Two uniform solid spheres, each with mass M = 0.800 kg and radius R = 0.0800 m, are connected by a short, light rod that is along a diameter of each sphere and are at rest on a horizontal tabletop. A spring with force constant k = 160 N/m has one end attached to the wall and the other end attached to a frictionless ring that passes over the rod at the center of mass of the spheres, which is midway between the centers of the two spheres. The spheres are each pulled the same distance from the wall, stretching the spring, and released. There is sufficient friction between the tabletop and the spheres for the spheres to roll without slipping as they move back and forth on the end of the spring. Show that the motion of the center of mass of the spheres is simple harmonic and calculate the period.
> 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
> A large, 34.0-kg bell is hung from a wooden beam so it can swing back and forth with negligible friction. The bell’s center of mass is 0.60 m below the pivot. The bell’s moment of inertia about an axis at the pivot is 18.0 kg.m2. The clapper is a small,
> In Fig. P14.85 the upper ball is released from rest, collides with the stationary lower ball, and sticks to it. The strings are both 50.0 cm long. The upper ball has mass 2.00 kg, and it is initially 10.0 cm higher than the lower ball, which has mass 3.0
> A rifle bullet with mass 8.00 g and initial horizontal velocity 280 m/s strikes and embeds itself in a block with mass 0.992 kg that rests on a frictionless surface and is attached to one end of an ideal spring. The other end of the spring is attached to
> An object is moving with SHM of amplitude A on the end of a spring. If the amplitude is doubled, what happens to the total distance the object travels in one period? What happens to the period? What happens to the maximum speed of the object? Discuss how
> An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere
> While on a visit to Minnesota (“Land of 10,000 Lakes”), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the 1500-kg boat is tied, you find that the boat is bobbing up and down in the waves, executing simple
> A 40.0-N force stretches a vertical spring 0.250 m. (a) What mass must be suspended from the spring so that the system will oscillate with a period of 1.00 s? (b) If the amplitude of the motion is 0.050 m and the period is that specified in part (a), whe
> A spring of negligible mass and force constant k = 400 N/m is hung vertically, and a 0.200-kg pan is suspended from its lower end. A butcher drops a 2.2-kg steak onto the pan from a height of 0.40 m. The steak makes a totally inelastic collision with the
> A 0.0200-kg bolt moves with SHM that has an amplitude of 0.240 m and a period of 1.500 s. The displacement of the bolt is +0.240 m when t = 0. Compute (a) the displacement of the bolt when t = 0.500 s; (b) the magnitude and direction of the force acting
> A 5.00-kg partridge is suspended from a pear tree by an ideal spring of negligible mass. When the partridge is pulled down 0.100 m below its equilibrium position and released, it vibrates with a period of 4.20 s. (a) What is its speed as it passes throug
> A uniform beam is suspended horizontally by two identical vertical springs that are attached between the ceiling and each end of the beam. The beam has mass 225 kg, and a 175-kg sack of gravel sits on the middle of it. The beam is oscillating in SHM with
> A 2.00-kg bucket containing 10.0 kg of water is hanging from a vertical ideal spring of force constant 450 N/m and oscillating up and down with an amplitude of 3.00 cm. Suddenly the bucket springs a leak in the bottom such that water drops out at a stead
> An object with mass 0.200 kg is acted on by an elastic restoring force with force constant 10.0 N/m. (a) Graph elastic potential energy U as a function of displacement x over a range of x from -0.300 m to +0.300 m. On your graph, let 1 cm = 0.05 J vertic
> A square object of mass m is constructed of four identical uniform thin sticks, each of length L, attached together. This object is hung on a hook at its upper corner (Fig. P14.73). If it is rotated slightly to the left and then released, at what frequen
> What would Kepler’s third law be for circular orbits if an amendment to Newton’s law of gravitation made the gravitational force inversely proportional to r3? Would this change affect Kepler’s other two laws? Explain.
> An object with height h, mass M, and a uniform cross-sectional area A floats upright in a liquid with density r. (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force
> An apple weighs 1.00 N. When you hang it from the end of a long spring of force constant 1.50 N/m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency
> A 10.0-kg mass is traveling to the right with a speed of 2.00 m/s on a smooth horizontal surface when it collides with and sticks to a second 10.0-kg mass that is initially at rest but is attached to a light spring with force constant 170.0 N/m. (a) Find
> A 1.50-kg, horizontal, uniform tray is attached to a vertical ideal spring of force constant 185 N>m and a 275-g metal ball is in the tray. The spring is below the tray, so it can oscillate up and down. The tray is then pushed down to point A, which is 1
> A block with mass M rests on a frictionless surface and is connected to a horizontal spring of force constant k. The other end of the spring is attached to a wall (Fig. P14.68). A second block with mass m rests on top of the first block. The coefficient
> At the end of a ride at a winter-theme amusement park, a sleigh with mass 250 kg (including two passengers) slides without friction along a horizontal, snow-covered surface. The sleigh hits one end of a light horizontal spring that obeys Hooke’s law and
> Four passengers with combined mass 250 kg compress the springs of a car with worn-out shock absorbers by 4.00 cm when they get in. Model the car and passengers as a single body on a single ideal spring. If the loaded car has a period of vibration of 1.92
> An object is undergoing SHM with period 1.200 s and amplitude 0.600 m. At t = 0 the object is at x = 0 and is moving in the negative x-direction. How far is the object from the equilibrium position when t = 0.480 s ?
> An object is undergoing SHM with period 0.300 s and amplitude 6.00 cm. At t = 0 the object is instantaneously at rest at x = 6.00 cm. Calculate the time it takes the object to go from x = 6.00 cm to x = -1.50 cm.
> For each of the eight planets Mercury to Neptune, the semi-major axis a of their orbit and their or bital period T are as follows: (a) Explain why these values, when plotted as T2 versus a3, fall close to a straight line. Which of Keplerâ€&#
> A musical interval of an octave corresponds to a factor of 2 in frequency. By what factor must the tension in a guitar or violin string be increased to raise its pitch one octave? To raise it two octaves? Explain your reasoning. Is there any danger in at
> For transverse waves on a string, is the wave speed the same as the speed of any part of the string? Explain the difference between these two speeds. Which one is constant?
> What is the best explanation for the observation that the electric charge on the stem became positive as the charged bee approached (before it landed)? (a). Because air is a good conductor, the positive charge on the bee’s surface flowed through the air
> Consider a bee with the mean electric charge found in the experiment. This charge represents roughly how many missing electrons? (a). 1.9 × 108; (b). 3.0 × 108; (c). 1.9 × 1018; (d). 3.0 × 1018.
> Two thin rods of length L lie along the x-axis, one between x = 1 2 a and x = 1 2 a + L and the other between x = - 1 2 a and x = - 1 2 a - L. Each rod has positive charge Q distributed uniformly along its length. (a). Calculate the electric field p
> Two charges are placed as shown in Fig. P21.96. The magnitude of q1 is 3.00 µC, but its sign and the value of the charge q2 are not known. The direction of the net electric field
> Three charges are placed as shown in Fig. P21.95. The magnitude of q1 is 2.00 µC, but its sign and the value of the charge q2 are not known. Charge q3 is +4.00 µC, and the net force
> Positive charge Q is distributed uniformly around a very thin conducting ring of radius a, as in Fig. 21.23. You measure the electric field E at points on the ring axis, at a distance x from the center of the ring, over a wide range of values of x. Fig
> Two small spheres, each carrying a net positive charge, are separated by 0.400 m. You have been asked to perform measurements that will allow you to determine the charge on each sphere. You set up a coordinate system with one sphere (charge q1) at the or
> Inkjet printers can be described as either continuous or drop-on-demand. In a continuous inkjet printer, letters are built up by squirting drops of ink at the paper from a rapidly moving nozzle. You are part of an engineering group working on the design
> A thin disk with a circular hole at its center, called an annulus, has inner radius R1 and outer radius R2 (Fig. P21.91). The disk has a uniform positive surface charge density s on its surface. (a). Determine the total electric charge on the annulus.
> Two very large horizontal sheets are 4.25 cm apart and carry equal but opposite uniform surface charge densities of magnitude
> Suppose that the charge shown in Fig. 21.28a is fixed in position. A small, positively charged particle is then placed at some location and released. Will the trajectory of the particle follow an electric field line? Why or why not? Suppose instead that
> Repeat Problem 21.88 for the case where sheet B is positive. Problem 21.88: Two very large parallel sheets are 5.00 cm apart. Sheet A carries a uniform surface charge density of -8.80 µC/m2, and sheet B, which is to the right of A, carries a uniform ch
> Two very large parallel sheets are 5.00 cm apart. Sheet A carries a uniform surface charge density of -8.80 µC/m2, and sheet B, which is to the right of A, carries a uniform charge density of -11.6 µC/m2. Assume that the sheets are large enough to be tre
> Two 1.20-m nonconducting rods meet at a right angle. One rod carries +2.50 µC of charge distributed uniformly along its length, and the other carries -2.50 µC distributed uniformly along it (Fig. P21.87). Fig. P21.87: (a). F
> A semicircle of radius a is in the first and second quadrants, with the center of curvature at the origin. Positive charge +Q is distributed uniformly around the left half of the semicircle, and negative charge -Q is distributed uniformly around the righ
> Negative charge -Q is distributed uniformly around a quarter-circle of radius a that lies in the first quadrant, with the center of curvature at the origin. Find the x- and y-components of the net electric field at the origin.
> A small sphere with mass m carries a positive charge q and is attached to one end of a silk fiber of length L. The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density s. Show that when the sp
> A uniformly charged disk like the disk in Fig. 21.25 has radius 2.50 cm and carries a total charge of 7.0 × 10-12 C. Fig. 21.25: (a). Find the electric field (magnitude and direction) on the x-axis at x = 20.0 cm. (b). Show that for x &
> Positive charge Q is distributed uniformly along the positive y-axis between y = 0 and y = a. A negative point charge -q lies on the positive x-axis, a distance x from the origin (Fig. P21.82). Fig. P21.82: (a). Calculate the x- and y-components of t
> A negative point charge q1 = -4.00 nC is on the x-axis at x = 0.60 m. A second point charge q2 is on the x-axis at x = -1.20 m. What must the sign and magnitude of q2 be for the net electric field at the origin to be (a). 50.0 N/C in the +x-direction an
> In a region where there is a uniform electric field that is upward and has magnitude 3.60 × 104 N/C, a small object is projected upward with an initial speed of 1.92 m/s. The object travels upward a distance of 6.98 cm in 0.200 s. What is the object’s ch
> If the electric field of a point charge were proportional to 1/r3 instead of 1/r2, would Gauss’s law still be valid? Explain your reasoning. (Hint: Consider a spherical Gaussian surface centered on a single point charge.)
> Positive charge Q is distributed uniformly along the x-axis from x = 0 to x = a. A positive point charge q is located on the positive x-axis at x = a + r, a distance r to the right of the end of Q (Fig. P21.79). Fig. P21.79: (a). Calculate the x- and
> A small object with mass m, charge q, and initial speed v0 = 5.00 × 103 m/s is projected into a uniform electric field between two parallel metal plates of length 26.0 cm (Fig. P21.78). The electric field between the plates is directed downw
> A proton is projected into a uniform electric field that points vertically upward and has magnitude E. The initial velocity of the proton has a magnitude v0 and is directed at an angle
> The earth has a downward-directed electric field near its surface of about 150 N/C. If a raindrop with a diameter of 0.020 mm is suspended, motionless, in this field, how many excess electrons must it have on its surface?
