College Physics: A Strategic Approach Volume 1

Randall D. Knight, Brian Jones, Stuart Field

Chapter 14

Oscillations - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

Problem 1

When a guitar string plays the note "A," the string vibrates at $440 \mathrm{~Hz}$. What is the period of the vibration?

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

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?

Sachin Rao

Numerade Educator

Problem 3

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?

Supratim Pal

Numerade Educator

Problem 4

Make a table with 3 columns and 8 rows. In row $1,$ label the columns $\theta\left(^{\circ}\right), \theta(\mathrm{rad}),$ and $\sin \theta .$ In the left column, starting in row $2,$ write $0,2,4,6,8,10,$ and 12
a. Convert each of these angles, in degrees, to radians. Put the results in column 2 . Show four decimal places.
b. Calculate the sines. Put the results, showing four decimal places, in column $3 .$
c. What is the first angle for which $\theta$ and $\sin \theta$ differ by more than $0.0010 ?$
d. Over what range of angles does the small-angle approximation appear to be valid?

Manish Jain

Numerade Educator

Problem 5

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 $5^{\circ}$ angle with the vertical. Holding the ball in place takes a force of $20 \mathrm{~N}$. If the ball is pulled farther to the side so that the cord makes a $10^{\circ}$ angle, what force is required to hold the ball?

Sachin Rao

Numerade Educator

Problem 6

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

Sachin Rao

Numerade Educator

Problem 7

An air-track glider is attached to a spring. The glider is pulled to the right and released from rest at $t=0 \mathrm{~s}$. It then oscillates with a period of $2.0 \mathrm{~s}$ and a maximum speed of $40 \mathrm{~cm} / \mathrm{s}$.
a. What is the amplitude of the oscillation?
b. What is the glider's position at $t=0.25 \mathrm{~s} ?$

Sachin Rao

Numerade Educator

Problem 8

What are the (a) amplitude and (b) frequency of the oscillation shown in Figure $\mathrm{P} 14.8 ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

What are the (a) amplitude and (b) frequency of the oscillation shown in Figure $\mathrm{P} 14.9 ?$

Sachin Rao

Numerade Educator

Problem 10

An object in simple harmonic motion has an amplitude of $6.0 \mathrm{~cm}$ and a frequency of $0.50 \mathrm{~Hz}$. Draw a position graph showing two cycles of the motion.

Vishal Gupta

Numerade Educator

Problem 11

During an earthquake, the top of a building oscillates with an amplitude of $30 \mathrm{~cm}$ at $1.2 \mathrm{~Hz}$. What are the magnitudes of
(a) the maximum displacement,
(b) the maximum velocity, and
(c) the maximum acceleration of the top of the building?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

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 $1 \mathrm{~m}$ with a period of $15 \mathrm{~s}$
a. To one significant figure, what is the maximum acceleration of the passengers during this motion?
b. What fraction is this of $g$ ?

Sachin Rao

Numerade Educator

Problem 13

A passenger car traveling down a rough road bounces up and down at $1.3 \mathrm{~Hz}$ with a maximum vertical acceleration of $0.20 \mathrm{~m} / \mathrm{s}^{2}$, both typical values. What are the (a) amplitude and (b) maximum speed of the oscillation?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

The New England Merchants Bank Building in Boston is $152 \mathrm{~m}$ high. On windy days it sways with a frequency of 0.17 $\mathrm{Hz},$ 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?

Sachin Rao

Numerade Educator

Problem 15

a. When the displacement of a mass on a spring is $\frac{1}{2} A,$ 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?

Sachin Rao

Numerade Educator

Problem 16

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

Sachin Rao

Numerade Educator

Problem 17

A block attached to a spring with unknown spring constant oscillates with a period of $2.00 \mathrm{~s}$. 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 $\mathrm{d}$ are independent questions, each referring to the initial situation.

Sachin Rao

Numerade Educator

Problem 18

A $200 \mathrm{~g}$ air-track glider is attached to a spring. The glider is pushed $10.0 \mathrm{~cm}$ against the spring, then released. A student with a stopwatch finds that 10 oscillations take $12.0 \mathrm{~s}$. What is the spring constant?

Sachin Rao

Numerade Educator

Problem 19

The position of a $50 \mathrm{~g}$ oscillating mass is given by $x(t)=$ $(2.0 \mathrm{~cm}) \cos (10 t),$ 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 $t=0.40 \mathrm{~s}$.

Sachin Rao

Numerade Educator

Problem 20

A $200 \mathrm{~g}$ mass attached to a horizontal spring oscillates at a frequency of $2.0 \mathrm{~Hz}$. At one instant, the mass is at $x=5.0 \mathrm{~cm}$ and has $v_{x}=-30 \mathrm{~cm} / \mathrm{s} .$ Determine:
a. The period.
b. The amplitude.
c. The maximum speed.
d. The total energy.

Rashmi Sinha

Numerade Educator

Problem 21

A $507 \mathrm{~g}$ mass oscillates with an amplitude of $10.0 \mathrm{~cm}$ on a spring whose spring constant is $20.0 \mathrm{~N} / \mathrm{m}$. Determine:
a. The period.
b. The maximum speed.
c. The total energy.

Sachin Rao

Numerade Educator

Problem 22

A 300 g oscillator has a speed of $95.4 \mathrm{~cm} / \mathrm{s}$ when its displacement is $3.00 \mathrm{~cm}$ and $71.4 \mathrm{~cm} / \mathrm{s}$ when its displacement is $6.00 \mathrm{~cm} .$ What is the oscillator's maximum speed?

Vishal Gupta

Numerade Educator

Problem 23

A mass on a string of unknown length oscillates as a pendulum with a period of $4.00 \mathrm{~s}$. 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 $\mathrm{d}$ are independent questions, each referring to the initial situation.

Sachin Rao

Numerade Educator

Problem 24

A $200 \mathrm{~g}$ ball is tied to a string. It is pulled to an angle of $8.00^{\circ}$ and released to swing as a pendulum. A student with a stopwatch finds that 10 oscillations take $12.0 \mathrm{~s}$. How long is the string?

Sachin Rao

Numerade Educator

Problem 25

The angle of a pendulum is given by $\theta(t)=(0.10 \mathrm{rad}) \cos (5 t)$ where $t$ is in seconds. Determine:
a. The amplitude.
b. The frequency.
c. The length of the string.
d. The angle at $t=2.0 \mathrm{~s}$.

Vishal Gupta

Numerade Educator

Problem 26

It is said that Galileo discovered a basic principle of the pendulum- that 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 $5.5 \mathrm{~s}$. How long is the lamp chain?

Vishal Gupta

Numerade Educator

Problem 27

The free-fall acceleration on the moon is $1.62 \mathrm{~m} / \mathrm{s}^{2}$. What is the length of a pendulum whose period on the moon matches the period of a $2.00-\mathrm{m}$ -long pendulum on the earth?

Sachin Rao

Numerade Educator

Problem 28

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

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

A building is being knocked down with a wrecking ball, which is a big metal sphere that swings on a $10-\mathrm{m}$ -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?

Sachin Rao

Numerade Educator

Problem 30

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 $5.0 \mathrm{~kg}$ lower leg was found to be $18 \mathrm{~cm}$ from the knee. When pivoted at the knee and allowed to swing, the oscillation frequency was $1.6 \mathrm{~Hz}$. What was the moment of inertia of the lower leg?

Sachin Rao

Numerade Educator

Problem 31

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 $\theta(t)=(0.175 \mathrm{rad}) \sin (\pi t),$ where $t$ is in seconds.
a. What is the angular position of the rod at $t=0.250 \mathrm{~s} ?$
b. What is the period of oscillation?
c. How long is the rod?

Vishal Gupta

Numerade Educator

Problem 32

You and your friends find a rope that hangs down $15 \mathrm{~m}$ 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 $2.0 \mathrm{~m} / \mathrm{s}$ and grab the rope, launching yourself out over the river. How long must you hang on if you want to stay dry?

Vishal Gupta

Numerade Educator

Problem 33

A thin, circular hoop with a radius of $0.22 \mathrm{~m}$ 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 $I=2 M R^{2} .$ What is the period of oscillation of the hoop?

Vishal Gupta

Numerade Educator

Problem 34

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 $2.3 \mathrm{~m}$ 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?

Sachin Rao

Numerade Educator

Problem 35

The amplitude of an oscillator decreases to $36.8 \%$ of its initial value in $10.0 \mathrm{~s}$. What is the value of the time constant?

Averell Hause

Carnegie Mellon University

Problem 36

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 $\mathrm{Hz}$ and a time constant of $4.0 \mathrm{~s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

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

Sachin Rao

Numerade Educator

Problem 38

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 $\mathrm{P} 14.38$ shows real data for a car driven over a bump. Estimate the frequency and the time constant for this damped oscillation.

Manish Jain

Numerade Educator

Problem 39

A $25 \mathrm{~kg}$ child sits on a $2.0-\mathrm{m}$ -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?

Sachin Rao

Numerade Educator

Problem 40

Your car rides on springs, so it will have a natural frequency of oscillation. Figure $\mathrm{P} 14.40$ shows data for the amplitude of motion of a car driven at different frequencies. The car is driven at $20 \mathrm{mph}$ 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?

Prashant Bana

Numerade Educator

Problem 41

Vision is blurred if the head is vibrated at $29 \mathrm{~Hz}$ 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 $7.5 \mathrm{~g}$, a typical value, what is the effective spring constant of the musculature attached to the eyeball?

Sachin Rao

Numerade Educator

Problem 42

A spring has an unstretched length of $12 \mathrm{~cm}$. When an $80 \mathrm{~g}$ ball is hung from it, the length increases by $4.0 \mathrm{~cm}$. Then the ball is pulled down another $4.0 \mathrm{~cm}$ 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.

Sachin Rao

Numerade Educator

Problem 43

A $0.40 \mathrm{~kg}$ ball is suspended from a spring with spring constant $12 \mathrm{~N} / \mathrm{m}$. If the ball is pulled down $0.20 \mathrm{~m}$ from the equilibrium position and released, what is its maximum speed while it oscillates?

Sachin Rao

Numerade Educator

Problem 44

A spring is hanging from the ceiling. Attaching a $500 \mathrm{~g}$ mass to the spring causes it to stretch $20.0 \mathrm{~cm}$ in order to come to equilibrium.
a. What is the spring constant?
b. From equilibrium, the mass is pulled down $10.0 \mathrm{~cm}$ 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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

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

Sachin Rao

Numerade Educator

Problem 46

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

Sachin Rao

Numerade Educator

Problem 47

On your first trip to Planet X you happen to take along a $200 \mathrm{~g}$ mass, a $40.0-\mathrm{cm}$ -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 $31.2 \mathrm{~cm}$. You then pull the mass down $10.0 \mathrm{~cm}$ and release it. With the stopwatch you find that 10 oscillations take $14.5 \mathrm{~s} .$ Can you now satisfy your curiosity?

Averell Hause

Carnegie Mellon University

Problem 48

An object oscillating on a spring has the velocity graph shown in Figure $\mathrm{P} 14.48 .$ 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.

Sachin Rao

Numerade Educator

Problem 49

The two graphs in Figure $\mathrm{P} 14.49$ 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 $k_{\mathrm{A}} / k_{\mathrm{B}}$ of their spring constants?

Sachin Rao

Numerade Educator

Problem 50

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 $606 \mathrm{~N} / \mathrm{m}$. The empty chair oscillated with a period of $0.901 \mathrm{~s}$. What is the mass of an astronaut who oscillates with a period of $2.09 \mathrm{~s}$ when sitting in the chair?

Sachin Rao

Numerade Educator

Problem 51

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

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

The ultrasonic transducer used in a medical ultrasound imaging device is a very thin disk $(m=0.10 \mathrm{~g})$ driven back and forth in SHM at $1.0 \mathrm{MHz}$ by an electromagnetic coil.
a. The maximum restoring force that can be applied to the disk without breaking it is $40,000 \mathrm{~N}$. What is the maximum oscillation amplitude that won't rupture the disk?
b. What is the disk's maximum speed at this amplitude?

Sachin Rao

Numerade Educator

Problem 53

A compact car has a mass of $1200 \mathrm{~kg}$. 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 $70 \mathrm{~kg}$ passengers?

Averell Hause

Carnegie Mellon University

Problem 54

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

Vishal Gupta

Numerade Educator

Problem 55

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

Averell Hause

Carnegie Mellon University

Problem 56

A $1.00 \mathrm{~kg}$ block is attached to a horizontal spring with spring constant $2500 \mathrm{~N} / \mathrm{m}$. The block is at rest on a frictionless surface. A $10.0 \mathrm{~g}$ 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 $10.0 \mathrm{~cm} ?$
b. Could you determine the bullet's speed by measuring the oscillation frequency? If so, how? If not, why not?

Vishal Gupta

Numerade Educator

Problem 57

Figure $\mathrm{P} 14.57$ shows two springs, each with spring constant $20 \mathrm{~N} / \mathrm{m},$ connecting a $2.5 \mathrm{~kg}$ 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?

Vishal Gupta

Numerade Educator

Problem 58

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 $12,000 \mathrm{~kg}$, and its speed was $21.2 \mathrm{~m} / \mathrm{s}$. The bus came to a stop in $50.0 \mathrm{~m}$.
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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

Two $50 \mathrm{~g}$ blocks are held $30 \mathrm{~cm}$ above a table. As shown in Figure $\mathrm{P} 14.59,$ one of them is just touching a $30-\mathrm{cm}$ -long spring. The blocks are 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?

Tatiana Graham

Numerade Educator

Problem 60

The earth's free-fall acceleration varies from $9.78 \mathrm{~m} / \mathrm{s}^{2}$ at the equator to $9.83 \mathrm{~m} / \mathrm{s}^{2}$ at the poles. A pendulum whose length is precisely $1.000 \mathrm{~m}$ 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 $201 \mathrm{~s}$. What is $g$ on the mountain top?

Vishal Gupta

Numerade Educator

Problem 61

A pendulum clock has a heavy bob supported on a very thin steel rod that is $1.00000 \mathrm{~m}$ long at $20^{\circ} \mathrm{C}$
a. To 6 significant figures, what is the clock's period? Assume that $g$ is $9.80 \mathrm{~m} / \mathrm{s}^{2}$ exactly.
b. To 6 significant figures, what is the period if the temperature increases by $10^{\circ} \mathrm{C} ?$
c. The clock keeps perfect time at $20^{\circ} \mathrm{C}$. At $30^{\circ} \mathrm{C},$ after how many hours will the clock be off by $1.0 \mathrm{~s} ?$

Tatiana Graham

Numerade Educator

Problem 62

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 $5.0 \mathrm{~m} / \mathrm{s}$. If the pendulum later undergoes small-amplitude oscillations at the bottom of the arc, what will the frequency be?

Tatiana Graham

Numerade Educator

Problem 63

Two side-by-side pendulum clocks have heavy bobs at the ends of rigid, very lightweight arms. One pendulum has a $38.8-\mathrm{cm}$ -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?

Manish Jain

Numerade Educator

Problem 64

Orangutans can move by brachiation, swinging like a pendulum beneath successive handholds. If an orangutan has arms that are $0.90 \mathrm{~m}$ long and repeatedly swings to a $20^{\circ}$ angle, taking one swing immediately after another, estimate how fast it is moving in $\mathrm{m} / \mathrm{s}$.

Prashant Bana

Numerade Educator

Problem 65

The $15 \mathrm{~g}$ head of a bobble-head doll oscillates in SHM at a frequency of $4.0 \mathrm{~Hz}$.
a. What is the spring constant of the spring on which the head is mounted?
b. Suppose the head is pushed $2.0 \mathrm{~cm}$ 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 $0.50 \mathrm{~cm}$ in $4.0 \mathrm{~s}$. What is the head's time constant?

Tatiana Graham

Numerade Educator

Problem 66

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

Tatiana Graham

Numerade Educator

Problem 67

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 $0.50 \mathrm{~s}$. His older sister pulls the spring a bit more than intended. She pulls the animal $30 \mathrm{~cm}$ below its equilibrium position, then lets 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?

Sachin Rao

Numerade Educator

Problem 68

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 $\mathrm{P} 14.68$ 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 $1.2 \mathrm{~N} / \mathrm{m}$.
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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

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

Sachin Rao

Numerade Educator

Problem 70

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 $\mathrm{P} 14.70$ 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

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

Tatiana Graham

Numerade Educator

Problem 72

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 $60 \mathrm{~kg}$ 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 $2.5 \mathrm{~mm} .$
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 $0.10 \mathrm{~s},$ 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?

Manish Jain

Numerade Educator

Problem 73

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 $12 \mathrm{mg}$ fly lands in the center of a horizontal spider's web, causing the web to sag by $3.0 \mathrm{~mm}$.Assuming that the web acts like a spring, what is the spring constant of the web?
A. $0.039 \mathrm{~N} / \mathrm{m}$
B. $0.39 \mathrm{~N} / \mathrm{m}$
C. $3.9 \mathrm{~N} / \mathrm{m}$
D. $39 \mathrm{~N} / \mathrm{m}$

Vishal Gupta

Numerade Educator

Problem 74

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. $0.91 \mathrm{~Hz}$
B. $2.9 \mathrm{~Hz}$
C. $9.1 \mathrm{~Hz}$
D. $29 \mathrm{~Hz}$

Sachin Rao

Numerade Educator

Problem 75

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.

Sachin Rao

Numerade Educator

Problem 76

Spiders are more sensitive to oscillations at higher frequencies. For example, a low-frequency oscillation at $1 \mathrm{~Hz}$ can be detected for amplitudes down to $0.1 \mathrm{~mm}$, but a high-frequency oscillation at $1 \mathrm{kHz}$ can be detected for amplitudes as small as $0.1 \mu \mathrm{m} .$ 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator