Scientists arc using tiny, nanoscale cantilevers 4

Problem 3CQ An air-track cart bounces back and forth between the two ends of an air track. Is this motion periodic? Is it simple harmonic? Explain.

Problem 2P A person in a rocking chair completes 12 cycles in 21 s. What are the period and frequency of the rocking?

Problem 1P A small cart on a 5.0-m-long air track moves with a speed of 0.85 m/s. Bumpers at either end of the track cause the cart to reverse direction and maintain the same speed. Find the period and frequency of this motion.

Problem 1CQ A basketball player dribbles a ball with a steady period of T seconds. Is the motion of the ball periodic? Is it simple harmonic? Explain.

Problem 3P While fishing for catfish, a fisherman suddenly notices that the bobber (a floating device) attached to his line is bobbing up and down with a frequency of 2.6 Hz. What is the period of the bobber's motion?

Problem 106IP If the block's initial speed is increased, does the total time the block is in contact with the spring increase, decrease, or stay the same? (b) Find the total time of contact for v0 = 1.65 m/s, m = 0.980 kg, and k = 245 N/m.

Problem 4CQ If a mass m and a mass 2m oscillate on identical springs with identical amplitudes, they both have the same maximum kinetic energy. How can this be? Shouldn't the larger mass have more kinetic energy? Explain.

Problem 4P If you dribble a basketball with a frequency of 1.77 Hz, how long does it take for you to complete 12 dribbles?

Problem 5P You take your pulse and observe 74 heartbeats in a minute. What are the period and frequency of your heartbeat?

Problem 5CQ An object oscillating with simple harmonic motion completes a cycle in a time T. If the object's amplitude is doubled, the time required for one cycle is still T, even though the object covers twice the distance. How can this be? Explain.

Problem 6CQ The position of an object undergoing simple harmonic motion is given by x =A cos(Bt). Explain the physical significance of the constants A and B. What is the frequency of this object's motion?

Problem 7CQ The velocity of an object undergoing simple harmonic motion is given by v = ?C sin(Dt). Explain the physical significance of the constants C and D. What are the amplitude and period of this object's motion?

Problem 6P (a) Your heart beats with a frequency of 1.45 Hz. How many beats occur in a minute? (b) If the frequency of your heartbeat increases, will the number of beats in a minute increase, decrease, or stay the same? (c) How many beats occur in a minute if the frequency increases to 1.55 Hz?

Problem 9CQ Soldiers on the march are often ordered to break cadence in their step when crossing a bridge. Why is this a good idea?

The pendulum bob in Figure 13–12 leaks sand onto the strip chart. What effect does this loss of sand have on the period of the pendulum? Explain.

Problem 9P A mass moves back and forth in simple harmonic motion with amplitude A and period T. (a) In terms of T, how long does it take for the mass to move through a total distance of 2A? (b) How long does it take for the mass to move through a total distance of 3A?

Problem 7P You rev your car's engine to 2700 rpm (rev/min). (a) What are the period and frequency of the engine? (b) If you change the period of the engine to 0.044 s, how many rpms is it doing?

Problem 8P A mass moves back and forth in simple harmonic motion with amplitude A and period T.(a) In terms of A,through what distance does the mass move in the time T? (b) Through what distance does it move in the time 5T/2?

Problem 10P The position of a mass oscillating on a spring is given by x = (3.2 cm) cos[2?t/(0.58 s)]. (a) What is the period of this motion? (b) What is the first time the mass is at the position x = 0?

Problem 11P The position of a mass oscillating on a spring is given by x = (7.8 cm) cos[2?t/(0.68 s)] (a) What is the frequency of this motion? (b) When is the mass first at the position x = –7.8 cm?

Problem 13P Amass on a spring osculates with simple harmonic motion of amplitude A about the equilibrium position x = 0. Its maximum speed is vmax and its maximum acceleration is amax. (a) What is the speed of the mass at x =0? (b) What is the acceleration of the mass at x =0? (c) What is the speed of the mass at x = A? (d) What is the acceleration of the mass at x = A?

Problem 14P A mass oscillates on a spring with a period of 0.73 s and an amplitude of 5.4 cm. Write an equation giving x as a function of time, assuming the mass starts at x = A at time t = 0.

A position-versus-time plot for an object undergoing simple harmonic motion is given in Figure 13–21. Rank the six points indicated in the figure in order of increasing (a) speed, (b) velocity, and (c) acceleration. Indicate ties where necessary.

Molecular Oscillations An atom in a molecule oscillates about its equilibrium position with a frequency of \(2.00 \times 10^{14} \mathrm{~Hz}\) and a maximum displacement of \(3.50 \mathrm{~nm}\). (a) Write an expression giving \(x\) as a function of time for this atom, assuming that \(x=A\) at \(t=0\). (b) If, instead, we assume that \(x=0\) at \(t=0\), would your expression for position versus time use a sine function or a cosine function? Explain.

Problem 16P A mass oscillates on a spring with a period T and an amplitude 0.48 cm. The mass is at the equilibrium position x = 0 at t = 0, and is moving in the positive direction. Where is the mass at the times (a) t = T/8, (b) t = T/4, (c) t = T/2and (d) t = 3T/4? (e) Plot your results for parts (a) through (d) with the vertical axis representing position and the horizontal axis representing time.

Problem 17P The position of a mass on a spring is given by x = (6.5 cm) cos[2?t/(0.88 s)]. (a) What is the period, T, of this motion? (b) Where is the mass at t = 0.25 s? (c) Show that the mass is at the same location at 0.25 s + T seconds as it is at 0.25 s.

Problem 18P A mass attached to a spring oscillates with a period of 3.35 s. (a) If the mass starts from rest at x = 0.0440 m and time t = 0, where is it at time t = 6.37 s? (b) Is the mass moving in the positive or negative x direction at t =6.37 s? Explain.

Problem 19P An object moves with simple harmonic motion of period T and amplitude A. During one complete cycle, for what length of time is the position of the object greater than A/2?

Problem 20P An object moves with simple harmonic motion of period T and amplitude A. During one complete cycle, for what length of time is the speed of the object greater than vmax/2?

Problem 21P An object executing simple harmonic motion has a maximum speed vmax and a maximum acceleration amax. Find (a) the amplitude and (b) the period of this motion. Express your answers in terms of vmax and amax.

Problem 22P A ball rolls on a circular track of radius 0.62 m with a constant angular speed of 1.3 rad/s in the counterclockwise direction. If the angular position of the ball at t = 0 is ? = 0, find the x component of the ball's position at the times 2.5 s, 5.0 s, and 7.5 s. Let ? = 0 correspond to the positive x direction.

Problem 23P An object executing simple harmonic motion has a maximum speed of 4.3 m/s and a maximum acceleration of 0.65 m/s2. Find (a) the amplitude and (b) the period of this motion.

Problem 2CQ A person rides on a Ferris wheel that rotates with constant angular speed. If the Sun is directly overhead, does the person's shadow on the ground undergo periodic motion? Does it undergo simple harmonic motion? Explain.

Problem 24P A child rocks back and forth on a porch swing with an amplitude of 0.204 m and a period of 2.80 s. Assuming the motion is approximately simple harmonic, find the child's maximum speed.

A 30.0 g goldfinch lands on a slender branch, where it oscillates up and down with simple harmonic motion of amplitude 0.0335 m and period 1.65 s. (a) What is the maximum acceleration of the finch? Express your answer as a fraction of the acceleration of gravity, \(g\). (b) What is the maximum speed of the goldfinch? (c) At the time when the goldfinch experiences its maximum acceleration, is its speed a maximum or a minimum? Explain.

Problem 27P A vibrating structural beam in a spacecraft can cause problems if the frequency of vibration is fairly high. Even if the amplitude of vibration is only a fraction of a millimeter, the acceleration of the beam can be several times greater than the acceleration due to gravity. As an example, find the maximum acceleration of a beam that vibrates with an amplitude of 0.25 mm at the rate of 110 vibrations per second. Give your answer as a multiple of g.

Tuning Forks in NeurologyTuning forks are used in the diagnosis of nervous afflictions known as large-fiber polyneuropathies, which are often manifested in the form of reduced sensitivity to vibrations. Disorders that can result in this type of pathology include diabetes and nerve damage from exposure to heavy metals. The tuning fork in Figure 13–22 has a frequency of 128 Hz. If the tips of the fork move with an amplitude of 1.25 mm, find (a) their maximum speed and (b) their maximum acceleration. Give your answer to part (b) as a multiple of g.

Problem 29P The pistons in an internal combustion engine undergo a motion that is approximately simple harmonic. If the amplitude of motion is 3.5 cm, and the engine runs at 1700 rev/min, find (a) the maximum acceleration of the pistons and (b) their maximum speed.

Problem 30P A 0.84-kg air cart is attached to a spring and allowed to oscillate. If the displacement of the air cart from equilibrium is x = (10.0 cm) cos[(2.00 s?1)t + ?], find (a) the maximum kinetic energy of the cart and (b) the maximum force exerted on it by the spring.

A person rides on a mechanical bucking horse (see Figure 13–23) that oscillates up and down with simple harmonic motion. The period of the bucking is 0.74 s and the amplitude is slowly increasing. At a certain amplitude the rider must hang on to prevent separating from the mechanical horse. (a) Give a strategy that will allow you to calculate this amplitude. (b) Carry out your strategy and find the desired amplitude.

Problem 28P A peg on a turntable moves with a constant tangential speed of 0.77 m/s in a circle of radius 0.23 m. The peg casts a shadow on a wall. Find the following quantities related to the motion of the shadow: (a) the period, (b) the amplitude, (c) the maximum speed, and (d) the maximum magnitude of the acceleration.

Problem 32P If a mass m is attached to a given spring,. its period of oscillation is T. If two such springs are connected end to end and the same mass m is attached, (a) is the resulting period of oscillation greater than, less than, or equal to T? (b) Choose the best explanation from among the following: I. Connecting two springs together makes the spring suffer, which means that less time is required for an oscillation. II. The period of oscillation does not depend on the length of a spring, only on its force constant and the mass attached to it. III. The longer spring stretches more easily, and hence takes longer to complete an oscillation.

The two blocks in Figure 13–24 have the same mass, m. All the springs have the same force constant, k, and are at their equilibrium length. When the blocks are set into oscillation, (a) is the period of block 1 greater than, less than, or equal to the period of block 2? (b) Choose the best explanation from among the following: I. Springs in parallel are stiffer than springs in series; therefore the period of block 1 is smaller than the period of block 2. II. The two blocks experience the same restoring force for a given displacement from equilibrium, and hence they have equal periods of oscillation. III. The force of the two springs on block 2 partially cancel one another, leading to a longer period of oscillation.

Problem 33P An old car with worn-out shock absorbers oscillates with a given frequency when it hits a speed bump. If the driver adds a couple of passengers to the car and hits another speed bump, (a) is the car's frequency of oscillation greater than, less than, or equal to what it was before? (b) Choose the best explanation from among the following: I. Increasing the mass on a spring increases its period, and hence decreases its frequency. II. The frequency depends on the force constant of the spring but is independent of the mass. III. Adding mass makes the spring oscillate more rapidly, which increases the frequency.

Show that the units of the quantity \(\sqrt{k/m}\) are \(\mathrm s^-1\). ________________ Equation Transcription: Text Transcription: sqrt{k/m} s^-1

Problem 36P A 0.46-kg mass attached to a spring undergoes simple harmonic motion with a period of 0.77 s. What is the force constant of the spring?

Problem 37P System A consists of a mass m attached to a spring with a force constant k;system B has a mass 2m attached to a spring with a force constant k;system C has a mass 3m attached to a spring with a force constant 6k; and system D has a mass m attached to a spring with a force constant 4k. Rank these systems in order of Increasing period of oscillation.

Find the periods of block 1 and block 2 in Figure 13–24, given that \(k=49.2\ \mathrm{N/m}\) and \(m=1.25\ \mathrm{kg}\). ________________ Equation Transcription: Text Transcription: k=49.2 N/m m=1.25 kg

Problem 39P When a 0.50-kg mass is attached to a vertical spring, the spring stretches by 15 cm. How much mass must be attached to the spring to result in a 0.75-s period of oscillation?

Problem 41P Two people with a combined mass of 125 kg hop into an old car with worn-out shock absorbers. This causes the springs to compress by 8.00 cm. When the car hits a bump in the road, it oscillates up and down with a period of 1.65 s. Find (a) the total load supported by the springs and (b) the mass of the car.

Problem 42P A 0.85-kg mass attached to a vertical spring of force constant 150 N/m oscillates with a maximum speed of 0.35 m/s. Find the following quantities related to the motion of the mass: (a) the period, (b) the amplitude, (c) the maximum magnitude of the acceleration.

Problem 44P The springs of a 511-kg motorcycle have an effective force constant of 9130 N/m. (a) If a person sits on the motorcycle, does its period of oscillation increase, decrease, or stay the same? (b) By what percent and in what direction does the period of oscillation change when a 112-kg person rides the motorcycle?

Problem 45P If a mass m is attached to a given spring, its period of oscillation is T. If two such springs are connected end to end, and the same mass m is attached, (a) is its period greater than, less than, or the same as with a single spring? (b) Verify your answer to part (a) by calculating the new period, T',in terms of the old period T.

Problem 43P When a 0.213-kg mass is attached to a vertical spring, it causes the spring to stretch a distance d. If the mass is now displaced slightly from equilibrium, it is found to make 102 oscillations in 56.7 s. Find the stretch distance, d.

Problem 46P How much work is required to stretch a spring 0.133 m if its force constant is 9.17 N/m?

Problem 48P Find the total mechanical energy of the system described in the previous problem.

Problem 49P A 1.8-kg mass attached to aspring oscillates with an amplitude of 7.1 cm and a frequency of 2.6 Hz. What is its energy of motion?

Problem 53P What is the maximum speed of the grapes in the previous problem if their amplitude of oscillation is 2.3 cm?

Problem 54P A 0.505-kg block slides on a frictionless horizontal surface with a speed of 1.18 m/s. The block encounters an unstretched spring and compresses it 23.2 cm before coming to rest. ?(a) What is the force constant of this spring? ?(b) For what length of time is the block in contact with the spring before it comes to rest? ?(c) If the force constant of the spring is increased, docs the time required to stop the block increase, decrease, or stay the same? Explain.

Problem 51P (a) What is the maximum speed of the mass in the previous problem? (b) How far is the mass from the equilibrium position when its speed is half the maximum speed?

Problem 50P A 0.40-kg mass is attached to a spring with a force constant of 26 N/m and released from rest a distance of 3.2 cm from the equilibrium position of the spring. (a) Give a strategy that allows you to find the speed of the mass when it is halfway to the equilibrium position. (b) Use your strategy to find this speed.

Problem 52P A bunch of grapes is placed in a spring scale at a supermarket. The grapes oscillate up and down with a period of 0.48 s, and the spring in the scale has a force constant of 650 N/m. What are (a) the mass and (b) the weight of the grapes?

Problem 54P A 0.505-kg block slides on a frictionless horizontal surface with a speed of 1.18 m/s. The block encounters an unstretched spring and compresses it 23.2 cm before coming to rest. (a) What is the force constant of this spring? (b) For what length of time is the block in contact with the spring before it comes to rest? (c) If the force constant of the spring is increased, docs the time required to stop the block increase, decrease, or stay the same? Explain.

Problem 55P A 2.25-g bullet embeds itself in a 1.50-kg block, which is attached to a spring of force constant 785 N/m. If the maximum compression of the spring is 5.88 cm, find (a) the initial speed of the bullet and (b) the time for the bullet-block system to come to rest.

Problem 58P A pendulum of length L has a period T. How long must the pendulum be if its period is to be 2T?

Problem 62P Find the length of a simple pendulum that has a period oi 1.00 s. Assume that the acceleration of gravity is g = 9.81 m/s2.

Problem 63P If the pendulum in the previous problem were to be taken to the Moon, where the acceleration of gravity is g/6, (a) would its period increase, decrease, or stay the same? (b) Check your result in part (a) by calculating the period of the pendulum on the Moon.

Problem 60P A simple pendulum of length 2.5 m makes 5.0 complete swings in 16 s. What is the acceleration of gravity at the location of the pendulum?

An observant fan at a baseball game notices that the radio commentators have lowered a microphone from their booth to just a few inches above the ground, as shown in Figure 13–25. The microphone is used to pick up sound from the field and from the fans. The fan also notices that the microphone is slowly swinging back and forth like a simple pendulum. Using her digital watch, she finds that 10 complete oscillations take 60.0 s. How high above the field is the radio booth? (Assume the microphone and its cord can be treated as a simple pendulum.)

Problem 61P A large pendulum with a 200-lb gold-plated bob 12 inches in diameter is on display in the lobby of the United Nations building. The penduliun has a length of 75 ft. It is used to show the rotation of the Earth—for this reason it is referred to as a Foucault pendulum, What is the least amount of time it takes for the bob to swing from a position of maximum displacement to the equilibrium position of the pendulum? (Assume that the acceleration due to gravity is g = 9.81 m /s2 at the UN building.)

A fireman tosses his 0.98-kg hat onto a peg, where it oscillates as a physical pendulum (Figure 13–26). If the center of mass of the hat is 8.4 cm from the pivot point, and its period of oscillation is 0.73 s, what is the moment of inertia of the hat about the pivot point?

Problem 64P A hula hoop hangs from a peg. Find the period of the hoop as it gently rocks back and forth on the peg. (For a hoop with axis at the rim I = 2mR2, where R is the radius of the hoop.)

Problem 67P On the construction site for a new skyscraper, a uniform beam of steel is suspended from one end. If the beam swings back and forth with a period of 2.00 s, what is its length?

Problem 68P (a) Find the period of a child's leg as it swings about the hip joint. Assume the leg is 0.55 m long and can be treated as a uniform rod. (b) Estimate the child's walking speed.

Problem 66P Consider a meter stick that oscillates back and forth about a pivot point at one of its ends. (a) Is the period of a simple pendulum of length L =1.00 m greater than, less than, or the same as the period of the meterstick? Explain. (b) Find the length L of a simple pendulum that has a period equal to the period of the meterstick.

Problem 70GP An object undergoes simple harmonic motion with a period T. In the time 3T/2 the object moves through a total distance of 12D. In terms of D, what is the object's amplitude of motion?

Problem 69P Suspended from the ceiling of an elevator is a simple pendulum of length L. What is the period of this pendulum if the elevator (a) accelerates upward with an acceleration a, or (b) accelerates downward with an acceleration whose magnitude is greater than zero but less than g? Give your answer in terms of L, g,and a.

Problem 71GP A mass on a string moves with simple harmonic motion. If the period of motion is doubled, with the force constant and the amplitude remaining the same, by what multiplicative factor do the following quantities change: (a) angular frequency, (b) frequency, (c) maximum speed, (d) maximum acceleration, (e) total mechanical energy?

Problem 72GP If the amplitude of a simple harmonic oscillator is doubled, by what multiplicative factor do the following quantities change: (a) angular frequency, (b) frequency, (c) period, (d) maximum. speed, (e) maximum acceleration, (f) total mechanical energy?

Problem 73GP A mass m is suspended from the ceiling of an elevator by a spring of force constant k. When the elevator is at rest, the period of the mass is T. Does the period increase, decrease, or remain the same when the elevator (a) moves upward with constant speed or (b) moves upward with constant acceleration?

Problem 74GP A pendulum of length L is suspended from the ceiling of an elevator. When the elevator is at rest, the period of the pendulum is T. Does the period increase, decrease, or remain the same when the elevator (a) moves upward with constant speed or (b) moves upward with constant acceleration?

Problem 75GP A 1.8-kg mass is attached to a spring with a force constant of 59 N/m. If the mass is released with a speed of 0.25 m/s at a distance of 8.4 cm from the equilibrium position of the spring, what is its speed when it is halfway to the equilibrium position?

Problem 76GP An astronaut uses a Body Mass Measurement Device (BMMD) to determine her mass. What is the astronaut's mass, given that the force constant of the BMMD is 2600 N/m and the period of oscillation is 0.85 s? (See the discussion on page 427 for more details on the BMMD.)

Problem 77GP A typical atom in a solid might oscillate with a frequency of 1012 Hz and an amplitude of 0.10 angstrom (10?11 m). Find the maximum acceleration of the atom and compare it with the acceleration of gravity.

Problem 78GP Sunspots vary in number as a function of time, exhibiting an approximately 11-year cycle. Galileo made the first European observations of sunspots in 1610, and daily observations were begun in Zurich in 1749. At the present time we are well into the 23rd observed cycle. What is the frequency of the sunspot cycle? Give your answer in Hz.

Weighing a Bacterium Scientists are using tiny, nanoscale cantilevers 4 micrometers long and 500 nanometers wide—essentially miniature diving boards—as a sensitive way to measure mass. The cantilevers oscillate up and down with a frequency that depends on the mass placed near the tip, and a laser beam is used to measure the frequency. A single E. coli bacterium was measured to have a mass of \(\text {665 femtograms}=6.65\times 10^{-16}\ \mathrm {kg}\) with this device, as the cantilever oscillated with a frequency of 14.5 MHz. Treating the cantilever as an ideal, massless spring, find its effective force constant. ________________ Equation Transcription: Text Transcription: 665 femtograms=6.65x10^{-16} kg

Problem 80GP An object undergoing simple harmonic motion with a period T is at the position x = 0 at the time t =0. At the time t = 0.25T the position of the object is positive. State whether x is positive, negative, or zero at the following times: (a) t = 1.5T, (b) t = 2T, (c) t = 2.25T, and (d) t = 6.75T.

Problem 81GP The maximum speed of a 3.1-kg mass attached to aspring is 0.68 m/s, and the maximum force exerted on the mass is 11 N. (a) What is the amplitude of motion for this mass? (b) What is the force constant of the spring? (c) What is the frequency of this system?

Helioseismology In 1962, physicists at Cal Tech discovered that the surface of the Sun vibrates due to the violent nuclear reactions that roil within its core. This has led to a new field of solar science known as helioseismology. A typical vibration of the Sun is shown in Figure 13–27; it has a period of 5.7 minutes. The blue patches in Figure 13–27 are moving outward; the red patches are moving inward. (a) Find the angular frequency of this vibration. (b) The maximum speed at which a patch of the surface moves during a vibration is 4.5 m/s. What is the amplitude of the vibration, assuming it to be simple harmonic motion?

Problem 85GP A 1.44-g spider oscillates on its web, which has a damping constant of 3.30 × 10?5 kg/s. How long does it take for the spider's amplitude of osculation to decrease by 10.0 percent?

Problem 84GP A 9.50-g bullet, moving horizontally with an initial speed v0, embeds itself in a 1.45-kg pendulum bob that is initially at rest. The length of the pendulum is L = 0.745 m. After the collision, the pendulum swings to one side and comes to rest when it has gained a vertical height of 12.4 cm. (a) Is the kinetic energy of the bullet-bob system immediately after the collision greater than, less than, or the same as the kinetic energy of the system just before the collision? Explain. (b) Find the initial speed of the bullet. (c) How long does it take for the bullet-bob system to come to rest for the first time?

Problem 82GP The acceleration of a block attached to a spring is given by a = ?(0.302 m/s2) cos([2.41 rad/s]t). (a) What is the frequency of the block's motion? (b) What is the maximum, speed of the block? (c) What is the amplitude of the block's motion?

Problem 86GP An object undergoes simple harmonic motion with a period T and amplitude A. In terms of T, how long does it take the object to travel from x = A to x = A/2?

Problem 87GP Find the period of oscillation of a disk of mass 0.32 kg and radius 0.15 m if it is pivoted about a small hole drilled near its rim.

Problem 89GP A 0.363-kg mass slides on a frictionless floor with a speed of 1.24 m/s. The mass strikes and compresses a spring with a force constant of 44.5 N/m. (a) How far docs the mass travel after contacting the spring before it comes to rest? (b) How long does it take for the spring to stop the mass?

Problem 88GP Calculate the ratio of the kinetic energy to the potential energy of a simple harmonic oscillator when its displacement is half its amplitude.

Problem 90GP A large rectangular barge floating on a lake oscillates up and down with a period of 4.5 s. Find the damping constant for the barge, given that its mass is 2.44 × 105 kg and that its amplitude of oscillation decreases by a factor of 2.0 in 5.0 minutes.

Problem 93GP A 0.45-kg crow lands on a slender branch and bobs up and down with a period of 1.5 s. An eagle flies up to the same branch, scaring the crow away, and lands. The eagle now bobs up and down with a period of 4.8 s. Treating the branch as an ideal spring, find (a) the effective force constant of the branch and (b) the mass of the eagle.

Figure 13–28 shows a displacement-versus-time graph of the periodic motion of a 3.8-kg mass on a spring. (a) Referring to the figure, do you expect the maximum speed of the mass to be greater than, less than, or equal to 0.50 m/s? Explain. (b) Calculate the maximum speed of the mass. (c) How much energy is stored in this system?

A 3.8-kg mass on a spring oscillates as shown in the displacement-versus-time graph in Figure 13–28. (a) Referring to the graph, at what times between \(t=0\) and \(t=6.0\ \mathrm s\) does the mass experience a force of maximum magnitude? Explain. (b) Calculate the magnitude of the maximum force exerted on the mass. (c) At what times shown in the graph does the mass experience zero force? Explain. (d) How much force is exerted on the mass at the time \(t=0.50\ \mathrm s\)? ________________ Equation Transcription: Text Transcription: t=0 t=6.0 s t=0.50 s

A mass m is connected to the bottom of a vertical spring whose force constant is k. Attached to the bottom of the mass is a string that is connected to a second mass m, as shown in Figure 13–29. Both masses are undergoing simple harmonic vertical motion of amplitude A. At the instant when the acceleration of the masses is a maximum in the upward direction the string breaks, allowing the lower mass to drop to the floor. Find the resulting amplitude of motion of the remaining mass.

Consider the pendulum shown in Figure 13–30. Note that the pendulum’s string is stopped by a peg when the bob swings to the left, but moves freely when the bob swings to the right. (a) Is the period of this pendulum greater than, less than, or the same as the period of the same pendulum without the peg? (b) Calculate the period of this pendulum in terms of L and \(\ell\). (c) Evaluate your result for \(L=1.0\ \mathrm m\) and \(\ell=0.25\ \mathrm m\). ________________ Equation Transcription: ? Text Transcription: L=1.0 m ell=0.25 m

Problem 96GP When a mass m is attached to a vertical spring with a force constant k,it stretches the spring by the amount L. Calculate (a) the period of this mass and (b) the period of a simple pendulum of length L.

An object undergoes simple harmonic motion of amplitude A and angular frequency about \(\omega\) the equilibrium point \(x=0\). Use energy conservation to show that the speed of the object at the general position x is given by the following expression: \(v=\omega \sqrt{A^2-x^2}\) ________________ Equation Transcription: Text Transcription: omega x=0 v=omega sqrt{A^2-x^2}

A physical pendulum consists of a light rod of length L suspended in the middle. A large mass \(m_1\) is attached to one end of the rod, and a lighter mass \(m_2\) is attached to the other end, as illustrated in Figure 13–31. Find the period of oscillation for this pendulum. ________________ Equation Transcription: Text Transcription: m_1 m_2 m_2 L/2 L/2 m_1

A vertical hollow tube is connected to a speaker, which vibrates vertically with simple harmonic motion (Figure 13–32). The speaker operates with constant amplitude, A, but variable frequency, ????. A slender pencil is placed inside the tube. (a) At low frequencies the pencil stays in contact with the speaker at all times; at higher frequencies the pencil begins to rattle. Explain the reason for this behavior. (b) Find an expression for the frequency at which rattling begins.

If the temperature is increased by 10 degrees Fahrenheit, how many additional chirps are heard in a 13-s interval? A. 5 B. 10 C. 13 D. 39

What is the temperature in degrees Fahrenheit if a cricket is observed to give 35 chirps in 13 s? A. \(13 ^\circ \mathrm F\) B. \(35 ^\circ \mathrm F\) C. \(74 ^\circ \mathrm F\) D. \(90 ^\circ \mathrm F\) ________________ Equation Transcription: Text Transcription: 13 ^oF 35 ^oF 74 ^oF 90 ^oF

Problem 104IP Suppose we can change the plane's period of oscillation, while keeping its amplitude of motion equal to 30.0 m. (a) If we want to reduce the maximum acceleration of the plane, should we increase or decrease the period? Explain. (b) Find the period that results in a maximum acceleration of 1.0g.

Problem 105IP Suppose the force constant of the spring is doubled, but the mass and speed of the block are still 0.980 kg and 1.32 m/s, respectively. (a) By what multiplicative factor do you expect the maximum compression of the spring to change? Explain. (b) Find the new maximum compression of the spring. (c) Find the time required for the mass to come to rest after contacting the spring.

What is the frequency of the cricket’s chirping (in Hz) when the temperature is \(68\ ^\circ \mathrm F\)? A. 0.45 Hz B. 2.2 Hz C. 5.2 Hz D. 29 Hz ________________ Equation Transcription: Text Transcription: 68 ^oF

Suppose the temperature decreases uniformly from \(75\ ^\circ \mathrm F\) to \(63\ ^\circ \mathrm F\) in 12 minutes. How many chirps does the cricket produce during this time? A. 28 B. 1700 C. 3800 D. 22,000 ________________ Equation Transcription: Text Transcription: 75 ^oF 63 ^oF

Problem 40P A spring with a force constant of 69 N/m is attached to a 0.57-kg mass. Assuming that the amplitude of motion is 3.1 cm, determine the following quantities for this system: (a) ?, (b) vmax, (c) T.

Metronomes, such as the penguin shown in the photo, are useful devices for music students. If it is desired to have the metronome tick with a greater frequency, should the penguin’s bow tie be moved upward or downward?

Problem 47P A 0.321-kg mass is attached to a spring with a force constant of 13.3 N/m. If the mass is displaced 0.256 m from equilibrium and released, what is its speed when it is 0.128 m from equilibrium?