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physics
mechanics

Physics 2nd edition Alan Giambattista, Betty Richardson, Robert Richardson - Solutions

A grandfather clock is constructed so that it has a simple pendulum that swings from one side to the other, a distance of 20.0 mm, in 1.00 s. What is the maximum speed of the pendulum bob? Use two different methods. First, assume SHM and use the relationship between amplitude and maximum speed.
Christy has a grandfather clock with a pendulum that is 1.000 m long? (a) If the pendulum is modeled as a simple pendulum, what would be the period? (b) Christy observes the actual period of the clock, and finds that it is 1.00% faster than that for a simple pendulum that is 1.000 m long. If
A pendulum of length 120 cm swings with an amplitude of 2.0 cm. Its mechanical energy is 5.0 mJ. What is the mechanical energy of the same pendulum when it swings with an amplitude of 3.0 cm?
A thin circular hoop is suspended from a knife edge. Its rotational inertia about the rotation axis (along the knife) is I = 2 mr2. Show that it oscillates with the same frequency as a simple pendulum of length equal to the diameter of the hoop?
(a) What is the energy of a pendulum (L = 1.0 m, m = 0.50 kg) oscillating with an amplitude of 5.0 cm? (b) The pendulum's energy loss (due to damping) is replaced in a clock by allowing a 2.0-kg mass to drop 1.0 m in 1 week. What average percentage of the pendulum's energy is lost during one cycle?
The amplitude of oscillation of a pendulum decreases by a factor of 20.0 in 120 s. By what factor has its energy decreased in that time?
Because of dissipative forces, the amplitude of an oscillator decreases 5.00% in 10 cycles. By what percentage does its energy decrease in ten cycles?
A 0.50-m-long guitar string, of cross-sectional area 1.0 × 10−6 m2, has Young's modulus Y = 2.0 × 109 N/m2. By how much must you stretch the string to obtain a tension of 20 N?
Four people sit in a car. The masses of the people are 45 kg, 52 kg, 67 kg, and 61 kg. The car's mass is 1020 kg. When the car drives over a bump, its springs cause an oscillation with a frequency of 2.00 Hz. What would the frequency be if only the 45-kg person were present?
A pendulum passes x = 0 with a speed of 0.50 m/s; it swings out to A = 0.20 m. What is the period T of the pendulum? (Assume the amplitude is small.)
What is the length of a simple pendulum whose horizontal position is described by x = (4.00 cm) cos [(3.14 rad/s) t]? What assumption do you make when answering this question?
Martin caught a fish and wanted to know how much it weighed, but he didn't have a scale. He did, however, have a stopwatch, a spring, and a 4.90-N weight. He attached the weight to the spring and found that the spring would oscillate 20 times in 65 s. Next he hung the fish on the spring and found
A naval aviator had to eject from her plane before it crashed at sea. She is rescued from the water by helicopter and dangles from a cable that is 45 m long while being carried back to the aircraft carrier. What is the period of her vibration as she swings back and forth while the helicopter hovers
An object of mass m is hung from the base of an ideal spring that is suspended from the ceiling. The spring has a spring constant k. The object is pulled down a distance D from equilibrium and released. Later, the same system is set oscillating by pulling the object down a distance 2D from
A spider's web can undergo SHM when a fly lands on it and displaces the web. For simplicity, assume that a web obeys Hooke's law (which it does not really as it deforms permanently when displaced). If the web is initially horizontal, and a fly landing on the web is in equilibrium when it displaces
A mass-spring system oscillates so that the position of the mass is described by x = − 10 cos (1.57t), where x is in cm when t is in seconds. Make a plot that has a dot for the position of the mass at t = 0, t = 0.2 s, t = 0.4 s, . . . ,t = 4 s. The time interval between each dot should be 0.2 s.
A hedge trimmer has a blade that moves back and forth with a frequency of 28 Hz. The blade motion is converted from the rotation provided by the electric motor to an oscillatory motion by means of a Scotch yoke (see Conceptual Question 7). The blade moves 2.4 cm during each stroke. Assuming that
The simple pendulum can be thought of as a special case of the physical pendulum where all of the mass is at a distance L from the rotation axis. For a simple pendulum of mass m and length L, show that the expression for the period of a physical pendulum (Eq. 10-27) reduces to the expression for
It takes a flea 1.0 × 10−3 s to reach a peak speed of 0.74 m/s. (a) If the mass of the flea is 0.45 × 10−6 kg, what is the average power required? (b) Insect muscle has a maximum output of 60 W/kg. If 20% of the flea's weight is muscle, can the muscle provide the power needed? (c) The flea
Luke is trying to catch a pesky animal that keeps eating vegetables from his garden. He is building a trap and needs to use a spring to close the door to his trap. He has a spring in his garage and he wants to determine the spring constant of the spring. To do this, he hangs the spring from the
A 4.0-N body is suspended vertically from an ideal spring of spring constant 250 N / m. The spring is initially in its relaxed position. Write an equation to describe the motion of the body if it is released at t = 0? [Let y = 0 at the equilibrium point and take + y = up.]
Show, using dimensional analysis, that the frequency f at which a mass-spring system oscillates radical is independent of the amplitude A and proportional to √ k / m? [Start by assuming that f does depend on A (to some power).]
A horizontal spring with spring constant of 9.82 N/m is attached to a block with a mass of 1.24 kg that sits on a frictionless surface. When the block is 0.345 m from its equilibrium position, it has a speed of 0.543 m/s? (a) What is the maximum displacement of the block from the equilibrium
A steel piano wire (Y = 2.0 × 1011 Pa) has a diameter of 0.80 mm. At one end it is wrapped around a tuning pin of diameter 8.0 mm. The length of the wire (not including the wire wrapped around the tuning pin) is 66 cm. Initially, the tension in the wire is 381 N. To tune the wire, the tension must
When the tension is 402 N, what is the tensile stress in the piano wire in Problem 84? How does that compare to the elastic limit of steel piano wire (8.26 × 108 Pa)?
A tightrope walker who weighs 640 N walks along a steel cable. When he is halfway across, the cable makes an angle of 0.040 rad below the horizontal.(a) What is the strain in the cable? Assume the cable is horizontal with a tension of 80 N before he steps onto it. Ignore the weight of the cable
A gibbon, hanging onto a horizontal tree branch with one arm, swings with a small amplitude. The gibbon's cm is 0.40 m from the branch and its rotational inertia divided by its mass is I / m = 0.25 m2. Estimate the frequency of oscillation.
In Problem 8.41, we found that the force of the tibia (shinbone) on the ankle joint for a person (of weight 750 N) standing on the ball of one foot was 2800 N. The ankle joint therefore pushes upward on the bottom of the tibia with a force of 2800 N, while the top end of the tibia must feel a net
The maximum height of a cylindrical column is limited by the compressive strength of the material; if the compressive stress at the bottom were to exceed the compressive strength of the material, the column would be crushed under its own weight. (a) For a cylindrical column of height h and radius
Using the stress-strain graph for bone (Fig. 10.4c), calculate Young's moduli for tension and for compression. Consider only small stresses?
A bungee jumper leaps from a bridge and undergoes a series of oscillations. Assume g = 9.78 m/s2. (a) If a 60.0-kg jumper uses a bungee cord that has an unstretched length of 33.0 m and she jumps from a height of 50.0 m above a river, coming to rest just a few centimeters above the water surface on
Spider silk has a Young's modulus of 4.0 Ã— 109 N/m2 and can withstand stresses up to 1.4 Ã— 109 N/m2. A single web strand has a cross-sectional area of 1.0 Ã— 10ˆ’11 m2, and a web is made up of 50 radial strands. A bug lands in the center of a
What is the period of a pendulum formed by placing a horizontal axis:(a) Through the end of a meter stick (100-cm mark)?(b) Through the 75-cm mark?(c) Through the 60-cm mark?
The motion of a simple pendulum is approximately SHM only if the amplitude is small. Consider a simple pendulum that is released from a horizontal position ((i = 90° in Fig. 10.22). (a) Using conservation of energy, find the speed of the pendulum bob at the bottom of its swing. Express your answer
The gravitational potential energy of a pendulum is U = mgy? (a) Taking y = 0 at the lowest point, show that y = L (1 − cos (), where ( is the angle the string makes with the vertical. (b) If ( is small, (1 - cos () ≈ 1/2 (2 and ( ≈ x / L (Appendix A.7). Show that the potential energy can be
A pendulum is made from a uniform rod of mass m1 and a small block of mass m2 attached at the lower end? (a) If the length of the pendulum is L and the oscillations are small, find the period of the oscillations in terms of m1, m2, L, and g? (b) Check your answer to part (a) in the two special
Resilin is a rubber-like protein that helps insects to fly more efficiently. The resilin, attached from the wing to the body, is relaxed when the wing is down and is extended when the wing is up. As the wing is brought up, some elastic energy is stored in the resilin. The wing is then brought back
The piano strings that vibrate with the lowest frequencies consist of a steel wire around which a thick coil of copper wire is wrapped. Only the inner steel wire is under tension. What is the purpose of the copper coil?
Simple ear-protection devices use materials that reflect or absorb sound before it reaches the ears. A newer technology, sometimes called noise cancellation, uses a microphone to produce an electrical signal that mimics the noise. The signal is modified electronically, then fed to the speakers in a
When connecting speakers to a stereo, it is important to connect them with the correct polarity so that, if the same electrical signal is sent, they both move in the same direction. If the wires going to one speaker are reversed, the listener hears a noticeably weaker bass (low frequencies).
Is the vibration of a string in a piano, guitar, or violin a sound wave? Explain.
The wavelength of the fundamental standing wave on a cello string depends on which of these quantities: length of the string, mass per unit length of the string, or tension? The wavelength of the sound wave resulting from the string's vibration depends on which of the same three quantities?
If the length of a guitar string is decreased while the tension remains constant, what happens to each of these quantities? (a) The wavelength of the fundamental, (b) The frequency of the fundamental, (c) The time for a pulse to travel the length of the string, (d) The maximum velocity of a point
Why is it possible to understand the words spoken by two people at the same time?
A cello player can change the frequency of the sound produced by her instrument by (a) Increasing the tension in the string, (b) Pressing her finger on the string at different places along the fingerboard, (c) Bowing a different string. Explain how each of these methods affects the frequency
The drawing shows a complex wave moving to the right along a cord. Draw the shape of the cord an instant later and determine which parts of the cord are moving upward and which are moving downward. Indicate the directions on your drawing with arrows.
When an earthquake occurs, the S waves (transverse waves) are not detected on the opposite side of the Earth while the P waves (longitudinal waves) are. How does this provide evidence that the Earth's solid core is surrounded by liquid?
The intensity of sunlight that reaches Earth's atmosphere is 1400 W/m2. What is the intensity of the sunlight that reaches Jupiter? Jupiter is 5.2 times as far from the Sun as Earth. [Treat the Sun as an isotropic source of light waves.]
Two strings, each 15.0 m long, are stretched side by side. One string has a mass of 78.0 g and a tension of 180.0 N. The second string has a mass of 58.0 g and a tension of 160.0 N. A pulse is generated at one end of each string simultaneously. On which string will the pulse move faster? Once the
A uniform string of length 10.0 m and weight 0.25 N is attached to the ceiling. A weight of 1.00 kN hangs from its lower end. The lower end of the string is suddenly displaced horizontally. How long does it take the resulting wave pulse to travel to the upper end? [Is the weight of the string
What is the speed of a wave whose frequency and wavelength are 500.0 Hz and 0.500 m, respectively?
What is the wavelength of a wave whose speed and period are 75.0 m/s and 5.00 ms, respectively?
What is the frequency of a wave whose speed and wavelength are 120 m/s and 30.0 cm, respectively?
The speed of sound in air at room temperature is 340 m/s. (a) What is the frequency of a sound wave in air with wavelength 1.0 m? (b) What is the frequency of a radio wave with the same wavelength? (Radio waves are electromagnetic waves that travel at 3.0 × 108 m/s in air or in vacuum.)
Light visible to humans consists of electromagnetic waves with wavelengths (in air) in the range 400-700 nm (4.0 × 10−7 m to 7.0 × 10−7 m). The speed of light in air is 3.0 × 108 m/s. What are the frequencies of electromagnetic waves that are visible?
A fisherman notices a buoy bobbing up and down in the water in ripples produced by waves from a passing speedboat. These waves travel at 2.5 m/s and have a wavelength of 7.5 m. At what frequency does the buoy bob up and down?
You are swimming in the ocean as water waves with wavelength 9.6 m pass by. What is the closest distance that another swimmer could be so that his motion is exactly opposite yours (he goes up when you go down)?
What is the speed of the wave represented by y (x, t) = A sin (kx − (t), where k = 6.0 rad/cm and ( = 5.0 rad/s?
Michelle is enjoying a picnic across the valley from a cliff. She is playing music on her radio (assume it to be an isotropic source) and notices an echo from the cliff. She claps her hands and the echo takes 1.5 s to return. (a) Given that the speed of sound in air is 343 m/s on that day, how far
The equation of a wave isFind (a) the amplitude and (b) the wavelength of this wave.
A wave on a string has equation y(x, t) = (4.0 mm) sin (( t − kx) Where ( = 6.0 × 102 rad/s and k = 6.0 rad/m. (a) What is the amplitude of the wave? (b) What is the wavelength? (c) What is the period? (d) What is the wave speed? (e) In which direction does the wave travel?
A transverse wave on a string is described by the equation y (x, t) = (2.20 cm) sin [(130 rad/s) t + (15 rad/m) x]. (a) What is the maximum transverse speed of a point on the string? (b) What is the maximum transverse acceleration of a point on the string? (c) How fast does the wave move along the
Write an equation for a sine wave with amplitude 0.120 m, wavelength 0.300 m, and wave speed 6.40 m/s traveling in the -x-direction.
Write the equation for a transverse sinusoidal wave with a maximum amplitude of 2.50 cm and an angular frequency of 2.90 rad/s that is moving along the positive x -direction with a wave speed that is 5.00 times as fast as the maximum speed of a point on the string. Assume that at time t = 0, the
A sine wave is traveling to the right on a cord. The lighter line in the figure represents the shape of the cord at time t = 0; the darker line represents the shape of the cord at time t = 0.10 s. (The horizontal and vertical scales are different.) What are (a) the amplitude and (b) the wavelength
(a) Plot a graph for y(x, t) = (4.0 cm) sin [(378 rad/s)t − (314 rad/cm)x] versus x at t = 0 and at t = 1/480 s. From the plots determine the amplitude, wavelength, and speed of the wave. (b) For the same function, plot a graph of y (x, t) versus t at x = 0 and find the period of the vibration.
For a transverse wave on a string described by y(x, t) = (0.0050 m) cos [(4.0p rad/s)t − (1.0p rad/m)x] Find the maximum speed and the maximum acceleration of a point on the string. Plot graphs for one cycle of displacement y versus t, velocity vy versus t, and acceleration ay versus t at the
A transverse wave on a string is described by y(x, t) = (1.2 mm) sin [(2.0p rad/s)t − (0.50p rad/m)x] Plot the displacement y and the velocity vy versus t for one complete cycle of the point x = 0 on the string.
(a) Sketch graphs of y versus x for the function y(x, t) = (0.80 mm) sin (kx − (t) For the times t = 0, 0.96 s, and 1.92 s. Make all three graphs of the same axes, using a solid line for the first, a dashed line for the second, and a dotted line for the third. Use the values k = (/(5.0 cm) and w
The intensity of the sound wave from a jet airplane as it is taking off is 1.0 × 102 W/m2 at a distance of 5.0 m. What is the intensity of the sound wave that reaches the ears of a person standing at a distance of 120 m from the runway? Assume that the sound wave radiates from the airplane equally
The drawing shows a snapshot of a transverse wave traveling along a string at 10.0 m/s. The equation for the wave is y (x, t) = A cos (ωt + kx).(1) Is the wave moving to the right or to the left?(2) What are the numerical values of A, ω, and k?(3) At what times could this snapshot have been
The drawing shows a snapshot of a transverse wave traveling along a string at 10.0 m/s. The equation for the wave is y (x, t) = A cos ((t + kx). (a) Is the wave moving to the right or to the left? (b) What are the numerical values of A, (, and k? (c) At what times could this snapshot have been
Two pulses on a cord at time t = 0 are moving toward each other; the speed of each pulse is 40 cm/s. Sketch the shape of the cord at 0.15, 0.25, and 0.30 s.
Two pulses on a cord at time t = 0 are moving toward one another; the speed of each pulse is 2.5 m/s. Sketch the shape of the cord at 0.60, 0.80, and 0.90 s.
Using graph paper, sketch two identical sine waves of amplitude 4.0 cm that differ in phase by (a) (/3 rad (60() and (b) (/2 rad (90(). Find the amplitude of the superposition of the two waves in each case.
Two traveling sine waves, identical except for a phase difference (, add so that their superposition produces another traveling wave with the same amplitude as the two component waves. What is the phase difference between the two waves?
A traveling sine wave is the result of the superposition of two other sine waves with equal amplitudes, wavelengths, and frequencies. The two component waves each have amplitude 5.00 cm. If the superposition wave has amplitude 6.69 cm, what is the phase difference ( between the component waves?
Light of wavelength 0.500 μ m (in air) enters the water in a swimming pool. The speed of light in water is 0.750 times the speed in air. What is the wavelength of the light in water?
The pulse of the figure travels to the right on a string whose ends at x = 0 and x = 4.0 m are both fixed in place. Imagine a reflected pulse that begins to move onto the string at an endpoint at the same time the incident pulse reaches that endpoint. The superposition of the incident and reflected
Two waves with identical frequency but different amplitudes A 1 = 5.0 cm and A 2 = 3.0 cm, occupy the same region of space (are superimposed). (a) At what phase difference does the resulting wave have the largest amplitude? What is the amplitude of the resulting wave in that case? (b) At what phase
At what rate in watts does the jet airplane in Problem 3 radiate energy in the form of sound waves? The intensity of the sound wave from a jet airplane as it is taking off is 1.0 × 102 W/m2 at a distance of 5.0 m. What is the intensity of the sound wave that reaches the ears of a person standing
Two waves with identical frequency but different amplitudes A1 = 6.0 cm and A2 = 3.0 cm, occupy the same region of space (i.e., are superimposed). (a) At what phase difference will the resulting wave have the highest intensity? What is the amplitude of the resulting wave in that case? (b) At what
A sound wave with intensity 25 mW/m2 interferes constructively with a sound wave that has an intensity of 15 mW/m2. What is the intensity of the superposition of the two?
A sound wave with intensity 25 mW/m2 interferes destructively with a sound wave that has an intensity of 28 mW/m2. What is the intensity of the superposition of the two?
Two coherent sound waves have intensities of 0.040 W/m2 and 0.090 W/m2 where you are listening. (a) If the waves interfere constructively, what is the intensity that you hear? (b) What if they interfere destructively? (c) If they were incoherent, what would be the intensity? [If your answers are
While testing speakers for a concert, Tomás sets up two speakers to produce sound waves at the same frequency, which is between 100 Hz and 150 Hz. The two speakers vibrate in phase with one another. He notices that when he listens at certain locations, the sound is very soft (a minimum intensity
In order to decrease the fundamental frequency of a guitar string by 4.0%, by what percentage should you reduce the tension?
The tension in a guitar string is increased by 15%. What happens to the fundamental frequency of the string?
A standing wave has wave number 2.0 ( 102 rad/m. What is the distance between two adjacent nodes?
A harpsichord string of length 1.50 m and linear mass density 25.0 mg/m vibrates at a (fundamental) frequency of 450.0 Hz. (a) What is the speed of the transverse string waves? (b) What is the tension? (c) What are the wavelength and frequency of the sound wave in air produced by vibration of the
A cord of length 1.5 m is fixed at both ends. Its mass per unit length is 1.2 g/m and the tension is 12 N. (a) What is the frequency of the fundamental oscillation? (b) What tension is required if the n = 3 mode has a frequency of 0.50 kHz?
The Sun emits electromagnetic waves (including light) equally in all directions. The intensity of the waves at Earth's upper atmosphere is 1.4 kW/m2. At what rate does the Sun emit electromagnetic waves? (In other words, what is the power output?)
Tension is maintained in a string by attaching one end to a wall and by hanging a 2.20-kg object from the other end of the string after it passes over a pulley that is 2.00 m from the wall. The string has a mass per unit length of 3.55 mg/m. What is the fundamental frequency of this string?
A guitar's E-string has length 65 cm and is stretched to a tension of 82 N. It vibrates at a fundamental frequency of 329.63 Hz. Determine the mass per unit length of the string.
A string 2.0 m long is held fixed at both ends. If a sharp blow is applied to the string at its center, it takes 0.050 s for the pulse to travel to the ends of the string and return to the middle. What is the fundamental frequency of oscillation for this string?
A 1.6-m-long string fixed at both ends vibrates at resonant frequencies of 780 Hz and 1040 Hz, with no other resonant frequency between these values. (a) What is the fundamental frequency of this string? (b) When the tension in the string is 1200 N, what is the total mass of the string?
A certain string has a mass per unit length of 0.120 g/m. It is attached to a vibrating device and weight similar to that shown in Figure 11.22. The vibrator oscillates at a constant frequency of 110 Hz. How heavy should the weight be in order to produce standing waves in a string of length 42 cm?
The longest "string" (a thick metal wire) on a particular piano is 2.0 m long and has a tension of 300.0 N. It vibrates with a fundamental frequency of 27.5 Hz. What is the total mass of the wire?
Suppose that a string of length L and mass m is under tension F. (a) Show that (FL/m has units of speed. (b) Show that there is no other combination of L, m, and F with units of speed. [Of the dimensions of the three quantities L, m, and F, only F includes time.] Thus, the speed of transverse waves
The speed of waves on a lake depends on frequency. For waves of frequency 1.0 Hz, the wave speed is 1.56 m/s; for 2.0-Hz waves, the speed is 0.78 m/s. The 2.0-Hz waves from a speedboat's wake reach you 120 s after the 1.0-Hz waves generated by the same boat. How far away is the boat?

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