By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m2 at a distance of 4.3 m from the source. (a) What is the intensity at a distance of 3.1 m from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?
> In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (a) How far are these clumps from the center of the black hole? (b) What
> Two uniform spheres, each with mass M and radius R, touch each other. What is the magnitude of their gravitational force of attraction?
> Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 13.8). A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. (a) Determi
> Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be 1.0 × 10-15 m, what would be the mass of a mini black hole?
> The acceleration due to gravity at the north pole of Neptune is approximately 11.2 m>s2. Neptune has mass 1.02 × 1026 kg and radius 2.46 × 104 km and rotates once around its axis in about 16 h. (a) What is the gravitational force on a 3.00-kg object at t
> You are captured by Martians, taken into their ship, and put to sleep. You awake some time later and find yourself locked in a small room with no windows. All the Martians have left you with is your digital watch, your school ring, and your long silverch
> You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of 395.0 N in the rope. If Sneezy hangs fr
> Consider the ring- shaped body of Fig. E13.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of
> A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is placed a distance x from one end of the rod, along the axis of the rod (Fig. E13.34). (a) Calculate the gravitational potential energy of the rod–sphere s
> A uniform, solid, 1000.0-kg sphere has a radius of 5.00 m. (a) Find the gravitational force this sphere exerts on a 2.00-kg point mass placed at the following distances from the center of the sphere: (i) 5.01 m, (ii) 2.50 m. (b) Sketch a qualitative grap
> A uniform, spherical, 1000.0-kg shell has a radius of 5.00 m. (a) Find the gravitational force this shell exerts on a 2.00-kg point mass placed at the following distances from the center of the shell: (i) 5.01 m, (ii) 4.99 m, (iii) 2.72 m. (b) Sketch a q
> On October 15, 2001, a planet was discovered orbiting around the star HD 68988. Its orbital distance was measured to be 10.5 million kilometers from the center of the star, and its orbital period was estimated at 6.3 days. What is the mass of HD 68988? E
> In 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term “hot Jupiter”). The orbit was just 1 9 the distance of Mercury from our sun, and it takes the planet only 3.09 days to ma
> 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 kg and the other a mass of 72 kg, and they start from rest 20.0 m apart. (a) Make a free-body diagram of each
> The dwarf planet Pluto has an elliptical orbit with a semimajor axis of 5.91 × 1012 m and eccentricity 0.249. (a) Calculate Pluto’s orbital period. Express your answer in seconds and in earth years. (b) During Pluto’s orbit around the sun, what are its c
> In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at 19,600 km with an orbital period of 6.39 days. Assuming t
> Two identical gliders on an air track are connected by an ideal spring. Could such a system undergo SHM? Explain. How would the period compare with that of a single glider attached to a spring whose other end is rigidly attached to a stationary object? E
> A spaceship makes a circular orbit with period T around a star. If it were to orbit, at the same distance, a star with three times the mass of the original star, would the new period (in terms of T) be (a) 3T, (b) T√3 (c) T, (d) T/√3, or (e) T/3?
> The star Rho1 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 Rho1 Cancri with an orbital radius equal to 0.11 times the radius of the earth’s orbit around the sun.
> Suppose that a planet was discovered between the sun and Mercury, with a circular orbit of radius equal to 2 3 of the average orbit radius of Mercury. What would be the orbital period of such a planet? (Such a planet was once postulated, in part to expla
> Deimos, a moon of Mars, is about 12 km in diameter with mass 1.5 × 1015 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! (a) With what speed would you have
> In its orbit each day, the International Space Station makes 15.65 revolutions around the earth. Assuming a circular orbit, how high is this satellite above the surface of the earth?
> Two satellites are in circular orbits around a planet that has radius 9.00 × 106m. One satellite has mass 68.0 kg, orbital radius 7.00 × 107m, and orbital speed 4800 m/s. The second satellite has mass 84.0 kg and orbital radius 3.00 × 107m. What is the
> On July 15, 2004, NASA launched the Aura spacecraft to study the earth’s climate and atmosphere. This satellite was injected into an orbit 705 km above the earth’s surface. Assume a circular orbit. (a) How many hours does it take this satellite to make o
> For a satellite to be in a circular orbit 890 km above the surface of the earth, (a) what orbital speed must it be given, and (b) what is the period of the orbit (in hours)?
> An earth satellite moves in a circular orbit with an orbital speed of 6200 m/s. Find (a) the time of one revolution of the satellite; (b) the radial acceleration of the satellite in its orbit.
> In the Cavendish balance apparatus shown in Fig. 13.4, suppose that m1 = 1.10 kg, m2 = 25.0 kg, and the rod connecting the m1 pairs is 30.0 cm long. If, in each pair, m1 and m2 are 12.0 cm apart center to center, find (a) the net force and (b) the net to
> A planet orbiting a distant star has radius 3.24 × 106 m. The escape speed for an object launched from this planet’s surface is 7.65 × 103 m/s. What is the acceleration due to gravity at the surface of the planet?
> A glider is attached to a fixed ideal spring and oscillates on a horizontal, friction-free air track. A coin rests atop the glider and oscillates with it. At what points in the motion is the friction force on the coin greatest? The least? Justify your an
> Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 × 106 km from the earth and traveling at 1.20 × 104 km/h relative to the earth. At this time, what were (a) the spacecraft’s kinetic e
> Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the sp
> 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 km (or even higher) above the surface. Io has a mass of 8.93 × 1022 kg and a radius of 1821 km. For this calc
> Calculate the earth’s gravity force on a 75-kg astronaut who is repairing the Hubble Space Telescope 600 km above the earth’s surface, and then compare this value with his weight at the earth’s surface. In view of your result, explain why it is said that
> Rhea, one of Saturn’s moons, has a radius of 764 km and an acceleration due to gravity of 0.265 m/s2 at its surface. Calculate its mass and average density.
> Titania, the largest moon of the planet Uranus, has 1/8 the radius of the earth and 1/1700 the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the average density of Titania? (This is less than the de
> The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. (a) Compute the acceleration due to gravity on the surface of Venus from these data. (b) If a rock weighs 75.0 N on earth, what would it weigh at the surface of Venu
> At what distance above the surface of the earth is the acceleration due to the earth’s gravity 0.980 m/s2 if the acceleration due to gravity at the surface has magnitude 9.80 m/s2?
> The point masses m and 2m lie along the x-axis, with m at the origin and 2m at x = L. A third point mass M is moved along the x-axis. (a) At what point is the net gravitational force on M due to the other two masses equal to zero? (b) Sketch the x-compon
> What is the ratio of the gravitational pull of the sun on the moon to that of the earth on the moon? (Assume the distance of the moon from the sun can be approximated by the distance of the earth from the sun.) Use the data in Appendix F. Is it more accu
> A box containing a pebble is attached to an ideal horizontal spring and is oscillating on a friction-free air table. When the box has reached its maximum distance from the equilibrium point, the pebble is suddenly lifted out vertically without disturbing
> One of the 63.5-cm-long strings of an ordinary guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by 1.0%
> (a) A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed v, frequency f, amplitude A, and wavelength l. Calculate the maximum transverse velocity and maximum transverse acceleration of points located
> The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is 60.0 cm long, and this length of the string has mass 2.00 g. The string sounds an A4 note
> One string of a certain musical instrument is 75.0 cm long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 m/s. (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produ
> A flexible stick 2.0 m long is not fixed in any way and is free to vibrate. Make clear drawings of this stick vibrating in its first three harmonics, and then use your drawings to find the wavelengths of each of these harmonics.
> The wave function of a standing wave is y(x, t) = 4.44 mm sin[(32.5 rad/m)x] sin[(754 rad/s)t]. For the two traveling waves that make up this standing wave, find the (a) amplitude; (b) wavelength; (c) frequency; (d) wave speed; (e) wave functions. (f) Fr
> A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation y(x, t) = (5.60 cm) sin [(0.0340 rad / cm)x] sin [(50.0 rad / s)t], where the origin is at the left end of the string, the x-axis is along
> At a distance of 7.00 × 1012 m from a star, the intensity of the radiation from the star is 15.4 W/m2. Assuming that the star radiates uniformly in all directions, what is the total power output of the star?
> A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t) = 2.30 mm cos[(6.98 rad/m)x +(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.003
> For a simple pendulum, clearly distinguish between v (the angular speed) and v (the angular frequency). Which is constant and which is variable?
> You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore
> A jet plane at takeoff can produce sound of intensity 10.0 W/m2 at 30.0 m away. But you prefer the tranquil sound of normal conversation, which is 1.0
> A light wire is tightly stretched with tension F. Transverse traveling waves of amplitude A and wavelength l1 carry average power Pav,1 = 0.400 W. If the wavelength of the waves is doubled, so
> A horizontal wire is stretched with a tension of 94.0 N, and the speed of transverse waves for the wire is 406 m/s. What must the amplitude of a traveling wave of frequency 69.0 Hz be for the average power carried by the wave to be 0.365 W?
> A piano wire with mass 3.00 g and length 80.0 cm is stretched with a tension of 25.0 N. A wave with frequency 120.0 Hz and amplitude 1.6 mm travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power
> A simple harmonic oscillator at the point x = 0 generates a wave on a rope. The oscillator operates at a frequency of 40.0 Hz and with an amplitude of 3.00 cm. The rope has a linear mass density of 50.0 g/m and is stretched with a tension of 5.00 N. (a)
> A heavy rope 6.00 m long and weighing 29.4 N is attached at one end to a ceiling and hangs vertically. A 0.500-kg mass is suspended from the lower end of the rope. What is the speed of transverse waves on the rope at the (a) bottom of the rope, (b) middl
> A thin, 75.0-cm wire has a mass of 16.5 g. One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire. (a) To what tension (in newtons) must you adjust the screw so that a transverse wave of w
> A 1.50-m string of weight 0.0125 N is tied to the ceiling at its upper end, and the lower end supports a weight W. Ignore the very small variation in tension along the length of the string that is produced by the weight of the string. When you pluck the
> The upper end of a 3.80-m-long steel wire is fastened to the ceiling, and a 54.0-kg object is suspended from the lower end of the wire. You observe that it takes a transverse pulse 0.0492 s to travel from the bottom to the top of the wire. What is the ma
> When the amplitude of a simple pendulum increases, should its period increase or decrease? Give a qualitative argument; do not rely on Eq. (14.35). Is your argument also valid for a physical pendulum?
> With what tension must a rope with length 2.50 m and mass 0.120 kg be stretched for transverse waves of frequency 40.0 Hz to have a wavelength of 0.750 m?
> One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at 120 Hz. The other end passes over a pulley and supports a 1.50-kg mass. The linear mass density of the rope is 0.0480 kg/m. (
> A wave on a string is described by y1x, t2 = A cos (kx – vt). (a) Graph y, vy, and ay as functions of x for time t = 0. (b) Consider the following points on the string: (i) x = 0; (ii) x = p/4k; (iii) x = p/2k; (iv) x = 3p/4k; (v) x = p/k; (vi) x = 5p/4k
> A transverse wave on a string has amplitude 0.300 cm, wavelength 12.0 cm, and speed 6.00 cm/s. It is represented by y(x, t) as given in Exercise 15.12. (a) At time t = 0, compute y at 1.5-cm intervals of x (that is, at x = 0, x = 1.5 cm, x = 3.0 cm, and
> (a) Show that Eq. (15.3) may be written as (b) Use y(x, t) to find an expression for the transverse velocity vy of a particle in the string on which the wave travels. (c) Find the maximum speed of a particle of the string. Under what circumstances is t
> A sinusoidal wave is propagating along a stretched string that lies along the x-axis. The displacement of the string as a function of time is graphed in Fig. E15.11 for particles at x = 0 and at x = 0.0900 m. (a) What is the amplitude of the wave? (b) Wh
> A water wave traveling in a straight line on a lake is described by the equation where y is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattern to go past a fisherman in a
> Which of the following wave functions satisfies the wave equation, Eq. (15.12)? (a) y(x, t)= A cos (kx + vt); (b) y(x, t)= A sin(kx + vt); (c) y(x, t)= A(cos kx + cos vt). (d) For the wave of part (b), write the equations for the transverse velocity and
> A certain transverse wave is described by Determine the wave’s (a) amplitude; (b) wavelength; (c) frequency; (d) speed of propagation; (e) direction of propagation. У(х, 1) %3D (6.50 mm) cos 2m( 28.0 cm 0.0360 s
> Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and wavelength 0.320 m. The waves travel in the −x-direction, and at t = 0 the x = 0 end of the string has its maximum upward displacement. (a) Find the frequency, period, and wav
> A simple pendulum is mounted in an elevator. What happens to the period of the pendulum (does it increase, decrease, or remain the same) if the elevator (a) accelerates upward at 5.0 m/s2 ; (b) moves upward at a steady 5.0 m/s ; (c) accelerates downward
> A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total distance of 0.53 m. The fisherman sees that the wave cre
> (a) Audible wavelengths The range of audible frequencies is from about 20 Hz to 20,000 Hz. What is the range of the wavelengths of audible sound in air? (b) Visible light. The range of visible light extends from 380 nm to 750 nm. What is the range of vis
> Sound having frequencies above the range of human hearing (about 20,000 Hz) is called ultrasound. Waves above this frequency can be used to penetrate the body and to produce images by reflecting from surfaces. In a typical ultrasound scan, the waves trav
> On December 26, 2004, a great earthquake occurred off the coast of Sumatra and triggered immense waves (tsunami) that killed some 200,000 people. Satellites observing these waves from space measured 800 km from one wave crest to the next and a period bet
> Provided the amplitude is sufficiently great, the human ear can respond to longitudinal waves over a range of frequencies from about 20.0 Hz to about 20.0 kHz. (a) If you were to mark the beginning of each complete wave pattern with a red dot for the lon
> The speed of sound in air at 20°C is 344 m/s. (a) What is the wavelength of a sound wave with a frequency of 784 Hz, corresponding to the note G5 on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave o
> By what percentage does the frequency of oscillation change if ksurf = 5 N/m? (a) 0.1%; (b) 0.2%; (c) 0.5%; (d) 1.0%.
> In the model of Problem 14.94, what is the mechanical energy of the vibration when the tip is not interacting with the surface? (a) 1.2 × 10-18 J; (b) 1.2 × 10-16 J; (c) 1.2 × 10-9 J; (d) 5.0 × 10-8 J. Data from Problem 14.94: If we model the vibrating
> If we model the vibrating system as a mass on a spring, what is the mass necessary to achieve the desired resonant frequency when the tip is not interacting with the surface? (a) 25 ng; (b) 100 ng; (c) 2.5 mg; (d) 100 mg.
> 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?
> If a pendulum has a period of 2.5 s on earth, what would be its period in a space station orbiting the earth? If a mass hung from a vertical spring has a period of 5.0 s on earth, what would its period be in the space station? Justify your answers.
> How many times the acceleration due to gravity g near the earth’s surface is the acceleration due to gravity near the surface of this exoplanet? (a) About 0.29g; (b) about 0.65g; (c) about 1.5g; (d) about 7.9g.
> Based on these data, what is the most likely composition of this planet? (a) Mostly iron; (b) iron and rock; (c) iron and rock with some lighter elements; (d) hydrogen and helium gases.
> The wave speed is measured for different vibration frequencies. A graph of the wave speed as a function of frequency (Fig. P15.80) indicates that as the frequency increases, the wavelength (a) increases; (b) decreases; (c) doesn’t chang
> Which of these is a possible mathematical description of the wave in Problem 15.78? Data from Problem 15.78: What is the wavelength of the wave that travels on the surface of the vocal folds when they are vibrating at frequency f? (a) 2.0 mm; (b) 3.3 m
> What is the wavelength of the wave that travels on the surface of the vocal folds when they are vibrating at frequency f? (a) 2.0 mm; (b) 3.3 mm; (c) 0.50 cm; (d) 3.0 cm.
> (a) Calculate how much work is required to launch a spacecraft of mass m from the surface of the earth (mass mE, radius RE) and place it in a circular low earth orbit—that is, an orbit whose altitude above the earth’s surface is much less than RE. (As an
> A hammer with mass m is dropped from rest from a height h above the earth’s surface. This height is not necessarily small compared with the radius RE of the earth. Ignoring air resistance, derive an expression for the speed y of the hammer when it reache
> Some scientists are eager to send a remote-controlled submarine to Jupiter’s moon Europa to search for life in its oceans below an icy crust. Europa’s mass has been measured to be 4.80 × 1022 kg, its diameter is 3120 km, and it has no appreciable atmosph
> Many satellites are moving in a circle in the earth’s equatorial plane. They are at such a height above the earth’s surface that they always remain above the same point. (a) Find the altitude of these satellites above the earth’s surface. (Such an orbit
> At a certain instant, the earth, the moon, and a stationary 1250-kg spacecraft lie at the vertices of an equilateral triangle whose sides are 3.84 × 105 km in length. (a) Find the magnitude and direction of the net gravitational force exerted on the spac
> Many people believe that orbiting astronauts feel weightless because they are “beyond the pull of the earth’s gravity.” How far from the earth would a spacecraft have to travel to be truly beyond the earth’s gravitational influence? If a spacecraft were
> For a spherical planet with mass M, volume V, and radius R, derive an expression for the acceleration due to gravity at the planet’s surface, g, in terms of the average density of the planet, r = M/V, and the planet’s
> Experimenting with pendulums, you attach a light string to the ceiling and attach a small metal sphere to the lower end of the string. When you displace the sphere 2.00 m to the left, it nearly touches a vertical wall; with the string taut, you release t
> You hang various masses m from the end of a vertical, 0.250-kg spring that obeys Hooke’s law and is tapered, which means the diameter changes along the length of the spring. Since the mass of the spring is not negligible, you must replace m in the equati
> A mass m is attached to a spring of force constant 75 N/m and allowed to oscillate. Figure P14.89 shows a graph of its velocity component vx as a function of time t. Find (a) the period, (b) the frequency, and (c) the angular frequency of this motion. (d
> Two identical thin rods, each with mass m and length L, are joined at right angles to form an L-shaped object. This object is balanced on top of a sharp edge (Fig. P14.88). If the L-shaped object is deflected slightly, it oscillates. Find the frequency o
> A slender, uniform, metal rod with mass M is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant k is attached to the lower end of the rod, with the other end of the spring att
