A mass-spring system has where b is the damping constant and the natural frequency. How

Is a vertically bouncing ball an example of oscillatory motion? Of simple harmonic motion? Explain.

The vibration frequencies of molecules are much higher than those of macroscopic mechanical systems. Why?

What happens to the frequency of a simple harmonic oscillator when the spring constant is doubled? When the mass is doubled?

If the spring of a simple harmonic oscillator is cut in half, what happens to the frequency?

How does the frequency of a simple harmonic oscillator depend on its amplitude?

How would the frequency of a horizontal mass-spring system change if it were taken to the Moon? Of a vertical mass-spring system? Of a simple pendulum?

When in its cycle is the acceleration of an undamped simple harmonic oscillator zero? When is the velocity zero?

Explain how simple harmonic motion might be used to determine the masses of objects in an orbiting spacecraft.

One pendulum consists of a solid rod of mass m and length L, and another consists of a compact ball of the same mass m on the end of a massless string of the same length L. Which has the greater period? Why?

The x- and y-components of motion of a body are both simple harmonic with the same frequency and amplitude. What shape is the path of the body if the component motions are (a) in phase, (b) out of phase, and (c) out of phase?

Why is critical damping desirable in a cars suspension?

Explain why the frequency of a damped system is lower than that of the equivalent undamped system.

Opera singers have been known to break glasses with their voices. How?

What will happen to the period of a mass-spring system if its placed in a jetliner accelerating down a runway? What will happen to the period of a pendulum in the same situation?

How can a system have more than one resonant frequency?

A doctor counts 77 heartbeats in 1 minute. What are the corresponding period and frequency?

A violin string playing the note A oscillates at 440 Hz. Whats its oscillation period?

The vibration frequency of a hydrogen chloride molecule is How long does it take the molecule to complete one oscillation?

Write expressions for simple harmonic motion (a) with amplitude 10 cm, frequency 5.0 Hz, and maximum displacement at and (b) with amplitude 2.5 cm, angular frequency and maximum velocity at

The top of a skyscraper sways back and forth, completing 9 oscillation cycles in 1 minute. Find the period and frequency of its motion.

A hummingbirds wings vibrate at about 45 Hz. Whats the corresponding period? t 5 0. 5.0 s21, t 5 0, 8.6631013 Hz. p/2 p/4

A 200-g mass is attached to a spring of constant and set into oscillation with amplitude Determine (a) the frequency in hertz, (b) the period, (c) the maximum velocity, and (d) the maximum force in the spring.

An automobile suspension has an effective spring constant of 26 kN/m, and the cars suspended mass is 1900 kg. In the absence of damping, with what frequency and period will the car undergo simple harmonic motion?

The quartz crystal in a watch executes simple harmonic motion at 32,768 Hz. (This is Hz, chosen so that 15 divisions by 2 give a signal at 1.00000 Hz.) If each face of the crystal undergoes a maximum displacement of 100 nm, find the maximum velocity and acceleration of the crystal faces.

A 50-g mass is attached to a spring and undergoes simple harmonic motion. Its maximum acceleration is and its maximum speed is 3.5 m/s. Determine the (a) angular frequency, (b) spring constant, and (c) amplitude.

A particle undergoes simple harmonic motion with amplitude 25 cm and maximum speed 4.8 m/s. Find the (a) angular frequency, (b) period, and (c) maximum acceleration.

A particle undergoes simple harmonic motion with maximum speed 1.4 m/s and maximum acceleration Find the (a) angular frequency, (b) period, and (c) amplitude.

How long should you make a simple pendulum so its period is (a) 200 ms, (b) 5.0 s, and (c) 2.0 min?

At the heart of a grandfather clock is a simple pendulum 1.45 m long; the clock ticks each time the pendulum reaches its maximum displacement in either direction. Whats the time interval between ticks?

A 640-g hollow ball 21 cm in diameter is suspended by a wire and is undergoing torsional oscillations at 0.78 Hz. Find the torsional constant of the wire.

A meter stick is suspended from one end and set swinging. Find the period of the resulting small-amplitude oscillations.

A wheel rotates at 600 rpm. Viewed from the edge, a point on the wheel appears to undergo simple harmonic motion. What are (a) the frequency in Hz and (b) the angular frequency for this SHM?

The x- and y-components of an objects motion are harmonic with frequency ratio 1.75:1. How many oscillations must each component undergo before the object returns to its initial position?

A 450-g mass on a spring is oscillating at 1.2 Hz, with total energy 0.51 J. Whats the oscillation amplitude?

A torsional oscillator of rotational inertia and torsional constant has total energy 4.7 J. Find its maximum angular displacement and maximum angular speed.

Youre riding in a friends 1400-kg car with bad shock absorbers, bouncing down the highway at 20 m/s and executing vertical SHM with amplitude 18 cm and frequency 0.67 Hz. Concerned about fuel efficiency, your friend wonders what percentage of the cars kinetic energy is tied up in this oscillation. Make an estimate, neglecting the wheels rotational energy and the fact that not all of the cars mass participates in the oscillation.

The vibration of a piano string can be described by an equation analogous to Equation 13.17. If the quantity analogous to in that equation has the value how long will it take the amplitude to drop to half its original value?

A mass-spring system has where b is the damping constant and the natural frequency. How does its amplitude at compare with its amplitude when driven at frequencies 10% above and below ?

A cars front suspension has a natural frequency of 0.45 Hz. The cars front shock absorbers are worn and no longer provide critical damping. The car is driving on a bumpy road with bumps 40 m apart. At a certain speed, the driver notices that the car begins to shake violently. What is this speed?

A simple model for carbon dioxide consists of three mass points (atoms) connected by two springs (electric forces), as shown in Fig. 13.27. One way this system can oscillate is if the carbon atom stays fixed and the two oxygens move symmetrically on either side of it. If the frequency of this oscillation is Hz, whats the effective spring constant? (Note: The mass of an oxygen atom is 16 u.)

Two identical mass-spring systems consist of 430-g masses on springs of constant Both are displaced from equilibrium, and the first is released at time How much later should the second be released so their oscillations differ in phase by

The human eye and muscles that hold it can be modeled as a massspring system with typical values and Whats the resonant frequency of this system? Shaking your head at this frequency blurs vision, as the eyeball undergoes resonant oscillations.

A mass m slides along a frictionless horizontal surface at speed It strikes a spring of constant k attached to a rigid wall, as shown in Fig. 13.28. After an elastic encounter with the spring, the mass heads back in the direction it came from. In terms of and determine (a) how long the mass is in contact with the spring and (b) the springs maximum compression.

Show by substitution that is a solution to Equation 13.3.

A physics student, bored by a lecture on simple harmonic motion, idly picks up his pencil (mass 9.2 g, length 17 cm) by the tip with his frictionless fingers, and allows it to swing back and forth with small amplitude. If the pencil completes 6279 full cycles during the lecture, how long does the lecture last?

A pendulum of length L is mounted in a rocket. Find its period if the rocket is (a) at rest on its launch pad; (b) accelerating upward with acceleration ; (c) accelerating downward with ; and (d) in free fall.

The protein dynein powers the flagella that propel some unicellular organisms. Biophysicists have found that dynein is intrinsically oscillatory, and that it exerts peak forces of about 1.0 pN when it attaches to structures called microtubules. The resulting oscillations have amplitude 15 nm. (a) If this system is modeled as a mass-spring system, whats the associated spring constant? (b) If the oscillation frequency is 70 Hz, whats the effective mass?

A mass is attached to a vertical spring, which then goes into oscillation. At the high point of the oscillation, the spring is in the original unstretched equilibrium position it had before the mass was attached; the low point is 5.8 cm below this. Find the oscillation period.

Derive the period of a simple pendulum by considering the horizontal displacement x and the force acting on the bob, rather than the angular displacement and torque.

A solid disk of radius R is suspended from a spring of spring constant k and torsional constant as shown in Fig. 13.29. In terms of k and what value of R will give the same period for the vertical and torsional oscillations of this system?

A thin steel beam 8.0 m long is suspended from a crane and is undergoing torsional oscillations. Two 75-kg steelworkers leap onto opposite ends of the beam, as shown in Fig. 13.30. If the frequency of torsional oscillations diminishes by 20%, whats the beams mass?

A cyclist turns her bicycle upside down to repair it. She then notices that the front wheel is executing a slow, small-amplitude, back-andforth rotational motion with period 12 s. Treating the wheel as a thin ring of mass 600 g and radius 30 cm, whose only irregularity is the tire valve stem, determine the mass of the valve stem.

An object undergoes simple harmonic motion in two mutually perpendicular directions, its position given by (a) Show that the object remains a fixed distance from the origin (i.e., that its path is circular), and find that distance. (b) Find an expression for the objects velocity. (c) Show that the speed remains constant, and find its value. (d) Find the angular speed of the object in its circular path.

The muscles that drive insect wings minimize the energy needed for flight by choosing to move at the natural oscillation frequency of the wings. Biologists study this phenomenon by clipping an insects wings to reduce their mass. If the wing system is modeled as a simple harmonic oscillator, by what percent will the frequency change if the wing mass is decreased by 25%? Will it increase or decrease?

A pendulum consists of a 320-g solid ball 15.0 cm in diameter, suspended by an essentially massless string 80.0 cm long. Calculate the period of this pendulum, treating it first as a simple pendulum and then as a physical pendulum. Whats the error in the simple-pendulum approximation? (Hint: Remember the parallelaxis theorem.)

If Jane and Tarzan are initially 8.0 m apart in Fig. 13.12, and Janes mass is 60 kg, whats the maximum tension in the vine, and at what point does it occur?

A small mass measuring device (SMMD) used for research on the biological effects of spaceflight consists of a small springmounted cage. Rats or other small subjects are introduced into the cage, which is set into oscillation. Calibration of a SMMD gives a linear function for the square of the oscillation period versus the subjects mass m in kg: . Find (a) the spring constant and (b) the mass of the cage alone.

A thin, uniform hoop of mass M and radius R is suspended from a horizontal rod and set oscillating with small amplitude, as shown in Fig. 13.31. Show that the period of the oscillations is . (Hint: You may find the parallel-axis theorem useful.)

A mass m is mounted between two springs with constants and as shown in Figure 13.32. Show that the angular frequency of oscillation is v 5 21k1 1 k22/m.

The equation for an ellipse is . Show that two-dimensional simple harmonic motion whose components have different amplitudes and are out of phase gives rise to elliptical motion. How are constants a and b related to the amplitudes?

Show that the potential energy of a simple pendulum is proportional to the square of the angular displacement in the small-amplitude limit.

The total energy of a mass-spring system is the sum of its kinetic and potential energy: . Assuming E remains constant, differentiate both sides of this expression with respect to time and show that Equation 13.3 results. (Hint: Remember that v 5 dx/dt.)

A solid cylinder of mass M and radius R is mounted on an axle through its center. The axle is attached to a horizontal spring of constant k, and the cylinder rolls back and forth without slipping (Fig. 13.33). Write the statement of energy conservation for this system, and differentiate it to obtain an equation analogous to Equation 13.3 (see Problem 62). Comparing your result with Equation 13.3, determine the angular frequency of the motion.

A mass m is free to slide on a frictionless track whose height y as a function of horizontal position xis , where a is a constant with units of inverse length. The mass is given an initial displacement from the bottom of the track and then released. Find an expression for the period of the resulting motion.

A 250-g mass is mounted on a spring of constant The damping constant for this system is How many oscillations will the system undergo before the amplitude decays to 1/e of its original value?

A harmonic oscillator is underdamped if the damping constant b is less than , where is the natural frequency of undamped motion. Show that for an underdamped oscillator, Equation 13.19 has a maximum at a driving frequency less than

A massless spring with hangs from the ceiling. A 490-g mass is hooked onto the unstretched spring and allowed to drop. Find (a) the amplitude and (b) the period of the resulting motion.

A meter stick is suspended from a frictionless rod through a small hole at the 25-cm mark. Find the period of small-amplitude oscillations about the sticks equilibrium position.

A particle of mass m has potential energy given by where a is a constant and x is the particles position. Find an expression for the frequency of simple harmonic oscillations this particle undergoes.

Two balls each of unknown mass m are mounted on opposite ends of a 1.5-m-long rod of mass 850 g. The system is suspended from a wire attached to the center of the rod and set into torsional oscillations. If the wire has torsional constant 0.63 N m/rad and the period of the oscillations is 5.6 s, whats the unknown mass m?

Two mass-spring systems with the same mass are undergoing oscillatory motion with the same amplitudes. System 1 has twice the frequency of system 2. How do (a) their energies and (b) their maximum accelerations compare?

Two mass-spring systems have the same mass and the same total energy. The amplitude of system 1 is twice that of system 2. How do (a) their frequencies and (b) their maximum accelerations compare?

A 500-g mass is suspended from a thread 45 cm long that can sustain a tension of 6.0 N before breaking. Find the maximum allowable amplitude for pendulum motion of this system.

A 500-g block on a frictionless, horizontal surface is attached to a rather limp spring with A second block rests on the first, and the whole system executes simple harmonic motion with period 1.8 s. When the amplitude of the motion is increased to 35 cm, the upper block just begins to slip. Whats the coefficient of static friction between the blocks?

Repeat Problem 64 for a small solid ball of mass M and radius R that rolls without slipping on the parabolic track.

Youre working on the script of a movie whose plot involves a hole drilled straight through Earths center and out the other side. Youre asked to determine what will happen if a person falls into the hole. You find that the gravitational acceleration inside Earth points toward Earths center, with magnitude given approximately by , where is the surface value, r is the distance from Earths center, and is Earths radius. What do you report for the persons motion, including equations and values for any relevant parameters?

A 1.2-kg block rests on a frictionless surface and is attached to a horizontal spring of constant (Fig. 13.34). The block oscillates with amplitude 10 cm and phase constant . A block of mass 0.80 kg moves from the right at 1.7 m/s and strikes the first block when the latter is at the rightmost point in its oscillation. The two blocks stick together. Determine the frequency, amplitude, and phase constant (relative to the original t 5 0) of the resulting motion.

A disk of radius R is suspended from a pivot somewhere between its center and edge (Fig. 13.35). For what pivot point will the period of this physical pendulum be a minimum?

Show by direct substitution that satisfies Equation 13.18 with A given by Equation 13.19.

Youre a structural engineer working on a design for a steel beam, and you need to know its resonant frequency. You test the beam by clamping one end and deflecting the other so it bends, and you determine the associated potential energy. The table below gives the results: Using a spreadsheet or other software, plot U versus and find the best-fit line. Ignore any constant term or, if your software permits, constrain the line to go through since you know when Then, assuming an effective beam mass of 3750 kg, find the resonant frequency.

Show that represents simple harmonic motion, as in Equation 13.8, with and .

Youre working for the summer with an ornithologist who knows youve studied physics. She asks you for a noninvasive way to measure birds masses. You propose using a bird feeder in the shape of a 50-cm-diameter disk of mass 340 g, suspended by a wire with torsional constant (Fig. 13.36). Two birds land on opposite sides and the feeder goes into torsional oscillation at 2.6 Hz. Assuming the birds have the same mass, what is it?

While waiting for your plane to take off, you suspend your keys from a thread and set the resulting pendulum oscillating. It completes exactly 90 cycles in 1 minute. You repeat the experiment as the plane accelerates down the runway, and now measure exactly 91 cycles in 1 minute. Find the planes acceleration.

Youre working for a playground equipment company, which wants to know the rotational inertia of its swing with a child on board; the combined mass is 20 kg. You observe the child twirling around in the swing, twisting the ropes as shown in Fig. 13.37. As a result, child and swing rise slightly, with the rise h in cm equal to the square of the number of full turns. When the child stops twisting, the swing begins torsional oscillations. You measure the period at 6.91 s. What do you report for the rotational inertia of the child-swing system?

If a 90-kg astronaut is weighed with this BMMD, the time for three periods will be a. 50% longer. b. shorter by less than 50%. c. longer by less than 50%. d. longer by more than 50%.

If the same device were used on Earth, the results for a given astronaut (assuming mass hasnt yet been lost in space) would be a. the same. b. greater than in an orbiting spacecraft. c. less than in an orbiting spacecraft. d. meaningless, because the device wont work on Earth.

If an astronauts mass declines linearly with time while shes in orbit, the oscillation period of the BMMD will a. decrease at an ever-decreasing rate. b. decrease linearly with time. c. decrease at an ever-increasing rate. d. increase linearly with time.

The spring constant for the BMMD described here is a. b. c. d. e. none of the above.