Chapter 13

\(\bullet\) A particle oscillates in SHM with an amplitude \(A\). What is the total distance (in terms of \(A\) ) the particle travels in three periods?

\(\bullet\) If it takes a particle in SHM 0.50 s to travel from the equilibrium position to the maximum displacement (amplitude), what is the period of oscillation?

\(\bullet\) A \(0.75-\mathrm{kg}\) object oscillating on a spring completes a cycle every \(0.50 \mathrm{~s}\). What is the frequency of this oscillation?

\(\bullet\) A particle in simple harmonic motion has a frequency of \(40 \mathrm{~Hz}\). What is the period of this oscillation?

\(\bullet\) The frequency of a simple harmonic oscillator is doubled from \(0.25 \mathrm{~Hz}\) to \(0.50 \mathrm{~Hz}\). What is the change in its period?

\(\bullet\) An object of mass \(0.50 \mathrm{~kg}\) is attached to a spring with spring constant \(10 \mathrm{~N} / \mathrm{m}\). If the object is pulled down \(0.050 \mathrm{~m}\) from the equilibrium position and released, what is its maximum speed?

\(\bullet\) An object of mass \(1.0 \mathrm{~kg}\) is attached to a spring with spring constant \(15 \mathrm{~N} / \mathrm{m}\). If the object has a maximum speed of \(0.50 \mathrm{~m} / \mathrm{s}\), what is the amplitude of oscillation?

\(\bullet\bullet\) Atoms in a solid are in continuous vibrational motion due to thermal energy. At room temperature, the amplitude of these atomic vibrations is typically about \(10^{-9} \mathrm{~cm},\) and their frequency is on the order of \(10^{12} \mathrm{~Hz}\). (a) What is the approximate period of oscillation of a typical atom? (b) What is the maximum speed of such an atom?

\(\bullet\bullet\) A particle of mass \(0.10 \mathrm{~kg}\) is attached to a spring of spring constant \(10 \mathrm{~N} / \mathrm{m}\). If the maximum acceleration of the particle is \(5.0 \mathrm{~m} / \mathrm{s}^{2},\) what is the maximum speed of the particle?

\(\bullet\bullet\) (a) At what position is the magnitude of the force on a mass in a mass-spring system minimum: \((1) x=0\), (2) \(x=-A,\) or \((3) x=+A ?\) Why? \((b)\) If \(m=0.500 \mathrm{~kg}\) \(k=150 \mathrm{~N} / \mathrm{m},\) and \(A=0.150 \mathrm{~m},\) what are the magnitude of the force on the mass and the acceleration of the mass at \(x=0,0.050 \mathrm{~m},\) and \(0.150 \mathrm{~m} ?\)

\(\bullet\bullet\) (a) At what position is the speed of a mass in a mass-spring system maximum: \((1) x=0,(2) x=-A,\) or (3) \(x=+A ?\) Why? (b) If \(m=0.250 \mathrm{~kg}, k=100 \mathrm{~N} / \mathrm{m}\) and \(A=0.10 \mathrm{~m}\) for such a system, what is the mass's maximum speed?

\(\bullet\bullet\) A mass-spring system is in SHM in the horizontal direction. If the mass is \(0.25 \mathrm{~kg}\), the spring constant is \(12 \mathrm{~N} / \mathrm{m},\) and the amplitude is \(15 \mathrm{~cm}\) (a) what is the maximum speed of the mass, and (b) where does this occur? (c) What is the speed at a half-amplitude position?

\(\bullet\bullet\) A horizontal spring on a frictionless level air track has a \(0.150-\mathrm{kg}\) object attached to it and it is stretched \(6.50 \mathrm{~cm} .\) Then the object is given an outward initial velocity of \(2.20 \mathrm{~m} / \mathrm{s}\). If the spring constant is \(35.2 \mathrm{~N} / \mathrm{m}\) determine how much farther the spring stretches.

\(\bullet\bullet\) A \(\mathrm{} 0.25-\mathrm{kg}\) object is suspended on a light spring of spring constant \(49 \mathrm{~N} / \mathrm{m}\). The spring is then compressed to a position \(15 \mathrm{~cm}\) above the stretched equilibrium position. How much more energy does the system have at the compressed position than at the stretched equilibrium position?

\(\bullet\bullet\) A 0.25 -kg object is suspended on a light spring of spring constant \(49 \mathrm{~N} / \mathrm{m}\) and the system is allowed to come to rest at its equilibrium position. The object is then pulled down \(0.10 \mathrm{~m}\) from the equilibrium position and released. What is the speed of the object when it goes through the equilibrium position?

\(\bullet\bullet\) A \(0.350-\mathrm{kg}\) block moving vertically upward collides with a light vertical spring and compresses it \(4.50 \mathrm{~cm}\) before coming to rest. If the spring constant is \(50.0 \mathrm{~N} / \mathrm{m}\) what was the initial speed of the block? (Ignore energy losses to sound and other factors during the collision.)

\(\bullet\bullet\) A 75-kg circus performer jumps from a 5.0 -m height onto a trampoline and stretches it downward \(0.30 \mathrm{~m}\). Assuming that the trampoline obeys Hooke's law, (a) how far will it stretch if the performer jumps from a height of \(8.0 \mathrm{~m} ?\) (b) How far will the trampoline stretch if the performer stands still on it while taking a bow?

\(\bullet\bullet\) A vertical spring has a 0.200 -kg mass attached to it. The mass is released from rest and falls \(22.3 \mathrm{~cm}\) before stopping (a) Determine the spring constant. (b) Determine the speed of the mass when it has fallen only \(10.0 \mathrm{~cm}\).

\(\bullet\) A 0.50-kg mass oscillates in simple harmonic motion on a spring with a spring constant of \(200 \mathrm{~N} / \mathrm{m}\). What are (a) the period and (b) the frequency of the oscillation?

\(\bullet\) The simple pendulum in a tall clock is \(0.75 \mathrm{~m}\) long. What are (a) the period and (b) the frequency of this pendulum?

\(\bullet\) How much mass should be at the end of a spring \((k=100 \mathrm{~N} / \mathrm{m})\) in order to have a period of \(2.0 \mathrm{~s} ?\)

\(\bullet\) If the frequency of a mass-spring system is \(1.50 \mathrm{~Hz}\) and the mass on the spring is \(5.00 \mathrm{~kg}\), what is the spring constant?

\(\bullet\) A breeze sets a suspended lamp into oscillation. If the period is \(1.0 \mathrm{~s}\), what is the distance from the ceiling to the lamp at the lowest point? Assume that the lamp is a point mass and acts as a simple pendulum.

\(\bullet\) Write the general equation of motion for a mass that is on a horizontal frictionless surface and is connected to a spring at equilibrium (a) if the mass is initially pulled in the \(+x\) axis from the spring (stretched) and released, and (b) if the mass is pushed in the \(-x\) axis toward the spring (compressed) and released.

\(\bullet\) The equation of motion for an oscillator in vertical \(\mathrm{SHM}\) is given by \(y=(0.10 \mathrm{~m}) \sin [(100 \mathrm{rad} / \mathrm{s}) t] .\) What are the (a) amplitude, (b) frequency, and (c) period of this motion?

\(\bullet\) The displacement of an object is given by \(y=(5.0 \mathrm{~cm}) \cos [(20 \pi \mathrm{rad} / \mathrm{s}) t] .\) What are the object's (a) amplitude, (b) frequency, and (c) period of oscillation?

\(\bullet\) If the displacement of an oscillator in SHM is described by the equation \(y=(0.25 \mathrm{~m}) \cos [(314 \mathrm{rad} / \mathrm{s}) t],\) where \(y\) is in meters and \(t\) is in seconds, what is the position of the oscillator at (a) \(t=0,\) (b) \(t=5.0 \mathrm{~s},\) and \((\mathrm{c}) t=15 \mathrm{~s} ?\)

\(\bullet\bullet\) The equation of motion of a SHM oscillator is \(x=(0.50 \mathrm{~m}) \sin (2 \pi f) t,\) where \(x\) is in meters and \(t\) is in seconds. If the position of the oscillator is at \(x=0.25 \mathrm{~m}\) at \(t=0.25 \mathrm{~s},\) what is the frequency of the oscillator?

\(\bullet\bullet\) Show that the total energy of a mass-spring system in simple harmonic motion is given by \(\frac{1}{2} m \omega^{2} A^{2}\).

\(\bullet\bullet\) The velocity of a vertically oscillating mass-spring system is given by \(v=(0.650 \mathrm{~m} / \mathrm{s}) \sin [(4 \mathrm{rad} / \mathrm{s}) t]\) Determine (a) the amplitude and (b) the maximum acceleration of this oscillator.

\(\bullet\bullet\) (a) If the mass in a mass-spring system is halved, the new period is \((1) 2,(2) \sqrt{2},(3) 1 / \sqrt{2},(4) 1 / 2\) times the old period. Why? (b) If the initial period is \(3.0 \mathrm{~s}\) and the mass is reduced to \(1 / 3\) of its initial value, what is the new period?

\(\bullet\bullet\) (a) If the spring constant in a mass-spring system is halved, the new period is \((1) 2,(2) \sqrt{2},(3) 1 / \sqrt{2},\) (4) \(1 / 2\) times the old period. Why? (b) If the initial period is \(2.0 \mathrm{~s}\) and the spring constant is reduced to \(1 / 3\) of its initial value, what is the new period?

\(\bullet\bullet\) Students use a simple pendulum with a length of \(36.90 \mathrm{~cm}\) to measure the acceleration of gravity at the location of their school. If it takes \(12.20 \mathrm{~s}\) for the pendulum to complete ten oscillations, what is the experimental value of \(g\) at the school?

\(\bullet\bullet\) The equation of motion of a particle in vertical SHM is given by \(y=(10 \mathrm{~cm}) \sin [(0.50 \mathrm{rad} / \mathrm{s}) t]\). What are the particle's (a) displacement (b) velocity, and (c) acceleration at \(t=1.0 \mathrm{~s}\) ?

\(\bullet\bullet\) What is the maximum elastic potential energy of a simple horizontal mass-spring oscillator whose equation of motion is given by \(x=(0.350\mathrm{~m}) \sin [(7 \mathrm{rad} / \mathrm{s}) t] ?\) The mass on the end of the spring is \(0.900 \mathrm{~kg}\).

\(\bullet\bullet\) Two masses oscillate on light springs. The second mass is half of the first and its spring constant is twice that of the first. Which system will have the greater frequency, and what is the ratio of the frequency of the second mass to that of the first mass?

\(\bullet\bullet\) During an earthquake, the floor of an apartment building is measured to oscillate in approximately \(\operatorname{sim}-\) ple harmonic motion with a period of 1.95 seconds and an amplitude of \(8.65 \mathrm{~cm} .\) Determine the maximum speed and acceleration of the floor during this motion.

\(\bullet\bullet\) (a) If a pendulum clock were taken to the Moon, where the acceleration due to gravity is only one-sixth (assume the figure to be exact) that on the Earth, will the period of vibration (1) increase, (2) remain the same, or (3) decrease? Why? (b) If the period on the Earth is \(2.0 \mathrm{~s}\), what is the period on the Moon?

\(\bullet\bullet\bullet\) The forces acting on a simple pendulum are shown in \(\nabla\) Fig. \(13.26 .\) (a) Show that, for the small angle approximation \((\sin \theta \approx \theta),\) the force producing the motion has the same form as Hooke's law. (b) Show by analogy with a mass on a spring that the period of a simple pendulum is given by \(T=2 \pi \sqrt{L / g}\). [Hint: Think of the effective spring constant.]

\(\bullet\bullet\bullet\) The acceleration as a function of time of a mass- spring system is given by \(a=\left(0.60 \mathrm{~m} / \mathrm{s}^{2}\right) \sin [(2 \mathrm{rad} / \mathrm{s}) t]\). If the spring constant is \(10 \mathrm{~N} / \mathrm{m},\) what are \((\mathrm{a})\) the amplitude, \((\mathrm{b})\) the initial velocity and (c) the mass of the object?

\(\bullet\bullet\bullet\) A clock uses a pendulum that is \(75 \mathrm{~cm}\) long. The clock is accidentally broken, and when it is repaired, the length of the pendulum is shortened by \(2.0 \mathrm{~mm} .\) Consider the pendulum to be a simple pendulum. (a) Will the repaired clock gain or lose time? (b) By how much will the time indicated by the repaired clock differ from the correct time (taken to be the time determined by the original pendulum in \(24 \mathrm{~h}\) )? (c) If the pendulum rod were metal, would the surrounding temperature make a difference in the timekeeping of the clock? Explain.

\(\bullet\) A sound wave has a speed of \(340 \mathrm{~m} / \mathrm{s}\) in air. If this wave produces a tone with a frequency of \(1000 \mathrm{~Hz}\), what is its wavelength?

\(\bullet\) A wave on a rope that measures \(10 \mathrm{~m}\) long takes \(2.0 \mathrm{~s}\) to travel the whole rope. If the wavelength of the wave is \(2.5 \mathrm{~m},\) what is the frequency of oscillation of any piece of the rope?

\(\bullet\) A student reading his physics book on a lake dock notices that the distance between two incoming wave crests is about \(0.75 \mathrm{~m},\) and he then measures the time of arrival between the crests to be \(1.6 \mathrm{~s}\). What is the approximate speed of the waves?

\(\bullet\)Dolphins and bats determine the location of their prey using echolocation (see Conceptual Question 15\()\). If it takes \(15 \mathrm{~ms}\) for a bat to receive the ultrasonic sound wave reflected off a mosquito, how far is the mosquito from the bat? Take the speed of sound as \(345 \mathrm{~m} / \mathrm{s}\)

\(\bullet\) Light waves travel in a vacuum at a speed of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} .\) The frequency of blue light is about \(6 \times 10^{14} \mathrm{~Hz}\). What is the approximate wavelength of the light?

\(\bullet\bullet\) A sonar generator on a submarine produces periodic ultrasonic waves at a frequency of \(2.50 \mathrm{MHz}\). The wavelength of the waves in seawater is \(4.80 \times 10^{-4} \mathrm{~m}\). When the generator is directed downward, an echo reflected from the ocean floor is received 10.0 s later. How deep is the ocean at that point?

\(\bullet\bullet\) The range of sound frequencies audible to the human ear extends from about \(20 \mathrm{~Hz}\) to \(20 \mathrm{kHz}\). If the speed of sound in air is \(345 \mathrm{~m} / \mathrm{s},\) what are the wavelength limits of this audible range?

\(\bullet\bullet\) The AM frequencies on a radio dial range from \(550 \mathrm{kHz}\) to \(1600 \mathrm{kHz},\) and the FM frequencies range from \(88.0 \mathrm{MHz}\) to \(108 \mathrm{MHz}\). All of these radio waves travel at a speed of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (speed of light). (a) Compared with the FM frequencies, the AM frequencies have (1) longer, (2) the same, or (3) shorter wavelengths. Why? (b) What are the wavelength ranges of the AM band and the FM band?

\(\bullet\bullet\) Assume that \(P\) and \(S\) (primary and secondary) waves from an earthquake with a focus near the Earth's surface travel through the Earth at nearly constant but different average speeds. A monitoring station that is \(1000 \mathrm{~km}\) from the epicenter detected the S wave to arrive at \(42 \mathrm{~s}\) after the arrival of the P wave. If the P wave has an average speed of \(8.0 \mathrm{~km} / \mathrm{s},\) what is the average speed of the \(\mathrm{S}\) wave?