Figure 14-29 shows a uniform disk that has a radius a mass

True or false: (a) For a simple harmonic oscillator, the period is proportional to the square of the amplitude. (b) For a simple harmonic oscillator, the frequency does not depend on the amplitude. (c) If the net force on a particle undergoing one-dimensional motion is proportional to, and oppositely directed from, the displacement from equilibrium, the motion is simple harmonic.

If the amplitude of a simple harmonic oscillator is tripled, by what factor is the energy changed?

An object attached to a spring exhibits simple harmonic motion with an amplitude of 4.0 cm. When the object is 2.0 cm from the equilibrium position, what percentage of its total mechanical energy is in the form of potential energy? (a) one-quarter, (b) one-third, (c) one-half, (d) two-thirds, (e) three-quarters

An object attached to a spring exhibits simple harmonic motion with an amplitude of 10.0 cm. How far from equilibrium will the object be when the systems potential energy is equal to its kinetic energy? (a) 5.00 cm, (b) 7.07 cm, (c) 9.00 cm, (d) The distance cannot be determined from the data given.

Two identical systems each consist of a spring with one end attached to a block and the other end attached to a wall. The springs are horizontal, and the blocks are supported from below by a frictionless horizontal table. The blocks are oscillating in simple harmonic motions such that the amplitude of the motion of block A SSM is four times as large as the amplitude of the motion of block B. How do their maximum speeds compare? (a) (b) (c) (d) This comparison cannot be done by using the data given.

Two systems each consist of a spring with one end attached to a block and the other end attached to a wall. The springs are horizontal, and the blocks are supported from below by a frictionless horizontal table. The identical blocks are oscillating in simple harmonic motions with equal amplitudes. However, the force constant of spring Ais four times as large as the force constant of spring B. How do their maximum speeds compare? (a) (b) (c) (d) This comparison cannot be done by using the data given.

Two systems each consist of a spring with one end attached to a block and the other end attached to a wall. The identical springs are horizontal, and the blocks are supported from below by a frictionless horizontal table. The blocks are oscillating in simple harmonic motions with equal amplitudes. However, the mass of block A is four times as large as the mass of block B. How do their maximum speeds compare? (a) (b) (c) (d) This comparison cannot be done by using the data given.

Two systems each consist of a spring with one end attached to a block and the other end attached to a wall. The identical springs are horizontal, and the blocks are supported from below by a frictionless horizontal table. The blocks are oscillating in simple harmonic motions with equal amplitudes. However, the mass of block A is four times as large as the mass of block B. How do the magnitudes of their maximum accelerations compare? (a) (b) (c) (d) (e) This comparison cannot be done by using the data given.

In general physics courses, the mass of the spring in simple harmonic motion is usually neglected because its mass is usually much smaller than the mass of the object attached to it. However, this is not always the case. If you neglect the mass of the spring when it is not negligible, how will your calculation of the systems period, frequency, and total energy compare to the actual values of these parameters? Explain.

Two massspring systems oscillate with periods and If and the systems springs have identical force constants, it follows that the systems masses are related by (a) (b) (c) (d)

Two massspring systems oscillate at frequencies and If and the systems springs have identical force constants, it follows that the systems masses are related by (a) (b) (c) (d)

Two massspring systems and oscillate so that their total mechanical energies are equal. If which expression best relates their amplitudes? (a) (b) (c) (d) Not enough information is given to determine the ratio of the amplitudes.

Two mass–spring systems A and B oscillate so that their total mechanical energies are equal. If the force constant of spring A is two times as large as the force constant of spring B, then which expression best relates their amplitudes? (a) \(A_A=A_B/4\), (b) \(A_A=A_B/\sqrt 2\), (c) \(A_A=A_B\), (d) Not enough information is given to determine the ratio of the amplitudes.

The length of the string or wire supporting a pendulum bob increases slightly when the temperature of the string or wire increases. How does this affect a clock operated by a simple pendulum?

A lamp hanging from the ceiling of the club car in a train oscillates with period when the train is at rest. The period will be (match left and right columns) 1. greater than when A. The train moves horizontally at constant velocity. 2. less than when B. The train rounds a curve at constant speed. 3. equal to when C. The train climbs a hill at constant speed. D. The train goes over the crest of a hill at constant speed.

Two simple pendulums are related as follows. Pendulum A has a length and a bob of mass pendulum B has a length and a bob of mass If the period of A is twice that of B, then (a) and (b) and (c) whatever the ratio (d) whatever the ratio

Two simple pendulums are related as follows. Pendulum Ahas a length and a bob of mass pendulum B has a length and a bob of mass If the frequency of A is one-third the frequency of B, then (a) and (b) and (c) regardless of the ratio (d) regardless of the ratio

Two simple pendulums are related as follows. Pendulum A has a length and a bob of mass pendulum B has a length and a bob of mass They have the same period. If the only difference between their motions is that the amplitude of As motion is twice the amplitude of Bs motion, then (a) and (b) and (c) whatever the ratio (d) whatever the ratio mA>mBL .

True or false: (a) The mechanical energy of a damped, undriven oscillator decreases exponentially with time. (b) Resonance for a damped, driven oscillator occurs when the driving frequency exactly equals the natural frequency. (c) If the factor of a damped oscillator is high, then its resonance curve will be narrow. (d) The decay time for a spring-mass oscillator with linear damping is independent of its mass. (e) The factor for a driven spring-mass oscillator with linear damping is independent of its mass.

Two damped spring-mass oscillating systems have identical spring and damping constants. However, system As mass is four times system Bs mass . How do their decay times compare? (a) (b) (c) (d) Their decay times cannot be compared, given the information provided.

Two damped spring-mass oscillating systems have identical spring constants and decay times. However, system As mass is twice system Bs mass How do their damping constants, compare? (a) (b) (c) (d) (e) Their decay times cannot be compared, given the information provided.

Two damped, driven spring-mass oscillating systems have identical driving forces as well as identical spring and damping constants. However, the mass of system Ais four times the mass of system B. Assume both systems are very weakly damped. How do their resonant frequencies compare? (a) (b) (c) (d) (e) Their resonant frequencies cannot be compared, given the information provided.

Two damped, driven spring-mass oscillating systems have identical masses, driving forces, and damping constants. However, system As force constant is four times system Bs force constant Assume they are both very weakly damped. How do their resonant frequencies compare? (a) (b) (c) (d) (e) Their resonant frequencies cannot be compared, given the information provided.

Two damped, driven simple-pendulum systems have identical masses, driving forces, and damping constants. However, system As length is four times system Bs length. Assume they are both very weakly damped. How do their resonant frequencies compare? (a) (b) (c) (d) (e) Their resonant frequencies cannot be compared, given the information provided.

Estimate the width of a typical grandfather clocks cabinet relative to the width of the pendulum bob, presuming the desired motion of the pendulum is simple harmonic.

A small punching bag for boxing workouts is approximately the size and weight of a persons head and is suspended from a very short rope or chain. Estimate the natural frequency of oscillations of such a punching bag.

For a child on a swing, the amplitude drops by a factor of in about eight periods if no additional mechanical energy is given to the system. Estimate the factor for this system.

(a) Estimate the natural period of oscillation for swinging your arms as you walk, when your hands are empty. (b) Now estimate the natural period of oscillation when you are carrying a heavy briefcase. (c) Observe other people while they walk. Do your estimates seem reasonable?

The position of a particle is given by where is in seconds. What are (a) the frequency, (b) the period, and (c) the amplitude of the particles motion? (d) What is the first time after that the particle is at its equilibrium position? In what direction is it moving at that time?

What is the phase constant in (Equation 14.4) if the position of the oscillating particle at time is (a) 0, (b) (c) and (d)

A particle of mass begins at rest from and oscillates about its equilibrium position at with a period of 1.5 s. Write expressions for (a) the position as a function of (b) the velocity as a function of and (c) the acceleration as a function of

Find (a) the maximum speed, and (b) the maximum acceleration of the particle in Problem 29. (c) What is the first time that the particle is at and moving to the right?

Work Problem 31 for when the particle is initially at and moving with velocity

The period of a particle that is oscillating in simple harmonic motion is 8.0 s and its amplitude is 12 cm. At it is at its equilibrium position. Find the distance the particle travels during the intervals (a) to (b) to (c) to and (d) to

The period of a particle oscillating in simple harmonic motion is 8.0 s. At the particle is at rest at (a) Sketch as a function of (b) Find the distance traveled in the first, second, third, and fourth second after

ENGINEERING APPLICATION, CONTEXT-RICH Military specifications often call for electronic devices to be able to withstand accelerations of up to To make sure that your companys products meet this specification, your manager has told you to use a shaking table, which can vibrate a device at controlled and adjustable frequencies and amplitudes. If a device is placed on the table and made to oscillate at an amplitude of 1.5 cm, what should you adjust the frequency to in order to test for compliance with the military specification?

The position of a particle is given by where is in meters and is in seconds. (a) Find the maximum speed and maximum acceleration of the particle. (b) Find the speed and acceleration of the particle when

(a) Show that can be written as and determine and in terms of and (b) Relate and to the initial position and velocity of a particle undergoing simple harmonic motion.

A particle moves at a constant speed of in a circle of radius 40 cm centered at the origin. (a) Find the frequency and period of the component of its position. (b) Write an expression for the component of the particles position as a function of time assuming that the particle is located on the axis at time

A particle moves in a 15-cm-radius circle centered at the origin and completes 1.0 rev every 3.0 s. (a) Find the speed of the particle. (b) Find its angular speed (c)Write an equation for the component of the particles position as a function of time t, assuming that the particle is on the _x axis at time t _ 0.

A 2.4-kg object on a frictionless hoizontal surface is attached to one end of a horizontal spring of force constant The other end of the spring is held stationary. The spring is stretched 10 cm from equilibrium and released. Find the systems total mechanical energy.

Find the total energy of a system consisting of a 3.0-kg object on a frictionless horizontal surface oscillating with an amplitude of 10 cm and a frequency of 2.4 Hz at the end of a horizontal spring.

A 1.50-kg object on a frictionless horizontal surface oscillates at the end of a spring (force constant ). The objects maximum speed is (a) What is the systems total mechanical energy? (b) What is the amplitude of the motion?

A 3.0-kg object on a frictionless horizontal surface oscillating at the end of a spring that has a force constant equal to has a total mechanical energy of 0.90 J. (a) What is the amplitude of the motion? (b) What is the maximum speed?

An object on a frictionless horizontal surface oscillates at the end of a spring with an amplitude of 4.5 cm. Its total mechanical energy is 1.4 J. What is the force constant of the spring?

A 3.0-kg object on a frictionless horizontal surface oscillates at the end of a spring with an amplitude of 8.0 cm. Its maximum acceleration is Find the total mechanical energy.

A 2.4-kg object on a frictionless horizontal surface is attached to the end of a horizontal spring that has a force constant The spring is stretched 10 cm from equilibrium and released. What are (a) the frequency of the motion, (b) the period, (c) the amplitude, (d) the maximum speed, and (e) the maximum acceleration? (f) When does the object first reach its equilibrium position? What is its acceleration at this time?

A 5.00-kg object on a frictionless horizontal surface is attached to one end of a horizontal spring that has a force constant . The spring is stretched 8.00 cm from equilibrium and released. What are (a) the frequency of the motion, (b) the period, (c) the amplitude, (d) the maximum speed, and (e) the maximum acceleration? (f) When does the object first reach its equilibrium position? What is its acceleration at this time?

A 3.0-kg object on a frictionless horizonal surface is attached to one end of a horizontal spring and oscillates with an amplitude and a frequency (a) What is the force constant of the spring? (b) What is the period of the motion? (c) What is the maximum speed of the object? (d) What is the maximum acceleration of the object?

An 85.0-kg person steps into a car of mass 2400 kg, causing it to sink 2.35 cm on its springs. If started into vertical oscillation, and assuming no damping, at what frequency will the car and passenger vibrate on these springs?

A 4.50-kg object oscillates on a horizontal spring with an amplitude of 3.80 cm. The objects maximum acceleration is Find (a) the force constant of the spring, (b) the frequency of the object, and (c) the period of the motion of the object.

An object of mass is suspended from a vertical spring of force constant When the object is pulled down 2.50 cm from equilibrium and released from rest, the object oscillates at 5.50 Hz. (a) Find (b) Find the amount the spring is stretched from its unstressed length when the object is in equilibrium. (c) Write expressions for the displacement the velocity and the acceleration a as functions of time t. x vx

An object is hung on the end of a vertical spring and is released from rest with the spring unstressed. If the object falls 3.42 cm before first coming to rest, find the period of the resulting oscillatory motion.

A suitcase of mass 20 kg is hung from two bungee cords, as shown in Figure 14-27. Each cord is stretched 5.0 cm when the suitcase is in equilibrium. If the suitcase is pulled down a little and released, what will be its oscillation frequency?

A 0.120-kg block is suspended from a spring. When a small pebble of mass 30 g is placed on the block, the spring stretches an additional 5.0 cm. With the pebble on the block, the block oscillates with an amplitude of 12 cm. (a) What is the frequency of the motion? (b) How long does the block take to travel from its lowest point to its highest point? (c) What is the net force on the pebble when it is at the point of maximum upward displacement?

Referring to Problem 55, find the maximum amplitude of oscillation at which the pebble will remain in contact with the block.

An object of mass 2.0 kg is attached to the top of a vertical spring that is anchored to the floor. The unstressed length of the spring is 8.0 cm and the length of the spring when the object is in equilibrium is 5.0 cm. When the object is resting at its equilibrium position, it is given a sharp downward blow with a hammer so that its initial speed is (a) To what maximum height above the floor does the object eventually rise? (b) How long does it take for the object to reach its maximum height for the first time? (c) Does the spring ever become unstressed? What minimum initial velocity must be given to the object for the spring to be unstressed at some time?

ENGINEERING APPLICATION A winch cable has a crosssectional area of and a length of 2.5 m. Youngs modulus for the cable is A950-kg engine block is hung from the end of the cable. (a) By what length does the cable stretch? (b) If we treat the cable as a simple spring, what is the oscillation frequency of the engine block at the end of the cable?

Find the length of a simple pendulum if its frequency for small amplitudes is 0.75 Hz.

Find the length of a simple pendulum if its period for small amplitudes is 5.0 s.

What would the period of the pendulum in Problem 60 be if the pendulum were on the moon, where the acceleration due to gravity is one-sixth that on Earth?

If the period of a 70.0-cm-long simple pendulum is 1.68 s, what is the value of at the location of the pendulum?

A simple pendulum that is set up in the stairwell of a 10- story building consists of a heavy weight suspended on a 34.0-mlong wire. What is the period of oscillation?

Show that the total energy of a simple pendulum undergoing oscillations of small amplitude (in radians) is Hint: Use the approximation for small

A simple pendulum of length is attached to a massive cart that slides without friction down a plane inclined at angle with the horizontal, as shown in Figure 14-28. Find the period of oscillation for small oscillations of this pendulum. SSM

The bob at the end of a simple pendulum of length is released from rest from an angle (a) Model the pendulums motion as simple harmonic motion, and find its speed as it passes through by using the small angle approximation. (b) Using the conservation of energy, find this speed exactly for any angle (not just small angles). (c) Show that your result from Part (b) agrees with the approximate answer in Part (a) when is small. (d) Find the difference between the approximate and exact results for and (e) Find the difference between the approximate and exact results for and

A thin 5.0-kg uniform disk with a 20-cm radius is free to rotate about a fixed horizontal axis perpendicular to the disk and passing through its rim. The disk is displaced slightly from equilibrium and released. Find the period of the subsequent simple harmonic motion.

Acircular hoop that has a 50-cm radius is hung on a narrow horizontal rod and allowed to swing in the plane of the hoop. What is the period of its oscillation, assuming that the amplitude is small?

A 3.0-kg plane figure is suspended at a point 10 cm from its center of mass. When it is oscillating with small amplitude, the period of oscillation is 2.6 s. Find the moment of inertia I about an axis perpendicular to the plane of the figure through the pivot point.

ENGINEERING APPLICATION, CONTEXT-RICH, CONCEPTUAL You have designed a cat door that consists of a square piece of plywood that is 1.0 in. thick and 6.0 in. on a side, and is hinged at its top. To make sure the cat has enough time to get through it safely, the door should have a natural period of at least 1.0 s. Will your design work? If not, explain qualitatively what you would need to do to make it meet your requirements.

You are given a meterstick and asked to drill a small diameter hole through it so that, when the stick is pivoted about a horizontal axis through the hole, the period of the pendulum will be a minimum. Where should you drill the hole?

Figure 14-29 shows a uniform disk that has a radius a mass of 6.00 kg, and a small hole a distance from the disks center that can serve as a pivot point. (a) What should be the distance so that the period of this physical pendulum is 2.50 s? (b) What should be the distance so that this physical pendulum will have the shortest possible period? What is this shortest possible period?

Points and on a plane object (Figure 14-30), are distances and respectively, from the center of mass. The object oscillates with the same period when it is free to rotate about an axis through and when it is free to rotate about an axis through Both of these axes are perpendicular to the plane of the object. Show that where

A physical pendulum consists of a spherical bob of radius and mass suspended from a rigid rod of negligible mass, as in Figure 14-31. The distance from the center of the sphere to the point of support is When is much less than such a pendulum is often treated as a simple pendulum of length (a) Show that the period for small oscillations is given by where is the period of a simple pendulum of length (b) Show that when is smaller than the period can be approximated by (c) If and find the error in the calculated value when the approximation is used for this period. How large must the radius of the bob be for the error to be 1.00 percent?

Figure 14-32 shows the pendulum of a clock in your grandmothers house. The uniform rod of length has a mass Attached to the rod is a uniform disk of mass and radius 0.150 m. The clock is constructed to keep perfect time if the period of the pendulum is exactly 3.50 s. (a) What should the distance be so that the period of this pendulum is 2.50 s? (b) Suppose that the pendulum clock loses To make sure your grandmother will not be late for her quilting parties, you decide to adjust the clock back to its proper period. How far and in what direction should you move the disk to ensure that the clock will keep perfect time?

A 2.00-kg object oscillates on a spring with an initial amplitude of 3.00 cm. The force constant of the spring is Find (a) the period, and (b) the total initial energy. (c) If the energy decreases by 1.00 percent per period, find the linear damping constant b and the Q factor.

Show that the ratio of the amplitudes for two successive oscillations is constant for a linearly damped oscillator.

An oscillator has a period of 3.00 s. Its amplitude decreases by 5.00 percent during each cycle. (a) By how much does its mechanical energy decrease during each cycle? (b) What is the time constant (c) What is the factor?

A linearly damped oscillator has a factor of 20. (a) By what fraction does the energy decrease during each cycle? (b) Use Equation 14-40 to find the percentage difference between \(\omega^\prime\) and \(\omega_0\). Hint: Use the approximation \((1+x)^{1/2}\approx 1+\frac{1}{2}x\) for small x.

A linearly damped mass-spring system oscillates at 200 Hz. The time constant of the system is 2.0 s. At the amplitude of oscillation is 6.0 cm and the energy of the oscillating system is 60 J. (a) What are the amplitudes of oscillation at and (b) How much energy is dissipated in the first 2-s interval and in the second 2-s interval?

ENGINEERING APPLICATION Seismologists and geophysicists have determined that the vibrating Earth has a resonance period of 54 min and a factor of about 400. After a large earthquake, Earth will ring (continue to vibrate) for up to 2 months. (a) Find the percentage of the energy of vibration lost to damping forces during each cycle. (b) Show that after periods the vibrational energy is given by where is the original energy. (c) If the original energy of vibration of an earthquake is what is the energy after 2.0 d?

A pendulum that is used in your physics laboratory experiment has a length of 75 cm and a compact bob with a mass equal to 15 g. To start the bob oscillating, you place a fan next to it that blows a horizontal stream of air on the bob. While the fan is on, the bob is in equilibrium when the pendulum is displaced by an angle of from the vertical. The speed of the air from the fan is You turn the fan off, and allow the pendulum to oscillate. (a) Assuming that the drag force due to the air is of the form predict the decay time constant for this pendulum. (b) How long will it take for the pendulums amplitude to reach

ENGINEERING APPLICATION, CONTEXT-RICH You are in charge of monitoring the viscosity of oils at a manufacturing plant and you determine the viscosity of an oil by using the following method: The viscosity of a fluid can be measured by determining the decay time of oscillations for an oscillator that has known properties and operates while immersed in the fluid. As long as the speed of the oscillator through the fluid is relatively small, so that turbulence is not a factor, the drag force of the fluid on a sphere is proportional to the spheres speed relative to the fluid: where is the viscosity of the fluid and is the spheres radius. Thus, the constant is given by Suppose your apparatus consists of a stiff spring that has a force constant equal to and a gold sphere (radius 6.00 cm) hanging on the spring. (a) What viscosity of an oil do you measure if the decay time for this system is 2.80 s? (b) What is the factor for your system?

A linearly damped oscillator loses 2.00 percent of its energy during each cycle. (a) What is its factor? (b) If its resonance frequency is 300 Hz, what is the width of the resonance curve when the oscillator is driven?

Find the resonance frequency for each of the three systems shown in Figure 14-33.

A damped oscillator loses 3.50 percent of its energy during each cycle. (a) How many cycles elapse before half of its original energy is dissipated? (b) What is its factor? (c) If the natural frequency is 100 Hz, what is the width of the resonance curve when the oscillator is driven by a sinusoidal force?

A 2.00-kg object oscillates on a spring that has a force constant equal to The linear damping constant has a value of The system is driven by a sinusoidal force of maximum value 10.0 N and angular frequency (a) What is the amplitude of the oscillations? (b) If the driving frequency is varied, at what frequency will resonance occur? (c) What is the amplitude of oscillation at resonance? (d) What is the width of the resonance curve

ENGINEERING APPLICATION, CONTEXT-RICH Suppose you have the same apparatus that is described in Problem 83 and the same gold sphere hanging from a weaker spring that has a force constant of only You have studied the viscosity of ethylene glycol with this device, and found that ethylene glycol has a viscosity value of Now you decide to drive this system with an external oscillating force. (a) If the magnitude of the driving force for the device is 0.110 N and the device is driven at resonance, how large would be the amplitude of the resulting oscillation? (b) If the system were not driven, but were allowed to oscillate, what percentage of its energy would it lose per cycle?

MULTISTEP A particle’s displacement from equilibrium is given by \(x(t)=0.40 \cos(3.0t+ \pi/4)\), where x is in meters and t is in seconds. (a) Find the frequency f and period T of its motion. (b) Find an expression for the velocity of the particle as a function of time. (c) What is its maximum speed?

ENGINEERING APPLICATION An astronaut arrives at a new planet, and gets out his simple device to determine the gravitational acceleration there. Prior to his arrival, he noted that the radius of the planet was 7550 km. If his 0.500-m-long simple pendulum has a period of 1.0 s, what is the mass of the planet?

A pendulum clock keeps perfect time on Earths surface. In which case will the error be greater: if the clock is placed in a mine of depth or if the clock is elevated to a height Prove your answer and assume that

Figure 14-34 shows a pendulum of length with a bob of mass The bob is attached to a spring that has a force constant as shown. When the bob is directly below the pendulum support, the spring is unstressed. (a) Derive an expression for the period of this oscillating system for smallamplitude vibrations. (b) Suppose that and is such that in the absence of the spring the period is 2.00 s. What is the force constant if the period of the oscillating system is 1.00 s?

A block that has a mass equal to is supported from below by a frictionless horizontal surface. The block, which is attached to the end of a horizontal spring that has a force constant oscillates with an amplitude When the spring is at its greatest extension and the block is instantaneously at rest, a second block of mass is placed on top of it. (a) What is the smallest value for the coefficient of static friction such that the second object does not slip on the first? (b) Explain how the total mechanical energy the amplitude the angular frequency and the period of the system are affected by the placing of on assuming that the coefficient of friction is great enough to prevent slippage.

A 100-kg box hangs from the ceiling of a room suspended from a spring with a force constant of The unstressed length of the spring is 0.500 m. (a) Find the equilibrium position of the box. (b) An identical spring is stretched and attached to the ceiling and the box, and is parallel with the first spring. Find the frequency of the oscillations when the box is released. (c) What is the new equilibrium position of the box once it comes to rest?

ENGINEERING APPLICATION The acceleration due to gravity varies with geographical location because of Earths rotation and because Earth is not exactly spherical. This was first discovered in the seventeenth century, when it was noted that a pendulum clock carefully adjusted to keep correct time in Paris lost about near the equator. (a) Show by using the differential approximation that a small change in the acceleration of gravity produces a small change in the period of a pendulum given by (b) How large a change in is needed to account for a change in the period?

A small block that has a mass equal to rests on a piston that is vibrating vertically with simple harmonic motion described by the formula (a) Show that the block will leave the piston if (b)If and at what time will the block leave the piston?

Show that for the situations in Figure 14-35a and 14-35b, the object oscillates with a frequency where is given by (a) and (b) Hint: Find the magnitude of the net force on the object for a small displacement and write Note that in Part(b) the springs stretch by different amounts, the sum of which is x. SSM

CONTEXT-RICH During an earthquake, a horizontal floor oscillates horizontally in approximately simple harmonic motion. Assume it oscillates at a single frequency with a period of 0.80 s. (a) After the earthquake, you are in charge of examining the video of the floor motion and discover that a box on the floor started to slip when the amplitude reached 10 cm. From your data, determine the coefficient of static friction between the box and the floor (b) If the coefficient of friction between the box and floor were 0.40, what would be the maximum amplitude of vibration before the box would slip?

If we attach two blocks that have masses and to either end of a spring that has a force constant and set them into oscillation by releasing them from rest with the spring stretched, show that the oscillation frequency is given by where is the reduced mass of the system.

In one of your chemistry labs, you determine that one of the vibrational modes of the HCl molecule has a frequency of Using the result of Problem 99, find the effective spring constant between the H atom and the Cl atom in the HCl molecule.

If a hydrogen atom in HCl were replaced by a deuterium atom (forming DCl) in Problem 100, what would be the new vibration frequency of the molecule? Deuterium consists of 1 proton and 1 neutron.

SPREADSHEET A block of mass resting on a horizontal table is attached to a spring that has a force constant as shown in Figure 14-36. The coefficient of kinetic friction between the block and the table is The spring is unstressed if the block is at the origin and the direction is to the right. The spring is stretched a distance where and the block is released. (a) Apply Newtons second law to the block to obtain an equation for its acceleration for the first halfcycle, during which the block is moving to the left. Show that the resulting equation can be written as where and with (b) Repeat Part (a) for the second half-cycle as the block moves to the right, and show that where and has the same value. (c) Use a spreadsheet program to graph the first five half-cycles for Describe the motion, if any, after the fifth half-cycle.

Figure 14-37 shows a uniform solid half-cylinder of mass and radius resting on a horizontal surface. If one side of this cylinder is pushed down slightly and then released, the half-cylinder will oscillate about its equilibrium position. Determine the period of this oscillation.

A straight tunnel is dug through Earth, as shown in Figure 14-38. Assume that the walls of the tunnel are frictionless. (a) The gravitational force exerted by Earth on a particle of mass at a distance from the center of Earth when is where is the mass of Earth and is its radius. Show that the net force on a particle of mass at a distance from the middle of the tunnel is given by and that the motion of the particle is therefore simple harmonic motion. (b) Show that the period of the motion is independent of the length of the tunnel and is given by (c) Find its numerical value in minutes.

MULTISTEP In this problem, derive the expression for the average power delivered by a driving force to a driven oscillator (Figure 14-39). (a) Show that the instantaneous power input of the driving force is given by \(P=F v=-A \omega F_0 \cos \omega t \sin (\omega t-\delta)\). (b) Use the identity \(\sin \left(\theta_1-\theta_2\right)=\sin \theta_1 \cos \theta_2-\cos \theta_1 \sin \theta_2\) to show that the equation in Part (a) can be written as \(P=A \omega F_0 \sin \delta \cos ^2 \omega t \sin -A \omega F_0 \cos \delta \cos \omega t \sin \omega t\) (c) Show that the average value of the second term in your result for Part (b) over one or more periods is zero, and that therefore \(P_{\text {av }}=\frac{1}{2} A \omega F_0 \sin \delta\). (d) From Equation 14-56 for \(\tan \delta\), construct a right triangle in which the side opposite the angle \(\delta\) is \)b \omega\) and the side adjacent is \(m\left(\omega_0^2-\omega^2\right)\), and use this triangle to show that \(\sin \delta=\frac{b \omega}{\sqrt{m^2\left(\omega_0^2-\omega^2\right)^2+b^2 \omega^2}}=\frac{b \omega A}{F_0} \text {.ssm }\) (e) Use your result for Part (d) to eliminate \(\omega A\) from your result for Part (c), so that the average power input can be written as \(P_{\mathrm{av}}=\frac{1}{2} \frac{F_0^2}{b} \sin ^2 \delta=\frac{1}{2}\left[\frac{b \omega^2 F_0^2}{m^2\left(\omega_0^2-\omega^2\right)^2+b^2 \omega^2}\right]\)

MULTISTEP In this problem, you are to use the result of Problem 105 to derive Equation 14-51. At resonance, the denominator of the fraction in brackets in Problem 105(e) is and has its maximum value. For a sharp resonance, the variation in in the numerator in this equation can be neglected. Then, the power input will be half its maximum value at the values of for which the denominator is (a) Show that then satisfies (b) Using the approximation show that (c) Express in terms of (d) Combine the results of Part (b) and Part (c) to show that there are two values of for which the power input is half that at resonance and that they are given by Therefore, which is equivalent to Equation 14-51.

SPREADSHEET The Morse potential, which is often used to model interatomic forces, can be written in the form where is the distance between the two atomic nuclei. (a) Using a spreadsheet program or graphing calculator, make a graph of the Morse potential using and (b) Determine the equilibrium separation and force constant for small displacements from equilibrium for the Morse potential. (c) Determine an expression for the oscillation frequency for a homonuclear diatomic molecule (that is, two of the same atoms), where the atoms each have mass m.