A block oscillating on a spring has period T = 2.0 s.a.

Problem 1P When a guitar string plays the note “A,” the string vibrates at 440 Hz. What is the period of the vibration?

Problem 1CQ Give three real-world examples of oscillatory motion. (Note that circular motion is similar to, but not the same as oscillatory motion.)

Problem 2CQ A person’s heart rate is given in beats per minute. Is this a period or a frequency?

Problem 2P In the aftermath of an intense earthquake, the earth as a whole “rings” with a period of 54 minutes. What is the frequency (in Hz) of this oscillation?

Problem 3CQ Figure Q14.3 shows the position-versus-time graph of a particle in SHM. a. At what time or times is the particle moving to the right at maximum speed? b. At what time or times is the particle moving to the left at maximum speed? c. At what time or times is the particle instantaneously at rest?

Problem 3P In taking your pulse, you count 75 heartbeats in 1 min. What are the period (in s) and frequency (in Hz) of your heart’s oscillations?

Problem 4CQ A block oscillating on a spring has an amplitude of 20 cm. What will be the amplitude if the maximum kinetic energy is doubled?

Problem 4P A block oscillating on a spring has an amplitude of 20 cm. What will be the amplitude if the maximum kinetic energy is doubled?

Problem 5CQ A block oscillating on a spring has a maximum speed of 20 cm/s. What will be the block’s maximum speed if the amplitude of the oscillation is doubled?

Problem 5P A heavy steel ball is hung from a cord to make a pendulum. The ball is pulled to the side so that the cord makes a 5° angle with the vertical. Holding the ball in place takes a force of 20 N. If the ball is pulled farther to the side so that the cord makes a 10° angle, what force is required to hold the ball?

Problem 6CQ A block oscillating on a spring has a maximum kinetic energy of 2.0 J. What will be the maximum kinetic energy if the amplitude is doubled? Explain.

Problem 6P 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) amplitude, and (d) maximum speed of the glider?

Problem 7CQ A block oscillating on a spring has a maximum speed of 30 cm/s. What will be the block’s maximum speed if the initial elongation of the spring is doubled?

Problem 7P 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 glider’s position at t = 0.25 s?

Problem 8CQ For the graph in Figure Q14.9 , determine the frequency f and the oscillation amplitude A .

Problem 8P What are the (a) amplitude and (b) frequency of the oscillation shown in Figure P14.8 ?

Problem 9CQ For the graph in Figure Q14.10 , determine the frequency f and the oscillation amplitude A .

Problem 9P What are the (a) amplitude and (b) frequency of the oscillation shown in Figure P14.9 ?

Problem 10CQ A block oscillating on a spring has period T = 2.0 s. a. What is the period if the block’s mass is doubled? b. What is the period if the value of the spring constant is quadrupled? c. What is the period if the oscillation amplitude is doubled while m and k are unchanged? Note: You do not know values for either m or k . Do not assume any particular values for them. The required analysis involves thinking about ratios.

Problem 10P An object in simple harmonic motion has an amplitude of 6.0 cm and a frequency of 0.50 Hz. Draw a position graph showing two cycles of the motion.

Problem 11CQ A pendulum on Planet X, where the value of g is unknown, oscillates with a period of 2.0 s. What is the period of this pendulum if: a. Its mass is doubled? b. Its length is doubled? c. Its oscillation amplitude is doubled? Note: You do not know the values of m, L, or g, so do not assume any specific values

Problem 11P What are the (a) amplitude and (b) frequency of the oscillation shown in Figure P14.9 ? 10. | During an earthquake, the top of a building oscillates with an amplitude of 30 cm at 1.2 Hz. What are the magnitudes of (a) the maximum displacement, (b) the maximum velocity, and (c) the maximum acceleration of the top of the building?

Problem 12P Some passengers on an ocean cruise may suffer from motion sickness as the ship rocks back and forth on the waves. At one position on the ship, passengers experience a vertical motion of amplitude 1 m with a period of 15 s. a. To one significant figure, what is the maximum acceleration of the passengers during this motion? b. What fraction is this of g ?

Problem 12CQ Flies flap their wings at frequencies much too high for pure muscle action. A hypothesis for how they achieve these high frequencies is that the flapping of their wings is the driven oscillation of a mass-spring system. One way to test this is to trim a fly’s wings. If the oscillation of the wings can be modeled as a mass-spring system, how would this change the frequency of the wingbeats?

Problem 13CQ As we saw in Chapter 6, the free-fall acceleration is slightly less in Denver than in Miami. If a pendulum clock keeps perfect time in Miami, will it run fast or slow in Denver? Explain.

Problem 13P A passenger car traveling down a rough road bounces up and down at 1.3 Hz with a maximum vertical acceleration of , both typical values. What are the (a) amplitude and (b) maximum speed of the oscillation?

Problem 14CQ If you want to play a tune on wine glasses, you’ll need to adjust the oscillation frequencies by adding water to the glasses. This changes the mass that oscillates (more water means more mass) but not the restoring force, which is determined by the stiffness of the glass itself. If you need to raise the frequency of a particular glass, should you add water or remove water?

Problem 14P The New England Merchants Bank Building in Boston is 152 m high. On windy days it sways with a frequency of 0.17 Hz, and the acceleration of the top of the building can reach 2.0% of the free-fall acceleration, enough to cause discomfort for occupants. What is the total distance, side to side, that the top of the building moves during such an oscillation?

Problem 15CQ Sprinters push off from the ball of their foot, then bend their knee to bring their foot up close to the body as they swing their leg forward for the next stride. Why is this an effective strategy for running fast?

Problem 15P a. When the displacement of a mass on a spring is , what fraction of the mechanical 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?

Problem 16CQ Gibbons move through the trees by swinging from successive handholds, as we have seen. To increase their speed, gibbons may bring their legs close to their bodies. How does this help them move more quickly?

Problem 16P 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 block’s speed at the point where ?

Problem 17CQ Describe the difference between the time constant ? and the period T. Don’t just name them; say what is different about the physical concepts that they represent.

Problem 17P A block attached to a spring with unknown spring constant oscillates with a period of 2.00 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.

Problem 18CQ What is the difference between the driving frequency and the natural frequency of an oscillator?

Problem 18P A 200 g air-track glider is attached to a spring. The glider is pushed 10.0 cm against the spring, then released. A student with a stopwatch finds that 10 oscillations take 12.0 s. What is the spring constant?

Problem 19CQ Humans have a range of hearing of approximately 20Hz to 20kHz. Mice have auditory systems similar to humans, but all of the physical elements are smaller. Given this, would you expect mice to have a higher or lower frequency range than humans? Explain.

Problem 19P The position of a 50 g oscillating mass is given by x(t) = (2.0 cm) cos (10t), where t is in seconds. Determine: a. The amplitude. b. The period. c. The spring constant. d. The maximum speed. e. The total energy. f. The velocity at t = 0.40 s.

Problem 20CQ A person driving a truck on a “washboard” road, one with regularly spaced bumps, notices an interesting effect: When the truck travels at low speed, the amplitude of the vertical motion of the car is small. If the truck’s speed is increased, the amplitude of the vertical motion also increases, until it becomes quite unpleasant. But if the speed is increased yet further, the amplitude decreases, and at high speeds the amplitude of the vertical motion is small again. Explain what is happening.

Problem 21CQ We’ve seen that stout tendons in the legs of hopping kangaroos store energy. When a kangaroo lands, much of the kinetic energy of motion is converted to elastic energy as the tendons stretch, returning to kinetic energy when the kangaroo again leaves the ground. If a hopping kangaroo increases its speed, it spends more time in the air with each bounce, but the contact time with the ground stays approximately the same. Explain why you would expect this to be the case.

Problem 21P A 507 g mass oscillates with an amplitude of 10.0 cm on a spring whose spring constant is 20.0 N/m. Determine: a. The period. b. The maximum speed. c. The total energy.

Problem 22MCQ A spring has an unstretched length of 20 cm. A 100 g mass hanging from the spring stretches it to an equilibrium length of 30 cm. a. Suppose the mass is pulled down to where the spring’s length is 40 cm. When it is released, it begins to oscillate. What is the amplitude of the oscillation? A. 5.0 cm B. 10 cm C. 20 cm D. 40 cm b. For the data given above, what is the frequency of the oscillation? A. 0.10 Hz B. 0.62 Hz C. 1.6 Hz D. 10 Hz c. Suppose this experiment were done on the moon, where the free-fall acceleration is approximately 1/6 of that on the earth. How would this change the frequency of the oscillation? A. The frequency would decrease. B. The frequency would increase. C. The frequency would stay the same.

Problem 22P A 300 g 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 oscillator’s maximum speed?

Problem 23MCQ Figure Q14.24 represents the motion of a mass on a spring. a. What is the period of this oscillation? A. 12 s B. 24 s C. 36 s D. 48 s E. 50 s b. What is the amplitude of the oscillation? A. 1.0 cm B. 2.5 cm C. 4.5 cm D. 5.0 cm E. 9.0 cm c. What is the position of the mass at time t = 30 s? A. -4.5 cm B. -2.5 cm C. 0.0 cm D. 4.5 cm E. 30 cm d. When is the first time the velocity of the mass is zero? A. 0 s B. 2 s C. 8 s D. 10 s E. 13 s e. At which of these times does the kinetic energy have its maximum value? A. 0 s B. 8 s C. 13 s D. 26 s E. 30 s

Problem 23P A mass on a string of unknown length oscillates as a pendulum with a period of 4.00 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 halved? Parts a to d are independent questions, each referring to the initial situation.

Problem 24MCQ A ball of mass m oscillates on a spring with spring constant k = 200 N/m. The ball’s position is x= (0.350 m) cos (15.0 t), with t measured in seconds. a. What is the amplitude of the ball’s motion? A. 0.175 m B. 0.350 m C. 0.700 m D. 7.50 m E. 15.0 m b. What is the frequency of the ball’s motion? A. 0.35 Hz B. 2.39 Hz C. 5.44 Hz D. 6.28 Hz E. 15.0 Hz c. What is the value of the mass m? A. 0.45 kg B. 0.89 kg C. 1.54 kg D. 3.76 kg E. 6.33 kg d. What is the total mechanical energy of the oscillator? A. 1.65 J B. 3.28 J C. 6.73 J D. 10.1 J E. 12.2 J e. What is the ball’s maximum speed? A. 0.35 m/s B. 1.76 m/s C. 2.60 m/s D. 3.88 m/s E. 5.25 m/s

Problem 24P A 200 g ball is tied to a string. It is pulled to an angle of 8.00° and released to swing as a pendulum. A student with a stopwatch finds that 10 oscillations take 12.0 s. How long is the string?

Problem 25MCQ If you carry heavy weights in your hands, how will this affect the natural frequency at which your arms swing back and forth? A. The frequency will increase. B. The frequency will stay the same. C. The frequency will decrease.

Problem 25P The angle of a pendulum is given by ?(t) = (0.10 rad) cos (5t), where t is in seconds. Determine: a. The amplitude. b. The frequency. c. The length of the string, d. The angle at t = 2.0 s.

Problem 26MCQ A heavy brass ball is used to make a pendulum with a period of 5.5 s. How long is the cable that connects the pendulum ball to the ceiling? A. 4.7 m B. 6.2 m C. 7.5 m D. 8.7 m

Problem 26P It is said that Galileo discovered a basic principle of the pendulum—that the period is independent of the amplitude— by 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. How long is the lamp chain?

Problem 27MCQ Suppose you travel to the moon, and you take with you two timepieces: a pendulum clock and a wristwatch that runs with a wheel and a mainspring. (The wheel and spring work, essentially, like a mass on a spring, but the wheel rotates back and forth rather than moving up and down.) Which will keep good time on the moon? A. Only the pendulum clock B. Only the wristwatch C. Both timepieces D. Neither timepiece

Problem 27P The free-fall acceleration on the moon is . What is the length of a pendulum whose period on the moon matches the period of a 2.00-m-long pendulum on the earth?

Problem 28MCQ Very loud sounds can damage hearing by injuring the vibration-sensing hair cells on the basilar membrane. Suppose a person has injured hair cells on a segment of the basilar membrane close to the stapes. What type of sound is most likely to have produced this particular pattern of damage? A. Loud music with a mix of different frequencies B. A very loud, high-frequency sound C. A very loud, low-frequency sound

Problem 28P 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 Martian free-fall acceleration?

Problem 29P A building is being knocked down with a wrecking ball, which is a big metal sphere that swings on a 10-m-long cable. You are (unwisely!) standing directly beneath the point from which the wrecking ball is hung when you notice that the ball has just been released and is swinging directly toward you. How much time do you have to move out of the way?

Problem 30P Interestingly, there have been several studies using cadavers to determine the moment of inertia of human body parts by letting them swing as a pendulum about a joint. In one study, the center of gravity of a 5.0 kg lower leg was found to be 18 cm from the knee. When pivoted at the knee and allowed to swing, the oscillation frequency was 1.6 Hz. What was the moment of inertia of the lower leg?

Problem 31P A pendulum clock keeps time by the swinging of a uniform solid rod pivoted at one end. The angular position of the rod is given by ?(t) = (0.175 rad)sin(?t), where t is in seconds. a. What is the angular position of the rod at t = 0.250 s? ________________ b. What is the period of oscillation? c. How long is the rod?

Problem 32P You and your friends find a rope that hangs down 15 m from a high tree branch right at the edge of a river. You find that you can run, grab the rope, and swing out over the river. You run at 2.0 m/s and grab the rope, launching yourself out over the river. How long must you hang on if you want to stay dry?

Problem 33P A thin, circular hoop with a radius of 0.22 m is hanging from its rim on a nail. When pulled to the side and released, the hoop swings back and forth as a physical pendulum. The moment of inertia of a hoop for a rotational axis passing through its edge is . What is the period of oscillation of the hoop?

Problem 34P An elephant’s legs have a reasonably uniform cross section from top to bottom, and they are quite long, pivoting high on the animal’s body. When an elephant moves at a walk, it uses very little energy to bring its legs forward, simply allowing them to swing like pendulums. For fluid walking motion, this time should be half the time for a complete stride; as soon as the right leg finishes swinging forward, the elephant plants the right foot and begins swinging the left leg forward. a. An elephant has legs that stretch 2.3 m from its shoulders to the ground. How much time is required for one leg to swing forward after completing a stride? b. What would you predict for this elephant’s stride frequency? That is, how many steps per minute will the elephant take?

Problem 35P 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?

Problem 37P A small earthquake starts a lamppost vibrating back and forth. The amplitude of the vibration of the top of the lamppost is 6.5 cm at the moment the quake stops, and 8.0 s later it is 1.8 cm. a. What is the time constant for the damping of the oscillation? b. What was the amplitude of the oscillation 4.0 s after the quake stopped?

Problem 36P Calculate and draw an accurate displacement graph from t = 0 s to t = 10 s of a damped oscillator having a frequency of 1.0 Hz and a time constant of 4.0 s.

Problem 38P When you drive your car over a bump, the springs connecting the wheels to the car compress. Your shock absorbers then damp the subsequent oscillation, keeping your car from bouncing up and down on the springs. Figure P14.39 shows real data for a car driven over a bump. We can model this as a damped oscillation, although this model is far from perfect. Estimate the frequency and time constant in this model.

Problem 39P A 25 kg child sits on a 2.0-m-long rope swing. You are going to give the child a small, brief push at regular intervals. If you want to increase the amplitude of her motion as quickly as possible, how much time should you wait between pushes?

Problem 40P Your car rides on springs, so it will have a natural frequency of oscillation. Figure P14.42 shows data for the amplitude of motion of a car driven at different frequencies. The car is driven at 20 mph over a washboard road with bumps spaced 10 feet apart; the resulting ride is quite bouncy. Should the driver speed up or slow down for a smoother ride?

Problem 41P Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball held by the musculature 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 attached to the eyeball?

Problem 42GP A spring has an unstretched length of 12 cm. When an 80 g ball is hung from it, the length increases by 4.0 cm. Then the ball is pulled down another 4.0 cm and released. a. What is the spring constant of the spring? b. What is the period of the oscillation? c. Draw a position-versus-time graph showing the motion of the ball for three cycles of the oscillation. Let the equilibrium position of the ball be y = 0. Be sure to include appropriate units on the axes so that the period and the amplitude of the motion can be determined from your graph.

Problem 43GP A 0.40 kg ball is suspended from a spring with spring constant 12 N/m. If the ball is pulled down 0.20 m from the equilibrium position and released, what is its maximum speed while it oscillates?

Problem 44GP A spring is hanging from the ceiling. Attaching a 500 g mass to the spring causes it to stretch 20.0 cm in order to come to equilibrium. a. What is the spring constant? b. From equilibrium, the mass is pulled down 10.0 cm and released. What is the period of oscillation? c. What is the maximum speed of the mass? At what position or positions does it have this speed?

Problem 45GP A spring with spring constant 15.0 N/m hangs from the ceiling. A ball is suspended from the spring and allowed to come to rest. It is then pulled down 6.00 cm and released. If the ball makes 30 oscillations in 20.0 s, what are its (a) mass and (b) maximum speed?

Problem 46GP A spring is hung from the ceiling. When a coffee mug is attached to its end, the spring stretches 2.0 cm before reaching its new equilibrium length. The mug is then pulled down slightly and released. What is the frequency of oscillation?

Problem 47GP On your first trip to Planet X you happen to take along a 200 g mass, a 40.0-cm-long spring, a meter stick, and a stopwatch. You’re curious about the free-fall acceleration on Planet X, where ordinary tasks seem easier than on earth, but you can’t find this information in your Visitor’s 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. Can you now satisfy your curiosity?

Problem 48GP An object oscillating on a spring has the velocity graph shown in Figure P14.50. Draw a velocity graph if the following changes are made. a. The amplitude is doubled and the frequency is halved. b. The amplitude and spring constant are kept the same, but the mass is quadrupled. Parts a and b are independent questions, each starting from the graph shown.

Problem 49GP The two graphs in Figure P14.51 are for two different vertical mass-spring systems. a. What is the frequency of system A? What is the first time at which the mass has maximum speed while traveling in the upward direction? b. What is the period of system B? What is the first time at which the mechanical energy is all potential? c. If both systems have the same mass, what is the ratio kA/Kb of their spring constants?

Problem 50GP As we’ve seen, astronauts measure their mass by measuring the period of oscillation when sitting in a chair connected to a spring. The Body Mass Measurement Device on Skylab, a 1970s space station, had a spring constant of 606 N/m. The empty chair oscillated with a period of 0.901 s. What is the mass of an astronaut who oscillates with a period of 2.09 s when sitting in the chair?

Problem 51GP A 100g ball attached to a spring with spring constant 2.50 N/m oscillates horizontally on a frictionless table. Its velocity is 20.0 cm/s when x = -5.00 cm. a. What is the amplitude of oscillation? b. What is the speed of the ball when x = 3.00 cm?

Problem 52GP The ultrasonic transducer used in a medical ultrasound imaging device 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 won’t rupture the disk? b. What is the disk’s maximum speed at this amplitude?

Problem 53GP 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 car’s oscillation frequency while carrying four 70 kg passengers?

Problem 54GP Four people with a combined mass of 300 kg are riding in a 1100 kg car. When they drive down a washboard road with bumps spaced 5.0 m apart, they notice that the car bounces up and down with a maximum amplitude when the car is traveling at 6.0 m/s. The driver stops the car and everyone exits the vehicle. How much does the car rise up on its springs?

Problem 55GP 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?

Problem 56GP 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 10.0 g bullet is fired into the block, in the face opposite the spring, and sticks. a. What was the bullet’s speed if the subsequent oscillations have an amplitude of 10.0 cm? b. Could you determine the bullet’s speed by measuring the oscillation frequency? If so, how? If not, why not?

Problem 57GP Figure 57 shows two springs, each with spring constant 20 N/m, connecting a 2.5 kg block to two walls. The block slides on a frictionless surface. If the block is displaced from equilibrium, it will undergo simple harmonic motion. What is the frequency of that motion? FIGURE 57

Problem 58GP Bungee Man is a superhero who does super deeds with the help of Super Bungee cords. The Super Bungee cords act like ideal springs no matter how much they are stretched. One day, Bungee Man stopped a school bus that had lost its brakes by hooking one end of a Super Bungee to the rear of the bus as it passed him, planting his feet, and holding on to the other end of the Bungee until the bus came to a halt. (Of course, he then had to quickly release the Bungee before the bus came flying back at him.) The mass of the bus, including passengers, was 12,000 kg, and its speed was 21.2 m/s. The bus came to a stop in 50.0 m. a. What was the spring constant of the Super Bungee? b. How much time after the Super Bungee was attached did it take the bus to stop?

Problem 59GP Two 50 g blocks are held 30 cm above a table. As shown in Figure 59, one of them is just touching a 30-cm-long spring. The blocks arc released at the same time. The block on the left hits the table at exactly the same instant as the block on the right first comes to an instantaneous rest. What is the spring constant? FIGURE 59

Problem 61GP A pendulum clock has a heavy bob supported on a very thin steel rod that is 1.00000 m long at 20°C. a. To 6 significant figures, what is the clock’s period? Assume that g is 9.80 m/s2 exactly. ________________ b. To 6 significant figures, what is the period if the temperature increases by 10°C? ________________ c. The clock keeps perfect time at 20°C. At 30°C, after how many hours will the clock be off by 1.0 s?

Problem 60GP The earth’s free-fall acceleration varies from at the poles. A pendulum whose length is precisely 1.000 m can be used to measure g. Such a device is called a gravimeter. a. How long do 100 oscillations take at the equator? b. How long do 100 oscillations take at the north pole? c. Suppose you take your gravimeter to the top of a high mountain peak near the equator. There you find that 100 oscillations take 201 s. What is g on the mountain top?

Problem 62GP 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 it can turn through a complete circle. The pendulum is inverted, so the mass is 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 the frequency be?

Problem 63GP Two side-by-side pendulum clocks have heavy bobs at the ends of rigid, very lightweight arms. One pendulum has a 38.8-cm-long rod, the other a 24.8-cm-long rod. Each clock makes one tick for each complete swing of its pendulum. a. Determine the frequencies and periods of the two clocks. ________________ b. Because the two pendulums have different frequencies, their ticks are usually “out of step.” However, you notice that they do get back into step (tick at the same instant) at regular intervals. How much time elapses between such events? ________________ c. The getting-into-step phenomenon is, itself, periodic. What is the frequency of this phenomenon? Can you see a relationship between its frequency and the frequencies of the two clocks?

Problem 64GP 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 immediately after another, estimate how fast it is moving in m/s.

Problem 66GP 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?

Problem 65GP 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. Suppose the head is pushed 2.0 cm against the spring, then released. What is the head’s maximum speed as it oscillates? ________________ c. The amplitude of the head’s oscillations decreases to 0.50 cm in 4.0 s. What is the head’s time constant?

Problem 67GP An infant’s toy has a 120 g wooden animal hanging from a spring. If pulled down gently, the animal oscillates up and down with a period of 0.50 s. His older sister pulls the spring a bit more than intended. She pulls the animal 30 cm below its equilibrium position, then let's go. The animal flies upward and detaches from the spring right at the animal’s equilibrium position. If the animal does not hit anything on the way up, how far above its equilibrium position will it go?

Problem 68GP A jellyfish can propel itself with jets of water pushed out of its bell, a flexible structure on top of its body. The elastic bell and the water it contains function as a mass-spring system, greatly increasing efficiency. Normally, the jellyfish emits one jet right after the other, but we can get some insight into the jet system by looking at a single jet thrust. Figure P14.64 shows a graph of the motion of one point in the wall of the bell for such a single jet; this is the pattern of a damped oscillation. The spring constant for the bell can be estimated to be 1.2 N/m. a. What is the period for the oscillation? b. Estimate the effective mass participating in the oscillation. This is the mass of the bell itself plus the mass of the water. c. Consider the peaks of positive displacement in the graph. By what factor does the amplitude decrease over one period? Given this, what is the time constant for the damping?

Problem 69GP 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?

Problem 71GP A 2.0 kg block oscillates up and down on a spring with spring constant 220 N/m. Its initial amplitude is 15 cm. If the time constant for damping of the oscillation is 3.0 s, how much mechanical energy has been dissipated from the block-spring system after 6.0 s?

Problem 70GP While seated on a tall bench, extend your lower leg a small amount and then let it swing freely about your knee joint, with no muscular engagement. It will oscillate as a damped pendulum. Figure P14.66 is a graph of the lower leg angle versus time in such an experiment. Estimate (a) the period and (b) the time constant for this oscillation.

Problem 72GP In Chapter 10, we saw that the Achilles tendon will stretch and then rebound, storing and returning energy during a step. We can model this motion as that of a mass on a spring. This is far from a perfect model, but it does give some insight. If a 60 kg person stands on a low wall with her full weight on the ball of one foot and the heel free to move, the stretch of the Achilles tendon will cause her center of gravity to lower by about 2.5 mm. a. What is the spring constant of her Achilles tendon? ________________ b. If she bounces a little, what is her oscillation period? ________________ c. When walking or running, the tendon spring begins to stretch as the ball of the foot takes the weight of a stride, transforming kinetic energy into elastic potential energy. Ideally, the- cycle of the motion will have advanced so that potential energy has just finished being converted back to kinetic energy as the foot leaves the ground. What fraction of an oscillation period should the time between landing and liftoff correspond to? Given the period you calculated above, what is this time? ________________ d. Sprinters running a short race keep their foot in contact with the ground for about 0.10 s, some of which corresponds to the heel strike and subsequent rolling forward of the foot. Given this, does the answer to part c make sense?

Problem 74PP Web Spiders and Oscillations All spiders have special organs that make them exquisitely sensitive to vibrations. Web spiders detect vibrations of their web to determine what has landed in their web, and where. In fact, spiders carefully adjust the tension of strands to “tune” their web. Suppose an insect lands and is trapped in a web. The silk of the web serves as the spring in a spring-mass system while the body of the insect is the mass. The frequency of oscillation depends on the restoring force of the web and the mass of the insect. Spiders respond more quickly to larger—and therefore more valuable—prey, which they can distinguish by the web’s oscillation frequency. Suppose a 12 mg fly lands in the center of a horizontal spider’s web, causing the web to sag by 3.0 mm. Modeling the motion of the fly on the web as a mass on a spring, at what frequency will the web vibrate when the fly hits it? A. 0.91 Hz B. 2.9 Hz C. 9.1 Hz D. 29 Hz

Problem 73PP Web Spiders and Oscillations All spiders have special organs that make them exquisitely sensitive to vibrations. Web spiders detect vibrations of their web to determine what has landed in their web, and where. In fact, spiders carefully adjust the tension of strands to “tune” their web. Suppose an insect lands and is trapped in a web. The silk of the web serves as the spring in a spring-mass system while the body of the insect is the mass. The frequency of oscillation depends on the restoring force of the web and the mass of the insect. Spiders respond more quickly to larger—and therefore more valuable—prey, which they can distinguish by the web’s oscillation frequency. Suppose a 12 mg fly lands in the center of a horizontal spider’s web, causing the web to sag by 3.0 mm. Assuming that the web acts like a spring, what is the spring constant of the web? A. 0.039 N/m B. 0.39 N/m C. 3.9 N/m D. 39 N/m

Problem 75PP Web Spiders and Oscillations All spiders have special organs that make them exquisitely sensitive to vibrations. Web spiders detect vibrations of their web to determine what has landed in their web, and where. In fact, spiders carefully adjust the tension of strands to “tune” their web. Suppose an insect lands and is trapped in a web. The silk of the web serves as the spring in a spring-mass system while the body of the insect is the mass. The frequency of oscillation depends on the restoring force of the web and the mass of the insect. Spiders respond more quickly to larger—and therefore more valuable—prey, which they can distinguish by the web’s oscillation frequency. Suppose a 12 mg fly lands in the center of a horizontal spider’s web, causing the web to sag by 3.0 mm. If the web were vertical rather than horizontal, how would the frequency of oscillation be affected? A. The frequency would be higher. B. The frequency would be lower. C. The frequency would be the same.

Problem 76PP Web Spiders and Oscillations All spiders have special organs that make them exquisitely sensitive to vibrations. Web spiders detect vibrations of their web to determine what has landed in their web, and where. In fact, spiders carefully adjust the tension of strands to “tune” their web. Suppose an insect lands and is trapped in a web. The silk of the web serves as the spring in a spring-mass system while the body of the insect is the mass. The frequency of oscillation depends on the restoring force of the web and the mass of the insect. Spiders respond more quickly to larger—and therefore more valuable—prey, which they can distinguish by the web’s oscillation frequency. Suppose a 12 mg fly lands in the center of a horizontal spider’s web, causing the web to sag by 3.0 mm. Spiders are more sensitive to oscillations at higher frequencies. For example, a low-frequency oscillation at 1 Hz can be detected for amplitudes down to 0.1 mm, but a high-frequency oscillation at 1 kHz can be detected for amplitudes as small as 0.1 mm. For these low- and high-frequency oscillations, we can say that A. The maximum acceleration of the low-frequency oscillation is greater. B. The maximum acceleration of the high-frequency oscillation is greater. C. The maximum accelerations of the two oscillations are approximately equal.