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.

A \(52 \mathrm{~kg}\) ice skater spins about a vertical axis through her body with her arms horizontally outstretched, 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.

I A flat (unbanked) curve on a highway has a radius of \(220 \mathrm{~m}\). A car successfully rounds the curve at a speed of \(35 \mathrm{~m} / \mathrm{s}\) but is on the verge of skidding out. (a) If the coefficient of static friction between the car's tires and the road surface were reduced by a factor of \(2,\) with what maximum speed could the car round the curve? (b) Suppose the coefficient of friction were increased by a factor of 2; what would be the maximum speed?

The "Giant Swing" at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. (See Figure 6.28.) Each arm supports a seat suspended from a 5.00-m-long rod, the upper end of which is fastened to the arm at a point \(3.00 \mathrm{~m}\) from the central shaft. (a) Make a free-body diagram of the seat, including the person in it. (b) Find the time of one revolution of the swing if the rod supporting the seat makes an angle of \(30.0^{\circ}\) with the vertical. (c) Does the angle depend on the weight of the passenger for a given rate of revolution?

A small button placed on a horizontal rotating platform with diameter \(0.320 \mathrm{~m}\) will revolve with the platform when it is brought up to a speed of 40.0 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 \(60.0 \mathrm{rev} / \mathrm{min} ?\)

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?

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.

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 \(9.50 \mathrm{~m} / \mathrm{s}\) what should the minimum radius of the circle be in order for the cen- tripetal 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?

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 \(2.5 g\) '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 \(2150 \mathrm{~kg}\) 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?

At a distance \(N \times R_{\mathrm{E}}\) from the earth's surface, where \(N\) is an integer, the gravitational force on an object is only \(1 / 25\) of its value at the earth's surface. What is \(N ?\)

An \(8.00 \mathrm{~kg}\) point mass and a \(15.0 \mathrm{~kg}\) point mass are held in place \(50.0 \mathrm{~cm}\) apart in deep space. They are both released at the same time. (a) Find the magnitude and direction of the acceleration of each particle the moment after they are released. (b) As the particles begin to move, is their acceleration constant? Explain.

Your spaceship lands on an unknown planet. To determine the characteristics of this planet, you drop a wrench from \(5.00 \mathrm{~m}\) above the ground and measure that it hits the ground 0.811 s later. (a) What is the acceleration of gravity near the surface of this planet? (b) Assuming that the planet has the same density as that of earth \(\left(5500 \mathrm{~kg} / \mathrm{m}^{3}\right),\) what is the radius of the planet?

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?

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 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 243 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 243 Ida. (b) What would an astronaut whose earth weight is \(650 \mathrm{~N}\) weigh on 243 Ida? (c) If you dropped a rock from a height of \(1.0 \mathrm{~m}\) on 243 Ida, how long would it take for the rock to reach the ground? (d) If you can jump \(60 \mathrm{~cm}\) straight up on earth, how high could you jump on 243 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)?

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 parts (a) and (b), are the astronauts weightless because gravity is so weak? Why are they apparently weightless?

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 \(g ?\)

Consider the fact that an object sitting at the equator is undergoing uniform circular motion due to the rotation of the earth. (a) Draw a free-body diagram for a box of mass \(m\) that is placed at the equator. (b) Now suppose that we allow the day to become shorter than 24 h. Use Newton's second law to calculate the maximum speed, due to the earth's rotation, that the box can have and still remain on the surface. (c) How long would this day be in hours?

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as \(500 \mathrm{~km}\) (or even higher) above the surface. Io has a mass of \(8.94 \times 10^{22} \mathrm{~kg}\) and a radius of \(1815 \mathrm{~km}\). Ignore any variation in gravity over the \(500 \mathrm{~km}\) range of the debris. How high would this material go on earth if it were ejected with the same speed as on lo?

A \(1125 \mathrm{~kg}\) car and a \(2250 \mathrm{~kg}\) pickup truck approach a curve on the expressway that has a radius of \(225 \mathrm{~m}\). (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 65.0 mph can safely round it regardless of the condition of their tires? Should the heavy truck go slower than the lighter car? (b) As the car and truck round the curve at \(65.0 \mathrm{mph}\), find the normal force on each one due to the highway surface.

Europa, a satellite of Jupiter, is believed to have an ocean of liquid water (with the possibility of life) beneath its icy surface. (See Figure \(6.31 .)\) Europa is \(3130 \mathrm{~km}\) in diameter and has a mass of \(4.78 \times 10^{22} \mathrm{~kg} .\) In the future, we will surely want to send astronauts to investigate Europa. In planning such a future mission, what is the fastest that such an astronaut could walk on the surface of Europa if her legs are \(1.0 \mathrm{~m}\) long?

The star Rho' 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 \({ }^{1}\) 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 \(^{1}\) Cancri?

As your bus rounds a flat curve at constant speed of \(20 \mathrm{~m} / \mathrm{s}\) 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 \(30.0^{8}\) with the vertical. (a) What is the radial acceleration of the bus? (b) What is the radius of the curve?

Observations of this planet over time show that it is in a nearly circular orbit around its star and completes one orbit in only 9.5 days. How many times the orbital radius \(r\) of the earth around our sun is this exoplanet's orbital radius around its sun? Assume that the earth is also in a nearly circular orbit. A. \(0.026 r\) B. \(0.078 r\) C. \(0.70 r\) D. \(2.3 r\)