Solved: A uniform rod of mass M and length L swings as a

What are oscillations?

What things undergo SHM?

How is SHM related to circular motion?

Is energy conserved in SHM?

What if theres friction?

Why is SHM important?

An air-track glider is attached to a spring, pulled 20.0 cm to the right, and released at t = 0 s. It makes 15 oscillations in 10.0 s. a. What is the period of oscillation? b. What is the objects maximum speed? c. What are the position and velocity at t = 0.800 s?

An object moves with simple harmonic motion. If the amplitude and the period are both doubled, the objects maximum speed is a. Quadrupled. b. Doubled. c. Unchanged. d. Halved. e. Quartere

A mass oscillating in simple harmonic motion starts at x = A and has period T. At what time, as a fraction of T, does the object first pass through x = 1 2 A?

The figure shows four oscillators at t = 0. Which one has the phase constant f0 = p/4 rad?

An object on a spring oscillates with a period of 0.80 s and an amplitude of 10 cm. At t = 0 s, it is 5.0 cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t = 2.0 s?

The four springs shown here have been compressed from their equilibrium position at x = 0 cm. When released, the attached mass will start to oscillate. Rank in order, from highest to lowest, the maximum speeds of the masses.

A 500 g block on a spring is pulled a distance of 20 cm and released. The subsequent oscillations are measured to have a period of 0.80 s. a. At what position or positions is the blocks speed 1.0 m/s? b. What is the spring constant?

This is the position graph of a mass on a spring. What can you say about the velocity and the force at the instant indicated by the dashed line? a. Velocity positive; force to the right. b. Velocity negative; force to the right. c. Velocity zero; force to the right. d. Velocity positive; force to the left. e. Velocity negative; force to the left. f. Velocity zero; force to the left. g. Velocity and force both zero.

At t = 0 s, a 500 g block oscillating on a spring is observed moving to the right at x = 15 cm. It reaches a maximum displacement of 25 cm at t = 0.30 s. a. Draw a position-versus-time graph for one cycle of the motion. b. What is the maximum force on the block, and what is the first time at which this occurs?

One person swings on a swing and finds that the period is 3.0 s. A second person of equal mass joins him. With two people swinging, the period is a. 6.0 s b. 73.0 s but not necessarily 6.0 s c. 3.0 s d. 63.0 s but not necessarily 1.5 s e. 1.5 s f. Cant tell without knowing the length

An 83 kg student hangs from a bungee cord with spring constant 270 N/m. The student is pulled down to a point where the cord is 5.0 m longer than its unstretched length, then released. Where is the student, and what is his velocity 2.0 s later?

Rank in order, from largest to smallest, the time constants ta to td of the decays shown in the figure. All the graphs have the same scale. t E t E t E t E (a) (b) (c) (d)

A 300 g mass on a 30-cm-long string oscillates as a pendulum. It has a speed of 0.25 m/s as it passes through the lowest point. What maximum angle does the pendulum reach?

Deposits of minerals and ore can alter the local value of the freefall acceleration because they tend to be denser than surrounding rocks. Geologists use a gravimeteran instrument that accurately measures the local free-fall accelerationto search for ore deposits. One of the simplest gravimeters is a pendulum. To achieve the highest accuracy, a stopwatch is used to time 100 oscillations of a pendulum of different lengths. At one location in the field, a geologist makes the following measurements: What is the local value of g?

A student in a biomechanics lab measures the length of his leg, from hip to heel, to be 0.90 m. What is the frequency of the pendulum motion of the students leg? What is the period?

A 500 g mass swings on a 60-cm-string as a pendulum. The amplitude is observed to decay to half its initial value after 35 oscillations. a. What is the time constant for this oscillator? b. At what time will the energy have decayed to half its initial value?

A pendulum consists of a massless, rigid rod with a mass at one end. The other end is pivoted on a frictionless pivot so that the rod can rotate in a complete circle. The pendulum is inverted, with the mass directly above the pivot point, then released. The speed of the mass as it passes through the lowest point is 5.0 m/s. If the pendulum later undergoes small-amplitude oscillations at the bottom of the arc, what will its frequency be?

A block oscillating on a spring has period T = 2 s. What is the period if: a. The blocks mass is doubled? Explain. Note that you do not know the value of either m or k, so do not assume any particular values for them. The required analysis involves thinking about ratios. b. The value of the spring constant is quadrupled? c. The oscillation amplitude is doubled while m and k are unchanged?

A pendulum on Planet X, where the value of g is unknown, oscillates with a period T = 2 s. What is the period of this pendulum if: a. Its mass is doubled? Explain. Note that you do not know the value of m, L, or g, so do not assume any specific values. The required analysis involves thinking about ratios. b. Its length is doubled? c. Its oscillation amplitude is doubled?

FIGURE Q15.3 shows a positionversus-time graph for a particle in SHM. What are (a) the amplitude A, (b) the angular frequency v, and (c) the phase constant f0?

FIGURE Q15.4 shows a position-versus-time graph for a particle in SHM. a. What is the phase constant f0? Explain. b. What is the phase of the particle at each of the three numbered points on the graph?

Equation 15.25 states that 1 2 kA2 = 1 2 m1vmax) 2 . What does this mean? Write a couple of sentences explaining how to interpret this equation

A block oscillating on a spring has an amplitude of 20 cm. What will the amplitude be if the total energy is doubled? Explain.

A block oscillating on a spring has a maximum speed of 20 cm/s. What will the blocks maximum speed be if the total energy is doubled? Explain

The solid disk and circular hoop in FIGURE Q15.8 have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at the top edge. Which, if either, has the larger period of oscillation

FIGURE Q15.9 shows the potential-energy diagram and the total energy line of a particle oscillating on a spring. a. What is the springs equilibrium length? b. Where are the turning points of the motion? Explain. c. What is the particles maximum kinetic energy? d. What will be the turning points if the particles total energy is doubled?

Suppose the damping constant b of an oscillator increases. a. Is the medium more resistive or less resistive? b. Do the oscillations damp out more quickly or less quickly? c. Is the time constant t increased or decreased?

a. Describe the difference between t and T. Dont just name them; say what is different about the physical concepts they represent. b. Describe the difference between t and t1/2

What is the difference between the driving frequency and the natural frequency of an oscillator?

An air-track glider attached to a spring oscillates between the 10 cm mark and the 60 cm mark on the track. The glider completes 10 oscillations in 33 s. What are the (a) period, (b) frequency, (c) angular frequency, (d) amplitude, and (e) maximum speed of the glider?

An air-track glider is attached to a spring. The glider is pulled to the right and released from rest at t = 0 s. It then oscillates with a period of 2.0 s and a maximum speed of 40 cm/s. a. What is the amplitude of the oscillation? b. What is the gliders position at t = 0.25 s?

When a guitar string plays the note A, the string vibrates at 440 Hz. What is the period of the vibration?

An object in SHM oscillates with a period of 4.0 s and an amplitude of 10 cm. How long does the object take to move from x = 0.0 cm to x = 6.0 cm?

What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation shown in FIGURE EX15.5?

What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation shown in FIGURE EX15.6?

FIGURE EX15.7 is the position-versus-time graph of a particle in simple harmonic motion. a. What is the phase constant? b. What is the velocity at t = 0 s? c. What is vmax?

FIGURE EX15.8 is the velocity-versus-time graph of a particle in simple harmonic motion. a. What is the amplitude of the oscillation? b. What is the phase constant? c. What is the position at t = 0 s?

An object in simple harmonic motion has an amplitude of 4.0 cm, a frequency of 2.0 Hz, and a phase constant of 2p/3 rad. Draw a position graph showing two cycles of the motion

An object in simple harmonic motion has an amplitude of 8.0 cm, a frequency of 0.25 Hz, and a phase constant of -p/2 rad. Draw a position graph showing two cycles of the motion.

An object in simple harmonic motion has amplitude 4.0 cm and frequency 4.0 Hz, and at t = 0 s it passes through the equilibrium point moving to the right. Write the function x1t2 that describes the objects position.

An object in simple harmonic motion has amplitude 8.0 cm and frequency 0.50 Hz. At t = 0 s it has its most negative position. Write the function x1t2 that describes the objects position.

An air-track glider attached to a spring oscillates with a period of 1.5 s. At t = 0 s the glider is 5.00 cm left of the equilibrium position and moving to the right at 36.3 cm/s. a. What is the phase constant? b. What is the phase at t = 0 s, 0.5 s, 1.0 s, and 1.5 s?

A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if a. The mass is doubled? b. The mass is halved? c. The amplitude is doubled? d. The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.

A 200 g air-track glider is attached to a spring. The glider is pushed in 10 cm and released. A student with a stopwatch finds that 10 oscillations take 12.0 s. What is the spring constant?

A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vx = -30 cm/s. Determine: a. The period. b. The angular frequency. c. The amplitude. d. The phase constant. e. The maximum speed. f. The maximum acceleration. g. The total energy. h. The position at t = 0.40 s.

The position of a 50 g oscillating mass is given by x1t2 = 12.0 cm2 cos110 t - p/42, where t is in s. Determine: a. The amplitude. b. The period. c. The spring constant. d. The phase constant. e. The initial conditions. f. The maximum speed. g. The total energy. h. The velocity at t = 0.40 s

A 1.0 kg block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40 cm/s. What are a. The amplitude of the subsequent oscillations? b. The blocks speed at the point where x = 1 2 A?

A student is bouncing on a trampoline. At her highest point, her feet are 55 cm above the trampoline. When she lands, the trampoline sags 15 cm before propelling her back up. For how long is she in contact with the trampoline?

FIGURE EX15.20 is a kinetic-energy graph of a mass oscillating on a very long horizontal spring. What is the spring constant?

A spring is hanging from the ceiling. Attaching a 500 g physics book to the spring causes it to stretch 20 cm in order to come to equilibrium. a. What is the spring constant? b. From equilibrium, the book is pulled down 10 cm and released. What is the period of oscillation? c. What is the books maximum speed?

A spring with spring constant 15 N/m hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 6.0 cm and released. If the ball makes 30 oscillations in 20 s, what are its (a) mass and (b) maximum speed?

A spring is hung from the ceiling. When a block is attached to its end, it stretches 2.0 cm before reaching its new equilibrium length. The block is then pulled down slightly and released. What is the frequency of oscillation?

A grandfather clock ticks each time the pendulum passes through the lowest point. If the pendulum is modeled as a simple pendulum, how long must it be for the ticks to occur once a second?

A 200 g ball is tied to a string. It is pulled to an angle of 8.0 and released to swing as a pendulum. A student with a stopwatch finds that 10 oscillations take 12 s. How long is the string?

A mass on a string of unknown length oscillates as a pendulum with a period of 4.0 s. What is the period if a. The mass is doubled? b. The string length is doubled? c. The string length is halved? d. The amplitude is doubled? Parts a to d are independent questions, each referring to the initial situation

What is the length of a pendulum whose period on the moon matches the period of a 2.0-m-long pendulum on the earth?

What is the period of a 1.0-m-long pendulum on (a) the earth and (b) Venus?

Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 s. The period on Mars turns out to be 2.45 s. What is the free-fall acceleration on Mars?

A 100 g mass on a 1.0-m-long string is pulled 8.0 to one side and released. How long does it take for the pendulum to reach 4.0 on the opposite side?

A uniform steel bar swings from a pivot at one end with a period of 1.2 s. How long is the bar?

A 2.0 g spider is dangling at the end of a silk thread. You can make the spider bounce up and down on the thread by tapping lightly on his feet with a pencil. You soon discover that you can give the spider the largest amplitude on his little bungee cord if you tap exactly once every second. What is the spring constant of the silk thread?

The amplitude of an oscillator decreases to 36.8% of its initial value in 10.0 s. What is the value of the time constant?

In a science museum, a 110 kg brass pendulum bob swings at the end of a 15.0-m-long wire. The pendulum is started at exactly 8:00 a.m. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulums damping constant is only 0.010 kg/s. At exactly 12:00 noon, how many oscillations will the pendulum have completed and what is its amplitude?

Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant of the musculature that holds the eyeball in the socket?

A 350 g mass on a 45-cm-long string is released at an angle of 4.5 from vertical. It has a damping constant of 0.010 kg/s. After 25 s, (a) how many oscillations has it completed and (b) how much energy has been lost?

The motion of a particle is given by x1t2 = 125 cm2cos110t2, where t is in s. What is the first time at which the kinetic energy is twice the potential energy?

a. When the displacement of a mass on a spring is 1 2 A, what fraction of the energy is kinetic energy and what fraction is potential energy? b. At what displacement, as a fraction of A, is the energy half kinetic and half potential?

For a particle in simple harmonic motion, show that vmax = 1p/22vavg, where vavg is the average speed during one cycle of the motion.

A 100 g block attached to a spring with spring constant 2.5 N/m oscillates horizontally on a frictionless table. Its velocity is 20 cm/s when x = -5.0 cm. a. What is the amplitude of oscillation? b. What is the blocks maximum acceleration? c. What is the blocks position when the acceleration is maximum? d. What is the speed of the block when x = 3.0 cm?

A 0.300 kg oscillator has a speed of 95.4 cm/s when its displacement is 3.00 cm and 71.4 cm/s when its displacement is 6.00 cm. What is the oscillators maximum speed?

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.10 g) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil. a. The maximum restoring force that can be applied to the disk without breaking it is 40,000 N. What is the maximum oscillation amplitude that wont rupture the disk? b. What is the disks maximum speed at this amplitude?

Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The springs length as a function of time is shown in FIGURE P15.43. a. What is her mass if the spring constant is 240 N/m? b. What is her speed when the springs length is 1.2 m?

Your lab instructor has asked you to measure a spring constant using a dynamic methodletting it oscillaterather than a static method of stretching it. You and your lab partner suspend the spring from a hook, hang different masses on the lower end, and start them oscillating. One of you uses a meter stick to measure the amplitude, the other uses a stopwatch to time 10 oscillations. Your data are as follows: Mass (g) Amplitude (cm) Time (s) 100 6.5 7.8 150 5.5 9.8 200 6.0 10.9 250 3.5 12.4 Use the best-fit line of an appropriate graph to determine the spring constant

A 5.0 kg block hangs from a spring with spring constant 2000 N/m. The block is pulled down 5.0 cm from the equilibrium position and given an initial velocity of 1.0 m/s back toward equilibrium. What are the (a) frequency, (b) amplitude, and (c) total mechanical energy of the motion?

A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the blocks a. Oscillation frequency? b. Distance from equilibrium when the speed is 50 cm/s? c. Distance from equilibrium at t = 1.0 s?

A block hangs in equilibrium from a vertical spring. When a second identical block is added, the original block sags by 5.0 cm. What is the oscillation frequency of the two-block system?

A 75 kg student jumps off a bridge with a 12-m-long bungee cord tied to his feet. The massless bungee cord has a spring constant of 430 N/m. a. How far below the bridge is the students lowest point? b. How long does it take the student to reach his lowest point? You can assume that the bungee cord exerts no force until it begins to stretch.

Scientists are measuring the properties of a newly discovered elastic material. They create a 1.5-m-long, 1.6-mm-diameter cord, attach an 850 g mass to the lower end, then pull the mass down 2.5 mm and release it. Their high-speed video camera records 36 oscillations in 2.0 s. What is Youngs modulus of the material?

A mass hanging from a spring oscillates with a period of 0.35 s. Suppose the mass and spring are swung in a horizontal circle, with the free end of the spring at the pivot. What rotation frequency, in rpm, will cause the springs length to stretch by 15%?

A compact car has a mass of 1200 kg. Assume that the car has one spring on each wheel, that the springs are identical, and that the mass is equally distributed over the four springs. a. What is the spring constant of each spring if the empty car bounces up and down 2.0 times each second? b. What will be the cars oscillation frequency while carrying four 70 kg passengers?

The two blocks in FIGURE P15.52 oscillate on a frictionless surface with a period of 1.5 s. The upper block just begins to slip when the amplitude is increased to 40 cm. What is the coefficient of static friction between the two blocks?

A 1.00 kg block is attached to a horizontal spring with spring constant 2500 N/m. The block is at rest on a frictionless surface. A 10 g bullet is fired into the block, in the face opposite the spring, and sticks. What was the bullets speed if the subsequent oscillations have an amplitude of 10.0 cm?

It has recently become possible to weigh DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.54 shows a thin rectangular cantilever etched out of silicon (density 2300 kg/m3) with a small gold dot (not visible) at the end. If pulled down and released, the end of the cantilever vibrates with SHM, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slightbut measurabledecrease in the oscillation frequency.A vibrating cantilever of mass M can be modeled as a block of mass 1 3 M attached to a spring. (The factor of 1 3 arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately perhaps to only two significant figuresbut the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly 12 MHz. Attachment of a DNA molecule caused the frequency to decrease by 50 Hz. What was the mass of the DNA?

It is said that Galileo discovered a basic principle of the pendulumthat the period is independent of the amplitudeby using his pulse to time the period of swinging lamps in the cathedral as they swayed in the breeze. Suppose that one oscillation of a swinging lamp takes 5.5 s. a. How long is the lamp chain? b. What maximum speed does the lamp have if its maximum angle from vertical is 3.0?

Orangutans can move by brachiation, swinging like a pendulum beneath successive handholds. If an orangutan has arms that are 0.90 m long and repeatedly swings to a 20 angle, taking one swing after another, estimate its speed of forward motion in m/s. While this is somewhat beyond the range of validity of the small-angle approximation, the standard results for a pendulum are adequate for making an estimate

The pendulum shown in FIGURE P15.57 is pulled to a 10 angle on the left side and released. a. What is the period of this pendulum? b. What is the pendulums maximum angle on the right side?

A uniform rod of mass M and length L swings as a pendulum on a pivot at distance L/4 from one end of the rod. Find an expression for the frequency f of small-angle oscillations.

Interestingly, there have been several studies using cadavers to determine the moments of inertia of human body parts, information that is important in biomechanics. In one study, the center of mass of a 5.0 kg lower leg was found to be 18 cm from the knee. When the leg was allowed to pivot at the knee and swing freely as a pendulum, the oscillation frequency was 1.6 Hz. What was the moment of inertia of the lower leg about the knee joint?

A 500 g air-track glider attached to a spring with spring constant 10 N/m is sitting at rest on a frictionless air track. A 250 g glider is pushed toward it from the far end of the track at a speed of 120 cm/s. It collides with and sticks to the 500 g glider. What are the amplitude and period of the subsequent oscillations?

A 200 g block attached to a horizontal spring is oscillating with an amplitude of 2.0 cm and a frequency of 2.0 Hz. Just as it passes through the equilibrium point, moving to the right, a sharp blow directed to the left exerts a 20 N force for 1.0 ms. What are the new (a) frequency and (b) amplitude?

FIGURE P15.62 is a top view of an object of mass m connected between two stretched rubber bands of length L. The object rests on a frictionless surface. At equilibrium, the tension in each rubber band is T. Find an expression for the frequency of oscillations perpendicular to the rubber bands. Assume the amplitude is sufficiently small that the magnitude of the tension in the rubber bands is essentially unchanged as the mass oscillates.

A molecular bond can be modeled as a spring between two atoms that vibrate with simple harmonic motion. FIGURE P15.63 shows an SHM approximation for the potential energy of an HCl molecule. Because the chlorine atom is so much more massive than the hydrogen atom, it is reasonable to assume that the hydrogen atom 1m = 1.67 * 10-27 kg2 vibrates back and forth while the chlorine atom remains at rest. Use the graph to estimate the vibrational frequency of the HCl molecule.

A penny rides on top of a piston as it undergoes vertical simple harmonic motion with an amplitude of 4.0 cm. If the frequency is low, the penny rides up and down without difficulty. If the frequency is steadily increased, there comes a point at which the penny leaves the surface. a. At what point in the cycle does the penny first lose contact with the piston? b. What is the maximum frequency for which the penny just barely remains in place for the full cycle?

On your first trip to Planet X you happen to take along a 200 g mass, a 40-cm-long spring, a meter stick, and a stopwatch. Youre curious about the free-fall acceleration on Planet X, where ordinary tasks seem easier than on earth, but you cant find this information in your Visitors Guide. One night you suspend the spring from the ceiling in your room and hang the mass from it. You find that the mass stretches the spring by 31.2 cm. You then pull the mass down 10.0 cm and release it. With the stopwatch you find that 10 oscillations take 14.5 s. Based on this information, what is g?

Suppose a large spherical object, such as a planet, with radius R and mass M has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance x R from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius r x; there is no net gravitational force from the mass in the spherical shell with r 7 x. a. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of x, R, m, M, and any necessary constants. b. You should have found that the gravitational force is a linear restoring force. Consequently, in the absence of air resistance, objects in the tunnel will oscillate with SHM. Suppose an intrepid astronaut exploring a 150-km-diameter, 3.5 * 1018 kg asteroid discovers a tunnel through the center. If she jumps into the hole, how long will it take her to fall all the way through the asteroid and emerge on the other side?

The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz. a. What is the spring constant of the spring on which the head is mounted? b. The amplitude of the heads oscillations decreases to 0.5 cm in 4.0 s. What is the heads damping constant?

An oscillator with a mass of 500 g and a period of 0.50 s has an amplitude that decreases by 2.0% during each complete oscillation. If the initial amplitude is 10 cm, what will be the amplitude after 25 oscillations?

A spring with spring constant 15.0 N/m hangs from the ceiling. A 500 g ball is attached to the spring and allowed to come to rest. It is then pulled down 6.0 cm and released. What is the time constant if the balls amplitude has decreased to 3.0 cm after 30 oscillations?

A captive James Bond is strapped to a table beneath a huge pendulum made of a 2.0-m-diameter uniform circular metal blade rigidly attached, at its top edge, to a 6.0-m-long, massless rod. The pendulum is set swinging with a 10 amplitude when its lower edge is 3.0 m above the prisoner, then the table slowly starts ascending at 1.0 mm/s. After 25 minutes, the pendulums amplitude has decreased to 7.0. Fortunately, the prisoner is freed with a mere 30 s to spare. What was the speed of the lower edge of the blade as it passed over him for the last time?

A 250 g air-track glider is attached to a spring with spring constant 4.0 N/m. The damping constant due to air resistance is 0.015 kg/s. The glider is pulled out 20 cm from equilibrium and released. How many oscillations will it make during the time in which the amplitude decays to e-1 of its initial value?

A 200 g oscillator in a vacuum chamber has a frequency of 2.0 Hz. When air is admitted, the oscillation decreases to 60% of its initial amplitude in 50 s. How many oscillations will have been completed when the amplitude is 30% of its initial value?

Prove that the expression for x1t2 in Equation 15.55 is a solution to the equation of motion for a damped oscillator, Equation 15.54, if and only if the angular frequency v is given by the expression in Equation 15.56

A block on a frictionless table is connected as shown in FIGURE P15.74 to two springs having spring constants k1 and k2. Show that the blocks oscillation frequency is given by f = 2f1 2 + f2 2 where f1 and f2 are the frequencies at which it would oscillate if attached to spring 1 or spring 2 alone

A block on a frictionless table is connected as shown in FIGURE P15.75 to two springs having spring constants k1 and k2. Find an expression for the blocks oscillation frequency f in terms of the frequencies f1 and f2 at which it would oscillate if attached to spring 1 or spring 2 alone.

A 15@cm@long, 200 g rod is pivoted at one end. A 20 g ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?

A solid sphere of mass M and radius R is suspended from a thin rod, as shown in FIGURE CP15.77. The sphere can swing back and forth at the bottom of the rod. Find an expression for the frequency f of smallangle oscillations.

A uniform rod of length L oscillates as a pendulum about a pivot that is a distance x from the center. a. For what value of x, in terms of L, is the oscillation period a minimum? b. What is the minimum oscillation period of a 15 kg, 1.0-m-long steel bar?

A spring is standing upright on a table with its bottom end fastened to the table. A block is dropped from a height 3.0 cm above the top of the spring. The block sticks to the top end of the spring and then oscillates with an amplitude of 10 cm. What is the oscillation frequenc

The analysis of a simple pendulum assumed that the mass was a particle, with no size. A realistic pendulum is a small, uniform sphere of mass M and radius R at the end of a massless string, with L being the distance from the pivot to the center of the sphere. a. Find an expression for the period of this pendulum. b. Suppose M = 25 g, R = 1.0 cm, and L = 1.0 m, typical values for a real pendulum. What is the ratio Treal /Tsimple, where Treal is your expression from part a and Tsimple is the expression derived in this chapter?

FIGURE CP15.81 shows a 200 g uniform rod pivoted at one end. The other end is attached to a horizontal spring. The spring is neither stretched nor compressed when the rod hangs straight down. What is the rods oscillation period? You can assume that the rods angle from vertical is always small.