Chapter 6

A stone with a mass of 0.80 \(\mathrm{kg}\) is attached to one end of a string 0.90 \(\mathrm{m}\) long. The string will break if its tension exceeds 60.0 \(\mathrm{N} .\) The stone is whirled in a horizontal circle on a frictionless tabletop; the other end of the string remains fixed. (a) Make a free-body diagram of the stone. (b) Find the maximum speed the stone can attain without breaking the string.

Force on a skater's wrist. A 52 \(\mathrm{kg}\) ice skater spins about a vertical axis through her body with her arms horizontally out-stretched, making 2.0 turns each second. The distance from one hand to the other is 1.50 \(\mathrm{m} .\) Biometric measurements indicate that each hand typically makes up about 1.25\(\%\) of body weight. (a) Draw a free-body diagram of one of her hands. (b) What horizontal force must her wrist exert on her hand? (c) Express the force in part (b) as a multiple of the weight of her hand.

A flat (unbanked) curve on a highway has a radius of 220 \(\mathrm{m}\) .A car rounds the curve at a speed of 25.0 \(\mathrm{m} / \mathrm{s}\) . (a) Make a free body diagram of the car as it rounds this curve. (b) What is the minimum coefficient of friction that will prevent sliding?

A small button placed on a horizontal rotating platform with diameter 0.320 \(\mathrm{m}\) will revolve with the plattorm when it is brought up to a speed of 40.0 \(\mathrm{rev} / \mathrm{min}\) , provided the button is no more than 0.150 \(\mathrm{m}\) from the axis. (a) What is the coefficient of static friction between the button and the platform? (b) How far from the axis can the button be placed, without slipping, if the platform rotates at 60.0 rev/min?

A highway curve with radius 900.0 \(\mathrm{ft}\) is to be banked so that a car traveling 55.0 mph will not skid sideways even in the absence of friction. (a) Make a free-body diagram of this car. (b) At what angle should the curve be banked?

The Indy 500 . The Indianapolis Speedway (home of the Indy 500 ) consists of a 2.5 mile track having four turns, each 0.25 mile long and banked at \(9^{\circ} 12^{\prime} .\) What is the no-friction-needed speed (in \(\mathrm{m} / \mathrm{s}\) and mph) for these turns? (Do you think drivers actually take the turns at that speed?

A bowling ball weighing 71.2 \(\mathrm{N}\) is attached to the ceiling by a 3.80 \(\mathrm{m}\) rope. The ball is pulled to one side and released; it then swings back and forth like a pendulum. As the rope swings through its lowest point, the speed of the bowling ball is measured at 4.20 \(\mathrm{m} / \mathrm{s}\) . At that instant, find (a) the magnitude and direction of the acceleration of the bowling ball and (b) the tension in the rope. Be sure to start with a free-body diagram.

\(\bullet\) The Cosmoclock 21 Ferris wheel in Yokohama City, Japan, has a diameter of 100 \(\mathrm{m}\) . Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 \(\mathrm{s} )\) . (a) Find the speed of the passengers when the Ferris wheel is rotating at this rate. (b) A passenger weighs 882 \(\mathrm{N}\) at the weight- guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel? (c) What would be the time for one revolution if the passenger's apparent weight at the highest point were zero? (d) What then would be the passenger's apparent weight at the lowest point?

A \(\mathrm{A} 50.0 \mathrm{kg}\) stunt pilot who has been diving her airplane vertically pulls out of the dive by changing her course to a circle in a vertical plane. (a) If the plane's speed at the lowest point of the circle is \(95.0 \mathrm{m} / \mathrm{s},\) what should the minimum radius of the circle be in order for the centripetal acceleration at this point not to exceed 4.00 \(\mathrm{g}\) ? (b) What is the apparent weight of the pilot at the lowest point of the pullout?

Effect on blood of walking. While a person is walking, his arms swing through approximately a \(45^{\circ}\) angle in \(\frac{1}{2}\) s. As a reasonable approximation, we can assume that the arm moves with constant speed during each swing. A typical arm is 70.0 \(\mathrm{cm}\) long, measured from the shoulder joint. (a) What is the acceleration of a 1.0 gram drop of blood in the fingertips at the bottom of the swing? (b) Make a free-body diagram of the drop of blood in part (a).(c) Find the force that the blood vessel must exert on the drop of blood in part (b). Which way does this force point? (d) What force would the blood vessel exert if the arm were not swinging?

Stay dry! You tie a cord to a pail of water, and you swing the pail in a vertical circle of radius 0.600 \(\mathrm{m} .\) What minimum speed must you give the pail at the highest point of the circle if no water is to spill from it? Start with a free-body diagram of the water at its highest point.

Stunt pilots and fighter pilots who fly at high speeds in a downward-curving arc may experience a "red out," in which blood is forced upward into the flier's head, potentially swelling or breaking capillaries in the eyes and leading to a reddening of vision and even loss of consciousness. This effect can occur at centripetal accelerations of about 2.5\(g^{\prime}\) s. For a stunt plane flying at a speed of 320 \(\mathrm{km} / \mathrm{h}\) , what is the minimum radius of downward curve a pilot can achieve without experiencing a red out at the top of the arc? (Hint: Remember that gravity provides part of the centripetal acceleration at the top of the arc; it's the acceleration required in excess of gravity that causes this problem.)

Rendezvous in space! A couple of astronauts agree to rendezvous in space after hours. Their plan is to let gravity bring them together. One of them has a mass of 65 \(\mathrm{kg}\) and the other a mass of \(72 \mathrm{kg},\) and they start from rest 20.0 \(\mathrm{m}\) apart. (a) Make a free-body diagram of each astronaut, and use it to find his or her initial acceleration. As a rough approximation, we can model the astronauts as uniform spheres. (b) If the astronauts' acceleration remained constant, how many days would they have to wait before reaching each other? (Careful! They both have acceleration toward each other.) (c) Would their acceleration, in fact, remain constant? If not, would it increase or decrease? Why?

A 2150 kg satellite used in a cellular telephone network is in a circular orbit at a height of 780 \(\mathrm{km}\) above the surface of the earth. What is the gravitational force on the satellite? What fraction is this force of the satellite's weight at the surface of the earth?

An 8.00 -kg point mass and a 15.0 -kg point mass are held in place 50.0 \(\mathrm{cm}\) apart. A particle of mass \(m\) is released from a point between the two masses 20.0 \(\mathrm{cm}\) from the 8.00 -kg mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.

Your spaceship lands on an unknown planet. To determine the characteristics of this planet, you drop a 1.30 \(\mathrm{kg}\) wrench from 5.00 \(\mathrm{m}\) above the ground and measure that it hits the ground 0.811 s later. You also do enough surveying to determine that the circumference of the planet is \(62,400 \mathrm{km}\) . (a) What is the mass of the planet, in kilograms? (b) Express the planet's mass in terms of the earth's mass.

If an object's weight is \(W\) on the earth, what would be its weight (in terms of \(W )\) if the earth had (a) twice its present mass, but was the same size, (b) half its present radius, but the same mass, (c) half its present radius and half its present mass, (d) twice its present radius and twice its present mass?

Huygens probe on Titan. In January 2005 the Huygens probe landed on Saturn's moon Titan, the only satellite in the solar system having a thick atmosphere. Titan's diameter is \(5150 \mathrm{km},\) and its mass is \(1.35 \times 10^{23} \mathrm{kg}\) . The probe weighed 3120 \(\mathrm{N}\) on the earth. What did it weigh on the surface of Titan?

The The mass of the moon is about 1\(/ 81\) the mass of the earth, its radius is \(\frac{1}{4}\) that of the earth, and the acceleration due to gravity at the earth's surface is 9.80 \(\mathrm{m} / \mathrm{s}^{2}\) . Without looking up either body's mass, use this information to compute the acceleration due to gravity on the moon's surface.

Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun, but a much smaller diameter. If you weigh 675 \(\mathrm{N}\) on the earth, what would you weigh on the surface of a neutron star that has the same mass as our sun and a diameter of 20.0 \(\mathrm{km} ?\)

The asteroid 234 Ida has a mass of about \(4.0 \times 10^{16} \mathrm{kg}\) and an average radius of about 16 \(\mathrm{km}\) (it's not spherical, but you can assume it is). (a) Calculate the acceleration of gravity on 234 Ida. (b) What would an astronaut whose earth weight is 650 \(\mathrm{N}\) weigh on 234 lda? (c) If you dropped a rock from a height of 1.0 \(\mathrm{m}\) on 234 Ida, how long would it take to reach the ground? (d) If you can jump 60 \(\mathrm{cm}\) straight up on earth, how high could you jump on 234 Ida? (Assume the asteroid's gravity doesn't weaken significantly over the distance of your jump.)

What is the period of revolution of a satellite with mass \(m\) that orbits the earth in a circular path of radius 7880 \(\mathrm{km}\) (about 1500 \(\mathrm{km}\) above the surface of the earth \() ?\)

In March \(2006,\) two small satellites were discovered orbiting Pluto, one at a distance of \(48,000 \mathrm{km}\) and the other at \(64,000 \mathrm{km} .\) Pluto already was known to have a large satellite Charon, orbiting at \(19,600 \mathrm{km}\) with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using the mass of Pluto.

Apparent weightlessness in a satellite. You have probably seen films of astronauts floating freely in orbiting satellites. People often think the astronauts are weightless because they are free of the gravity of the earth. Let us see if that explanation is correct. (a) Typically, such satellites orbit around 400 \(\mathrm{km}\) above the surface of the earth. If an astronaut weighs 750 \(\mathrm{N}\) on the ground, what will he weigh if he is 400 \(\mathrm{km}\) above the surface? (b) Draw the orbit of the satellite in part (a) to scale on a sketch of the earth. (c) In light of your answers to scale on a sketch of the earth. (c) In light of your answers to parts (a) and (b), are the astronauts weightless because gravity is so weak? Why are they apparently weightless?

Baseball on Deimos! Deimos! Deimos, a moon of Mars, is about 12 \(\mathrm{km}\) in diameter, with a mass of \(2.0 \times 10^{15} \mathrm{kg}\) . Suppose you are stranded alone on Deimos and want to play a one-person game of baseball. You would be the pitcher, and you would be the batter! With what speed would you have to throw a baseball so that it would go into orbit and return to you so you could hit it? Do you think you could actually throw it at that speed?

Artificial gravity. One way to create artificial gravity in a space station is to spin it. If a cylindrical space station 275 \(\mathrm{m}\) in diameter is to spin about its central axis, at how many revolutions per minute (rpm) must it turn so that the outermost points have an acceleration equal to \(g\) ?

You are driving with a friend who is sitting to your right on the passenger side of the front seat of your car. You would like to be closer to your friend, so you decide to use physics to achieve your romantic goal by making a quick turn. (a) Which way (to the left or the right) should you turn the car to get your friend to slide toward you? (b) If the coefficient of static friction between your friend and the car seat is 0.55 and you keep driving at a constant speed of \(15 \mathrm{m} / \mathrm{s},\) what is the maximum radius you could make your turn and still have your friend slide your way?

On the ride "Spindletop" at the amusement park Six Flags Over Texas, people stood against the inner wall of a hollow vertical cylinder with radius 2.5 \(\mathrm{m} .\) The cylinder started to rotate, and when it reached a constant rotation rate of \(0.60 \mathrm{rev} / \mathrm{s},\) the floor on which the people were standing dropped about 0.5 \(\mathrm{m}\) . The people remained pinned against the wall. (a) Draw a free-body diagram for a person on this ride after the floor has dropped. (b) What minimum coefficient of static friction is required if the person on the ride is not to slide downward to the new position of the floor? (c) Does your answer in part (b) depend on the mass of the passenger? (Note: When the ride is over, the cylinder is slowly brought to rest. As it slows down, people slide down the walls to the floor.)

Physical training. As part of a training program, an athlete runs while holding 8.0 kg weights in each hand. As he runs, the weights swing through a \(30.0^{\circ}\) arc in \(\frac{1}{3}\) s at essentially constant speed. His hands are 72 \(\mathrm{cm}\) from his shoulder joint, and they are light enough that we can neglect their weight compared with that of the 8.0 kg weight he is carrying. (a) Make a free body diagram of one of the 8.0 kg weights at the bottom of its swing. (b) What force does the runner's hand exert on each weight at the bottom of the swing?

The star Rho \(^{1}\) Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho I Cancri with an orbital radius equal to 0.11 times the radius of the earth's orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho "Cancri?

As your bus rounds a flat curve at constant speed, a package with mass \(0.500 \mathrm{kg},\) suspended from the luggage compartment of the bus by a string 45.0 \(\mathrm{cm}\) long, is found to hang at rest relative to the bus, with the string making an angle of \(30.0^{\circ}\) with the vertical. In this position, the package is 50.0 \(\mathrm{m}\) from the center of curvature of the curve. What is the speed of the bus?

Artificial gravity in space stations. One problem for humans living in outer space is that they are apparently weight- less. One way around this problem is to design a cylindrical space station that spins about an axis through its center at a constant rate. (See Figure \(6.32 . )\) This spin creates "artificial gravity" at the outside rim of the station. (a) If the diameter of the space station is \(800.0 \mathrm{m},\) how fast must the rim be moving in order for the "artificial gravity" acceleration to be \(g\) at the outer rim? (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface. How fast must the rim move in this case? (c) Make a free-body diagram of an astronaut at the outer rim.

BIO Weightlessness and artifial gravity. Astronauts who live under weightless (zero gravity) conditions for a prolonged time can experience health risks as a result. One way to avoid these adverse physiological effects is to provide an artificial gravity to simulate what is naturally experienced on the earth. In one design a space station is constructed in the shape of a long cylinder that spins at a constant rate about its longitudinal axis. Astronauts standing on the inside lateral surface of the cylinder experience a centripetal acceleration (due to their circular motion about the axis of the cylinder) that simulates the effect of gravity. The magnitude of the simulated gravity can be increased or decreased to the desired value by changing the rotation rate of the cylinder. If the diameter of the space station is \(1000 \mathrm{m},\) how fast must the outer edge of the space station move to give an astronaut the experience of a reduced "gravity" of 5 \(\mathrm{m} / \mathrm{s}^{2}\) (roughly 1\(/ 2\) earth normal)? You may assume that the astronaut is standing on the inner wall at a distance of nearly 1000 \(\mathrm{m}\) from the axis of the space station. A. 5000 \(\mathrm{m} / \mathrm{s}\) B. 2500 \(\mathrm{m} / \mathrm{s}\) C. 71 \(\mathrm{m} / \mathrm{s}\) D. 50 \(\mathrm{m} / \mathrm{s}\)