(II) Our Sun revolves about the center of our Galaxy at a

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) 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 \(250 \mathrm{~km}\) above the Earth's surface in terms of g.

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?

(II) What is the distance from the Earth's center to a point outside the Earth where the gravitational acceleration due to the Earth is \(\frac{1}{10}\) of its value at the Earth's surface? Equation Transcription: Text Transcription: \frac{1}{10}

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.

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 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 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 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 47GP A car with 58-cm-diameter tires accelerates uniformly from rest to 20 m/s in 10 s. How many times does each tire rotate?

(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.

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

Problem 48GP The cable lifting an elevator is wrapped around a 1.0-mdiameter cylinder that is turned by the elevator’s motor. The elevator is moving upward at a speed of 1.6 m/s. It then slows to a stop as the cylinder makes one complete turn at constant angular acceleration. How long does it take for the elevator to stop?

Problem 53GP The machinist's square shown in Figure 53 consists of a thin, rectangular blade connected to a rectangular handle. a. Determine the ?x and y coordinates of the center or gravity. Let the lower left c? ? ner be? ? 0, y = 0. b. Sketch how the tool would hang if it were allowed to freely pivot about the ? ? point ?x = 0,? = 0. c. When hanging from that point, what angle would the long side of the blade make with the vertical? FIGURE 53

Problem 60GP Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel’s energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. A motor spins up the flywheel with a constant torque of 50 N . m. How long does it take the flywheel to reach top angular speed of 1200 rpm?

Problem 58GP Starting from rest, a 12-cm-diameter compact disk takes 3.0 s to reach its operating angular velocity of 2000 rpm. Assume that the angular acceleration is constant. The disk’s moment of inertia is . a. How much torque is applied to the disk? b. How many revolutions does it make before reaching full speed?

Problem 61GP A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid, massless rod. The rod and balls are rotating clockwise about their center of gravity at 20 rpm. What torque will bring the balls to a halt in 5.0 s?

Problem 1PE A physics major is cooking breakfast when he notices that the frictional force between his steel spatula and his Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction between the two materials, he quickly calculates the normal force. What is it?

Problem 2CQ Problem Are the objects described here in static equilibrium, dynamic equilibrium, or not in equilibrium at all? a. A girder is lifted at constant speed by a crane. b. A girder is lowered by a crane. It is slowing down. c. You’re straining to hold a 200 lb barbell over your head. d. A jet plane has reached its cruising speed and altitude. e. A rock is falling into the Grand Canyon. f. A box in the back of a truck doesn’t slide as the truck stops.

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 2PE Problem (a) When rebuilding her car’s engine, a physics major must exert 300 N of force to insert a dry steel piston into a steel cylinder. What is the magnitude of the normal force between the piston and cylinder? (b) What is the magnitude of the force would she have to exert if the steel parts were oiled?

Problem 2Q Problem 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 3CQ Problem What forces are acting on you right now? What net force is acting on you right now?

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 3PE (a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous exercise it is possible to exert forces to the joints that are easily ten times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pain.

Problem 4CQ When you push a piece of chalk across a chalkboard, it sometimes screeches because it rapidly alternates between slipping and sticking to the board. Describe this process in more detail, in particular explaining how it is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.)

Problem 4PE Suppose you have a 120-kg wooden crate resting on a wood floor. (a) What maximum force can you exert horizontally on the crate without moving it? (b) If you continue to exert this force once the crate starts to slip, what will the magnitude of its acceleration then be?

Problem 4Q Problem 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 5CQ Problem Athletes such as swimmers and bicyclists wear body suits in competition. Formulate a list of pros and cons of such suits.

Problem 5PE Problem (a) If half of the weight of a small 1.00×103 kg utility truck is supported by its two drive wheels, what is the magnitude of the maximum acceleration it can achieve on dry concrete? (b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate? (c) Solve both problems assuming the truck has four-wheel drive.

Problem 6CQ Problem Two expressions were used for the drag force experienced by a moving object in a liquid. One depended upon the speed, while the other was proportional to the square of the speed. In which types of motion would each of these expressions be more applicable than the other one?

Problem 6PE Problem A team of eight dogs pulls a sled with waxed wood runners on wet snow (mush!). The dogs have average masses of 19.0 kg, and the loaded sled with its rider has a mass of 210 kg. (a) Calculate the magnitude of the acceleration starting from rest if each dog exerts an average force of 185 N backward on the snow. (b) What is the magnitude of the acceleration once the sled starts to move? (c) For both situations, calculate the magnitude of the force in the coupling between the dogs and the sled.

Problem 6Q Problem 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?

Problem 7CQ Problem As cars travel, oil and gasoline leaks onto the road surface. If a light rain falls, what does this do to the control of the car? Does a heavy rain make any difference?

Problem 7PE Problem Consider the 65.0-kg ice skater being pushed by two others shown in Figure 5.21. (a) Find the direction and magnitude of Ftot , the total force exerted on her by the others, given that the magnitudes F1 and F2 are 26.4 N and 18.6 N, respectively. (b) What is her initial acceleration if she is initially stationary and wearing steel-bladed skates that point in the direction of Ftot ? (c) What is her acceleration assuming she is already moving in the direction of Ftot ? (Remember that friction always acts in the direction opposite that of motion or attempted motion between surfaces in contact.)

Problem 8CQ Problem Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall?

Problem 9CQ Problem The elastic properties of the arteries are essential for blood flow. Explain the importance of this in terms of the characteristics of the flow of blood (pulsating or continuous).

Problem 8PE Problem Show that the acceleration of any object down a frictionless incline that makes an angle ? with the horizo?ntal?? = ? sin ?. (Note that this acceleration is independent of mass.)

Problem 9PE Problem Show that the acceleration of any object down an incline where friction behaves simply (that is,?wher?? k = ?µk?N )?? = ? ( si?n ? ? ?µk cos ?). Note that the acceleration is independent of mass and reduces to the expression found in the previous problem when friction becomes negligibly? small (?µk = 0).

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

Problem 10CQ Problem What are you feeling when you feel your pulse? Measure your pulse rate for 10 s and for 1 min. Is there a factor of 6 difference?

Problem 10PE Problem Calculate the deceleration of a snow boarder going up a 5.0º, slope assuming the coefficient of friction for waxed wood on wet snow. The result of Exercise 5.9 may be useful, but be careful to consider the fact that the snow boarder is going uphill. Explicitly show how you follow the steps in Problem-Solving Strategies.

Problem 11CQ Problem Examine different types of shoes, including sports shoes and thongs. In terms of physics, why are the bottom surfaces designed as they are? What differences will dry and wet conditions make for these surfaces?

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 11PE Problem (a) Calculate the acceleration of a skier heading down a 10.0º slope, assuming the coefficient of friction for waxed wood on wet snow. (b) Find the angle of the slope down which this skier could coast at a constant velocity. You can neglect air resistance in both parts, and you will find the result of Exercise 5.9 to be useful. Explicitly show how you follow the steps in the Problem-Solving Strategies.

Problem 11Q Problem 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 12CQ Problem Would you expect your height to be different depending upon the time of day? Why or why not?

Problem 12PE Problem If an object is to rest on an incline without slipping, the friction must be equal the component of the weight of the object parallel to the incline. This requires greater and greater friction for steeper slopes. Show that the maximum angle of an incline above the –1? horizontal for which an object will not slide down is ? = tan? ?? . Yos?may use the result of the previous problem. Assu? me that a ? = 0 and that static friction has reached its maximum value.

Problem 14CQ Problem Explain why pregnant women often suffer from back strain late in their pregnancy.

Problem 13PE Problem Calculate the maximum deceleration of a car that is heading down a 6º slope (one that makes an angle of 6º with the horizontal) under the following road conditions. You may assume that the weight of the car is evenly distributed on all four tires and that the coefficient of static friction is involved—that is, the tires are not allowed to slip during the deceleration. (Ignore rolling.) Calculate for a car: (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that µs = 0.100 , the same as for shoes on ice.

(II) 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 15CQ Problem An old carpenter’s trick to keep nails from bending when they are pounded into hard materials is to grip the center of the nail firmly with pliers. Why does this help?

Problem 15PE Problem Repeat Exercise 5.14 for a car with four-wheel drive.

Problem 16CQ Problem When a glass bottle full of vinegar warms up, both the vinegar and the glass expand, but vinegar expands significantly more with temperature than glass. The bottle will break if it was filled to its tightly capped lid. Explain why, and also explain how a pocket of air above the vinegar would prevent the break. (This is the function of the air above liquids in glass containers.)

Problem 16PE Problem A freight train consists of two 8.00×10? -kg engines and 45 cars with average masses of 5? 5.50×10? kg . (a) What force must each engine exert backward on the track to accelerate the train at a rate of 5.00×10? m / s? if the force of friction is 7.50×10? N , assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the magnitude of the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

Problem 17PE Problem Consider the 52.0-kg mountain climber in Figure 5.22. (a) Find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock face to remain stationary. Assume that the force is exerted parallel to her legs. Also, assume negligible force exerted by her arms. (b) What is the minimum coefficient of friction between her shoes and the cliff?

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

Problem 18PE Problem A contestant in a winter sporting event pushes a 45.0-kg block of ice across a frozen lake as shown in Figure 5.23(a). (a) Calculate the minimum force ?F? he must exert to get the block moving. (b) What is the magnitude of its acceleration once it starts to move, if that force is maintained?

Problem 19PE Problem Repeat Exercise 5.18 with the contestant pulling the block of ice with a rope over his shoulder at the same angle above the horizontal as shown in Figure 5.23(b).

Problem 20PE Problem The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of 0.140 m2

Problem 21PE Problem A 60-kg and a 90-kg skydiver jump from an airplane at an altitude of 6000 m, both falling in the pike position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits.

Problem 22PE Problem A 560-g squirrel with a surface area of 930 cm2 falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?

Problem 23Q Problem 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. [/Vote: 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 23PE Problem To maintain a constant speed, the force provided by a car’s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the magnitudes of drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Drag area is 0.70 m? ) (b) What is the magnitude of drag force at 70 km/h and 100 km/h for a Hummer H2? (Drag area is 2.44 m2 ) Assume all values are accurate to three significant digits.

Problem 24PE Problem By what factor does the drag force on a car increase as it goes from 65 to 110 km/h?

Problem 24Q Problem 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 25PE Problem Calculate the speed a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be 1.00×10? kg/m3 , and the surface area to be ?r2 .

Problem 26PE Problem Using Stokes’ law, verify that the units for viscosity are kilograms per meter per second.

Problem 27PE Problem Find the terminal velocity of a spherical bacterium (diameter 2.00 ?m ) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be 1.10×10? kg/m3

Problem 28PE Problem Stokes’ law describes sedimentation of particles in liquids and can be used to measure viscosity. Particles in liquids achieve terminal velocity quickly. One can measure the time it takes for a particle to fall a certain distance and then use Stokes’ law to calculate the viscosity of the liquid. Suppose a steel ball 3? bearing (density 7.8×10? kg/m3 , diameter 3.0 mm ) is dropped in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity of the oil.

Problem 29PE Problem During a circus act, one performer swings upside down hanging from a trapeze holding another, also upside-down, performer by the legs. If the upward force on the lower performer is three times her weight, how much do the bones (the femurs) in her upper legs stretch? You may assume each is equivalent to a uniform rod 35.0 cm long and 1.80 cm in radius. Her mass is 60.0 kg.

Problem 30P Problem (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 30PE Problem During a wrestling match, a 150 kg wrestler briefly stands on one hand during a maneuver designed to perplex his already moribund adversary. By how much does the upper arm bone shorten in length? The bone can be represented by a uniform rod 38.0 cm in length and 2.10 cm in radius.

Problem 32PE Problem TV broadcast antennas are the tallest artificial structures on Earth. In 1987, a 72.0-kg physicist placed himself and 400 kg of equipment at the top of one 610-m high antenna to perform gravity experiments. By how much was the antenna compressed, if we consider it to be equivalent to a steel cylinder 0.150 m in radius?

Problem 31PE Problem (a) The “lead” in pencils is a graphite composition with a Young’s modulus of 9? about 1×10? N / m2 . Calculate the change in length of the lead in an automatic pencil if you tap it straight into the pencil with a force of 4.0 N. The lead is 0.50 mm in diameter and 60 mm long. (b) Is the answer reasonable? That is, does it seem to be consistent with what you have observed when using pencils?

Problem 33PE Problem (a) By how much does a 65.0-kg mountain climber stretch her 0.800-cm diameter nylon rope when she hangs 35.0 m below a rock outcropping? (b) Does the answer seem to be consistent with what you have observed for nylon ropes? W? ould it make sense if the rope were actually a bungee cord?

Problem 34PE Problem 20.0-m tall hollow aluminum flagpole is equivalent in stiffness to a solid cylinder 4.00 cm in diameter. A strong wind bends the pole much as a horizontal force of 900 N exerted at the top would. How far to the side does the top of the pole flex?

Problem 35PE Problem As an oil well is drilled, each new section of drill pipe supports its own weight and that of the pipe and drill bit beneath it. Calculate the stretch in a new 6.00 m length of steel pipe that supports 3.00 km of pipe having a mass of 20.0 kg/m and a 100-kg drill bit. The pipe is equivalent in stiffness to a solid cylinder 5.00 cm in diameter.

Problem 36P Problem (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 36PE Problem Calculate the force a piano tuner applies to stretch a steel piano wire 8.00 mm, if the wire is originally 0.850 mm in diameter and 1.35 m long.

Problem 37PE Problem A vertebra is subjected to a shearing force of 500 N. Find the shear deformation, taking the vertebra to be a cylinder 3.00 cm high and 4.00 cm in diameter.

Problem 38PE Problem A disk between vertebrae in the spine is subjected to a shearing force of 600 9? N. Find its shear deformation, taking it to have the shear modulus of 1×10? N / m2 . The disk is equivalent to a solid cylinder 0.700 cm high and 4.00 cm in diameter.

Problem 39PE Problem When using a pencil eraser, you exert a vertical force of 6.00 N at a distance of 2.00 cm from the hardwood-eraser joint. The pencil is 6.00 mm in diameter and is held at an angle of 20.0º to the horizontal. (a) By how much does the wood flex perpendicular to its length? (b) How much is it compressed lengthwise?

(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 \(M_{V}=0.815 M_{E}, M_{j}=318 M_{E}, M_{s}=95.1 M_{E}\), and their mean distances from the Sun are , and 1430 million , respectively. What fraction of the Sun's force on the Earth is this? Equation Transcription: Text Transcription: MV=0.815ME, Mj=318ME, Ms=95.1ME

Problem 41PE Problem A farmer making grape juice fills a glass bottle to the brim and caps it tightly. The juice expands more than the glass when it warms up, in such a way that the volume increases by 0.2% (that is, ?V / V? = 2×100?) relative to the space available. Calculate the magnitude of the normal force exerted by the juice per 9? square centimeter if its bulk modulus is 1.8×10? N/m2 , assuming the bottle does not break. In view of your answer, do you think the bottle will survive?

Problem 40PE Problem To consider the effect of wires hung on poles, we take data from Example 4.8, in which tensions in wires supporting a traffic light were calculated. The left wire made an angle 30.0º below the horizontal with the top of its pole and carried a tension of 108 N. The 12.0 m tall hollow aluminum pole is equivalent in stiffness to a 4.50 cm diameter solid cylinder. (a) How far is it bent to the side? (b) By how much is it compressed?

Problem 42PE Problem (a) When water freezes, its volume increases by 9.05% (that is, ?V / V? = 0? 9.05×10? ). What force per unit area is water capable of exerting on a container when it freezes? (It is acceptable to use the bulk modulus of water in this problem.) (b) Is it surprising that such forces can fracture engine blocks, boulders, and the like?

Problem 43PE Problem This problem returns to the tightrope walker studied in Example 4.6, who 3? created a tension of 3.94×10? N in a wire making an angle 5.0º below the horizontal with each supporting pole. Calculate how much this tension stretches the steel wire if it was originally 15 m long and 0.50 cm in diameter. Example 4.6: Calculate the tension in the wire supporting the 70.0-kg tightrope walker shown in Figure 4.17. Figure 4.17 The weight of a tightrope walker causes a wire to sag by 5.0 degrees. The system of interest here is the point in the wire at which the tightrope walker is standing.

Problem 44PE Problem The pole in Figure 5.24 is at a 90.0º bend in a power line and is therefore subjected to more shear force th4? poles in straight parts of the line. The tension in each line is 4.00×10? N , at the angles shown. The pole is 15.0 m tall, has an 18.0 cm diameter, and can be considered to have half the stiffness of hardwood. (a) Calculate the compression of the pole. (b) Find how much it bends and in what direction. (c) Find the tension in a guy wire used to keep the pole straight if it is attached to the top of the pole at an angle of 30.0º with the vertical. (Clearly, the guy wire must be in the opposite direction of the bend.)

Problem 5Q Problem 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 19P Problem Aflat puck (mass M) is revolved in a circle on a frictionless air hockey table top, and is held in this orbit by a massless cord which is connected to a dangling mass (mass m) through a central hole as shown in Fig. 5-46.

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.

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 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?

A horizontal force of \(210 \mathrm{~N}\) is exerted on a \(2.0-\mathrm{kg}\) discus as it rotates uniformly in a horizontal circle (at arm's length) of radius \(0.90 \mathrm{~m}\). Calculate the speed of the discus.

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.

(II) 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?

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 decrease as he goes over the hill. Explain this decrease using Newton's second law.

Problem 8P A 0.45-kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3m 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?

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

(II) What is the maximum speed with which a \(1050-\mathrm{kg}\) car can round a turn of radius \(77 \mathrm{~m}\) 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?

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 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 41P Problem (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.

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?

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

(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 46P Problem (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 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 48P Problem 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?

(II) 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 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 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 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) In a “Rotor-ride” at a carnival, people are rotated in a cylindrically walled “room.” (See 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.]

(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 \(\rho=m / V=3 \pi / G T^{2}\). (b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of 85 min. Equation Transcription: Text Transcription: \rho=m / V=3 \pi / G T^2

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.

Problem 56P Problem (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.

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?

Problem 58P Problem (I) Neptune is an average distance of 4.5 X109 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.

(II) Redo Example 5-3, precisely this time, by not ignoring the weight of the ball which revolves on a string \(0.600 \mathrm{~m}\) long. In particular, find the magnitude of \(\overrightarrow{\mathbf{F}}_{\mathrm{T}}\), and the angle it makes with the horizontal. [Hint: Set the horizontal component of \(\overrightarrow{\mathbf{F}}_{\mathrm{T}}\) equal to \(m a_{\mathrm{R}}\); also, since there is no vertical motion, what can you say about the vertical component of \(\overrightarrow{\mathbf{F}}_{\mathrm{T}}\) ?]

Problem 59P Problem (II) Haffefs comet orbits the Sun roughly once every 76 years. It comes very close to the surface of the Sun on its closest approach (Fig. 5-45). 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?

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

(II) Our Sun revolves about the center of our Galaxy \(\left(M_{G} \approx 4 X 10^{14} \mathrm{~kg}\right)\) at a distance of about \(3 X 10^{4}\) light-years \(\left(1 \mathrm{ly}=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} \times 3.16 \times 10^{7} \mathrm{~s} / \mathrm{y} X 1 \mathrm{y}\right)\). What is the period of our orbital motion about the center of the Galaxy? Equation Transcription: Text Transcription: (MG \approx 4 X 1014 kg) 3 X 104 (1 ly=3.00 X 108 m/s X 3.16 X 107 s/y X 1 y)

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?

(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 data for Io. (b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?

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

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?

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

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

(III) Two blocks, of masses \(m_{1} \text { and } m_{2}\), 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 f (revolutions per second) on a frictionless horizontal surface at distances \(\mathrm{r}_{1} \text { and } \mathrm{r}_{2}\) from the post. Derive an algebraic expression for the tension in each segment of the cord. Equation Transcription: Text Transcription: m1 and m2 r1 and r2

A science-fiction tale describes an artificial "planet" in the form of a band completely encircling a sun (Fig. 5-50). 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 planet’s year, in Earth days?

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

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 \(15 \mathrm{~m} / \mathrm{s}\). The car's mass is \(1100 \mathrm{~kg}\).

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?

A particle revolves in a horizontal circle of radius \(2.90 \mathrm{~m}\). At a particular instant, its acceleration is \(1.05 \mathrm{~m} / \mathrm{s}^2\), in a direction that makes an angle of \(32.0^{\circ}\) to its direction of motion. Determine its speed (a) at this moment, and (b) \(2.00 \mathrm{~s}\) later, assuming constant tangential acceleration.

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

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 can't survive more than a few g's. Calculate the number of g 's a person would experience at the equator of such a planet. Use the following data for Jupiter: mass \(=1.9 \times 10^{27} \mathrm{~kg}\), equatorial radius \(=7.1 \times 10^4 \mathrm{~km}\), rotation period \(=9 \mathrm{hr} 55 \mathrm{~min}\). Take the centripetal acceleration into account.

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.

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 11,000 nautical miles [1 nautical mile = \(1.852 \mathrm{~km}=6076 \mathrm{ft}\). (a) Determine the speed of each satellite. (b) Determine the period of each satellite.

(III) 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.]

(I) 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 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 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.

(II) A Ferris wheel 24.0 m in diameter rotates once every 15.5s (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?

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) 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)?

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 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?

Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. 5–41). If his arms are 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.

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 didn't 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 the 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 jet pilot takes his aircraft in a vertical loop (Fig. 5-43). (a) If the jet is moving at a speed of \(1300 \mathrm{~km} / \mathrm{h}\) 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 \(6.0 \mathrm{~g}^{\prime}\) 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).

A projected space station consists of a circular tube that will rotate about its center (like a tubular bicycle tire) (Fig. ). 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?

Problem 74GP Consider a train that rounds a curve with a radius of 570 m at a speed of 160 kmih (approximately 100 miih ).(,?) Calculate the friction force needed on a train passenger of mass 55 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 at an angle of 8.0° toward the center of the curve.

Problem 75GP Two equal-mass stars maintain a constant distance apart of 8.0 % 1011 m and revolve 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 76GP How far above the Earth's surface will the acceleration of gravity be half what it is at the surface?

Two equal-mass stars maintain a constant distance apart of \(8.0 \times 10^{10} \mathrm{~m}\) 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?

A train traveling at a constant speed rounds a curve of radius \(235 \mathrm{~m}\). A lamp suspended from the ceiling swings out to an angle of \(17.5^{\circ}\) throughout the curve. What is the speed of the train?

Problem 82GP The Near Earth Asteroid Rendezvous {NEAR) spacecraft, after traveling 2.1 billion km, is meant to orbit the asteroid Eros with an orbital radius of about 20 km. Eros is roughly 40 km % 6 km x 6 km. Assume Eros has a density of about 2.3 x 103 kg/m3. (a) If Eros were a sphere with the same mass and density, what would its radius be? (b) What would g be at the surface of a spherical Eros? (c) Estimate the orbital period of NEAR as it orbits Eros, as if Eros were a sphere.

The Near Earth Asteroid Rendezvous (NEAR), after traveling billion , is meant to orbit the asteroid Eros at a height of about . Eros is roughly 40 \mathrm{~km} \times \(6 \mathrm{~km} \times 6 \mathrm{~km}\). Assume Eros has a density (mass/volume) of about \(2.3 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). (a) What will be the period of as it orbits Eros? ( ) If Eros were a sphere with the same mass and density, what would its radius be? What would be at the surface of this spherical Eros? Equation Transcription: Text Transcription: 40 km x 6 km x 6 km 2.3 x 103 kg/m3

Problem 85GP A satellite of mass 5500 kg orbits the Earth and has a period of 6600 s. Determine (a) the radius of its circular orbit, (£) the magnitude of the Earth's gravitational force on the satellite, and (c) the attitude of the satellite.

Problem 86GP Astronomers using the Hubble Space Telescope deduced the presence of an extremely massive core in the distant ga/axy 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 (= 5.7x 1017 m) from the core. Deduce the mass of the core, and compare it to the mass of our Sun.

Problem 87GP Suppose all the mass of the Earth were compacted into a small spherical ball. What radius must the sphere have so that the acceleration due to gravity at the Earth's new surface would equal the acceleration due to gravity at the surface of the Sun?

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 \(\left(1 / y=9.5 \times 10^{15} \mathrm{~m}\right)\). 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 \(\left(2 \times 10^{30} \mathrm{~kg}\right)\), how many stars would there be in our Galaxy? Equation Transcription: Text Transcription: ( 1/y=9.5 x 1015 m) ( 2 x 1030 kg)

Problem 89GP An asteroid of mass m is in a circular orbit of radius r around the Sun with a speed v. It has an impact with another asteroid of mass M and is kicked into a new circular orbit with a speed of1.5 v. What is the radius of the new orbit in terms of r?

A satellite of mass orbits the Earth \(\left(\text { mass }=6.0 \times 10^{24} \mathrm{~kg}\right)\) and has a period of . Find the magnitude of the Earth's gravitational force on the satellite, the altitude of the satellite. Equation Transcription: Text Transcription: (mass = 6.0 x 1024 kg)

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

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

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.

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 \(8.0^\circ\) toward the center of the curve.