makes with the horizontal. [Hint: Set the horizontal component of equal to also, since there is no vertical motion, what can you say about the vertical component of ?]

Problem 64P The asteroid belt between Mars and Jupiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destroyed). (a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun? (b) Can we use these data to deduce the mass of this planet?

(III) A science-fiction tale describes an artificial "planet" in the form of a band completely encircling a sun (Fig. ). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of as on Earth. What will be the period of revolution, this planet's year, in Earth days? FIGURE 5-40 Problem 65

Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. ). If his arms are capable of exerting a force of on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is , and the vine is long. FIGURE 5-41 Problem 66.

Problem 67GP How far above the Earth’s surface will the acceleration of gravity be half what it is at the surface?

Problem 68GP On an ice rink, two skaters of equal mass grab hands and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?

Problem 20Q What keeps a satellite up in its orbit around the Earth?

Astronauts who spend long periods in outer space could be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5-32). Explain how this simulates gravity. Consider (a) how objects fall, (b) the force we feel on our feet, and (c) any other aspects of gravity you can think of. Figure 5-32

Problem 22P A 1200-kg car rounds a curve of radius 67 m banked at an angle of 12°. If the car is traveling at 95 km/h, will a friction force be required? If so, how much and in what direction?

Problem 22Q Explain how a runner experiences “free fall” or “apparent weightlessness” between steps.

Problem 69GP Because the Earth rotates once per day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this effect. What fraction of g is this?

Problem 70GP At what distance from the Earth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forces?

Problem 71GP You know your mass is 65 kg, but when you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, and in which direction?

A projected space station consists of a circular tube that will rotate about its center (like a tubular bicycle tire) (Fig. 5-42). The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt? Figure 5-42

A jet pilot takes his aircraft in a vertical loop (Fig. 5-43). (a) If the jet is moving at a speed of 1300 km/h at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s. (b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle, and (c) at the top of the circle (assume the same speed). Figure 5-43

Problem 21P If a curve with a radius of 88 m is perfectly banked for a car traveling 75 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 95 km/h?