Deck 13: Vibrations and Waves

Question 1

Does Simple Harmonic Motion occur when the force, acting on a mass, is positively proportional to the displacement (F = kx)? Explain.

Answer 1

No. F = kx (k positive) leads to unconstrained motion. One must have the force in opposite direction, F = -kx, to produce SHM.

Question 2

What is the equation of motion of a 30. gram mass on a spring of stiffness 3.0 N/m if it is initially displaced 7.7 cm and released?

Answer 2

y = 7.7 cos(10 t) where y is in cm and t in seconds.

Question 3

SHM may be written y = A sin( ?t + ? ) What is the value of ?, the phase constant, when:
(a) the initial displacement is zero?
(b) the initial displacement is A?
(c) the initial velocity is zero?

Answer 3

(a) ? = 0
(b) ? = ?
(c) ? = +? or -?

Question 4

Spock has landed on a newly discovered planet and is instructed to determine its gravitational strength. He constructs a simple pendulum with a length of 700. mm and observes 20. full swings in 1 minute and 16.7 seconds. What does he deduce the "acceleration of gravity" to be from this?

Question 5

Shown in Figure 13-3 is a graph of position vs. time for a system undergoing simple harmonic motion. Which of the other graphs represents the system's acceleration as a function of time?

Question 6

How would you "weigh" the astronauts in orbit (where they feel weightless) so that you can keep them in good health?

Question 7

Imagine hitting a heavy anvil with a hammer. The hammer is in contact with the metal for a short period of time. How do you suppose the time of contact depends upon how hard you hit the anvil (i.e., does a hard hit remain in contact much longer or shorter than a light tap?) [It is reasonable to assume Hooke's law of elasticity to hold.]

Question 8

Name 5 different type of waves or wave motion.

Question 9

Why can longitudinal earthquake waves go straight through the center of the Earth but transverse waves cannot?

Question 10

When the speed of a wave depends on the wavelength (or frequency), the waves are said to exhibit diffraction.

Question 11

The condition of driving a system at a natural frequency is referred to as resonance.

Question 12

The total energy stored in simple harmonic motion is proportional to the

A) square of the amplitude.
B) frequency of motion.
C) reciprocal of the spring constant.
D) square of the spring constant.
E) amplitude.

Question 13

Simple Harmonic Motion is characterized by

A) acceleration proportional to velocity.
B) constant acceleration.
C) acceleration inversely proportional to velocity.
D) acceleration proportional to the acceleration of gravity.
E) acceleration proportional to displacement.

Question 14

Doubling only the amplitude of a vibrating mass-and-spring system produces what effect on the system's mechanical energy?

A) increases the energy by a factor of three
B) increases the energy by a factor of two
C) produces no change
D) increases the energy by a factor of

E) increases the energy by a factor of four

Question 15

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a maximum?

A) at either A or B
B) one-fourth of the way between A and B
C) midway between A and B

Question 16

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a minimum?

A) one-fourth of the way between A and B
B) midway between A or B
C) at either A or B

Question 17

Two masses, A and B, are attached to different springs. Mass A vibrates with an amplitude of 8 cm at a frequency of 10 Hz and mass B vibrates with an amplitude of 5 cm at a frequency of 16 Hz. How does the maximum speed of A compare to the maximum speed of B?

A) Mass B has the greater maximum speed.
B) Mass A has the greater maximum speed.
C) They are equal.

Question 18

A mass swinging on the end of a massless string, as shown in Figure 13-1, undergoes SHM.

Where is the instantaneous acceleration of the mass greatest?

A) C
B) A and D
C) A and C
D) A and B
E) B

Question 19

Doubling only the amplitude of a vibrating mass-on-a-spring system produces what effect on the system frequency?

A) increases the frequency by a factor of 2
B) increases the frequency by a factor of 3
C) produces no change
D) increases the frequency by a factor of 4

Question 20

Simple pendulum A swings back and forth at twice the frequency of simple pendulum B. Which statement is correct?

A) Amplitude of B is twice of A.
B) Amplitude of pendulum A is twice that of B.
C) Amplitude of A is 1.41 that of B.
D) Amplitude can not be determined from data given.

Question 21

When the mass of a simple pendulum is tripled, the time required for one complete vibration

A) decreases to 3/4 of its original value.
B) decreases to one-third of its original value.
C) increases to 4/3 of its original value.
D) does not change.
E) increases by a factor of 3.

Question 22

When the length of a simple pendulum is tripled, the time for one complete vibration increases by a factor of

A) 1/3.
B)
C) 9.
D) 3.
E) 1/9.

Question 23

Shown in Figure 13-2 is a graph of position vs. time for a system undergoing simple harmonic motion. Which of the other graphs represents the system's velocity as a function of time?

A) graph a
B) graph b
C) graph c
D) graph d

Question 24

Simple pendulum A swings back and forth at twice the frequency of simple pendulum B. Which statement is correct?

A) Pendulum B is twice as long as A.
B) The length of A is times the length of B.
C) The mass of B is times the mass of A.
D) Pendulum B is twice as massive as A.
E) The length of B is times the length of A.

Question 25

What happens to a simple pendulum's frequency if both its length and mass are increased?

A) It increases.
B) It decreases.
C) It remains constant.
D) It could remain constant, increase, or decrease; it depends on the mass to length ratio (mass density).

Question 26

Increasing the mass m of a mass-and-spring system causes what kind of change on the resonant frequency of the system?

A) There is no change in the frequency.
B) The frequency increases if the ratio k/m is greater than or equal to 1 and decreases if the ratio k/m is less than 1.
C) The frequency increases if the ratio k/m is less than or equal to 1 and decreases if the ratio k/m is greater than 1.
D) The frequency decreases.
E) The frequency increases.

Question 27

Increasing the spring constant k of a mass-and-spring system causes what kind of change in the resonant frequency of the system? (Assume no change in the system's mass m.)

A) The frequency increases.
B) The frequency increases if the ratio k/m is less than or equal to 1 and decreases if the ratio k/m is greater than 1.
C) The frequency increases if the ratio k/m is greater than or equal to 1 and decreases if the ratio k/m is less than 1.
D) The frequency decreases.
E) There is no change in the frequency.

Question 28

Consider the wave shown in Figure 13-4

The amplitude is

A) 4 m.
B) 8 m.
C) 2 m.
D) cannot be determined from given information

Question 29

Consider the wave shown in Figure 13-5 below for the following item(s).


-The wavelength in Figure 13-5

A) is 2 m.
B) is 4 m.
C) is 8 m.
D) cannot be determined from the given information.

Question 30

Consider the wave shown in Figure 13-5 below for the following item(s).


-The frequency in Figure 13-5

A) is 4 Hz.
B) is 1 Hz.
C) is 0.5 Hz.
D) is 2 Hz.
E) cannot be determined from the given information.

Question 31

The total distance traveled by an object in one complete cycle of Simple Harmonic Motion is ________ times the amplitude.

A) one
B) two
C) four
D) three
E) half

Question 32

What is the amplitude (in meters) of a wave whose displacement is given by y= 50.sin(0.20x + 120t) cm?

A) 6 m
B) 0.50 m
C) 10 m
D) 1.0 m
E) 0.20 m

Question 33

The lower the frequency of a sound wave, the

A) shorter its period.
B) lower its velocity.
C) greater its velocity.
D) greater its wavelength.
E) smaller its amplitude.

Question 34

When waves interfere, the result is ________ original waves ("they interfere").

A) always the same as
B) always larger than
C) sometimes smaller than, sometimes larger than
D) always smaller than

Question 35

Total constructive interference occurs when two waves with similar frequency and wavelength

A) are completely out of phase.
B) have 90 degrees phase difference.
C) are completely in phase.
D) have a phase difference of -90 degrees.

Question 36

When a wave obliquely crosses a boundary into another medium, it is

A) diffracted.
B) refracted.
C) reflected.
D) always slowed down.

Question 37

In music the 3rd harmonic is which overtone?

A) 5th
B) 4th
C) 3rd
D) 1st
E) 2nd

Question 38

The natural frequencies for a stretched string of length L, and wave speed v, are Nv/(2L) where N equals

A) 0, 1, 3, 5, ...
B) 1, 3, 4, 5, ...
C) 1, 2, 3, 4, ...
D) 2, 4, 6, 8, ...
E) 0, 1, 2, 3, ...

Question 39

Compare two vibrating strings, each has the same tension, but the second has four times the mass density of the first. The wave speed along the 2nd string is how many times the speed along the 1st?

A)

B) 4
C) 1/4
D) 1/2
E) 2

Question 40

If one doubles the tension in a violin string, the fundamental frequency of that string will be how many times the original frequency?

A) 4
B) 1/

C) 2
D)

E)

Question 41

Suppose you wish to lower the pitch of a violin string by 2 octaves. This could be done by

A) replacing the string with one of the same material only half as massive.
B) replacing the string with one of the same material but with quadruple the diameter.
C) increasing the string tension a factor of four.
D) reducing the string tension by a factor of four.
E) shortening the string length to a quarter as much.

Question 42

What is the spring constant of a spring that stretches 2.0 cm when a mass of 0.60 kg is suspended from it?

Question 43

A mountain climber of mass 60. kg slips and falls a distance of 4.0 m, at which time he reaches the end of his elastic safety rope. The rope then stretches an additional 2.0 m before the climber comes to rest. What is the spring constant of the rope, assuming it obeys Hooke's law?

Question 44

A spring-driven dart gun propels a 10. g dart. It is cocked by exerting a force of 20. N over a distance of 5.0 cm. With what speed will the dart leave the gun, assuming the spring has negligible mass?

Question 45

A 0.30 kg mass is suspended on a string. In equilibrium the mass stretches the spring 2.0 cm downward. The mass is then pulled an additional distance of 1.0 cm down and released from rest.
(a) Calculate the period of oscillation.
(b) Calculate the total energy of the system.

Question 46

Tarzan swings back and forth on a long vine. His friend Jane notices 30. swings in 2.4 minutes.
(a) The frequency of the swing is ________.
(b) The vine length is ________ meters.

Question 47

A 0.30 kg mass is suspended on a string. In equilibrium the mass stretches the spring 2.0 cm downward. The mass is then pulled an additional distance of 1.0 cm down and released from rest.
(a) Write down its equation of motion.
(b) Calculate the velocity of the mass at T/3 s.

Question 48

A simple pendulum has a length of 0.80 m and a bob of mass 0.20 kg. It is initially displaced to an angle of 18° from the vertical and released from rest. At t = 1.2 s:
(a) what angle does it make with the vertical?
(b) what is its velocity?

Question 49

An object oscillates such that its displacement is x = (0.222 m) sin(314.t) where t is in seconds.
(a) In one period, the object moves what distance?
(b) What is the frequency?
(c) What is the position at t = 1.00 s?

Question 50

A spring has 0.770 kg oscillating on it with a period of 1.25 seconds.
(a) What is the frequency?
(b) Determine the spring constant.
(c) If the mass is doubled and the spring made 3 times stiffer, what is the new period?

Question 51

Suppose you want to set up a simple pendulum with a period of 0.250 s.
(a) What length is required if g = 9.80 m/s² on Earth?
(b) What length would be required on the moon assuming g is 6 (assume exact) times LESS than on Earth?

Question 52

The equation of motion of a particle undergoing simple harmonic motion is y = 2.0 sin(0.60t) where y is in cm. At t = 0.60 s determine the particle's:
(a) displacement.
(b) velocity.
(c) acceleration.

Question 53

A 1.33 kg dense mass is on the end of a 4.50 m "string".
(a) What would be the period of the pendulum on Earth?
(b) If the mass were doubled, what would be the frequency of oscillation?
(c) What would be the period on the planet Mars where gravity is 38.% that of the Earth?

Question 54

A string of linear density 6.00 g/m is under a tension of 180. N. What is the velocity of propagation of transverse waves along the string?

Question 55

A stretched string is observed to have three equal segments in a standing wave driven at a frequency of 480. Hz. What driving frequency will set up a standing wave with four equal segments?

Question 56

What are the first three harmonics of a string of linear mass density 2.00 g/m and length 0.600 m when it is subjected to tension of 50.0 N?

Question 57

A string fixed at both ends is vibrating in its 8th harmonic.
(a) How many nodes are on the string?
(b) The 8th harmonic wavelength is what fraction of the string length?
(c) If the 8th harmonic frequency is 976. Hz, what is the frequency of the 2nd overtone?

Question 58

A long rope of linear density 0.25 kg/m is under tension. Transverse waves travel on it at 30. m/s. How much energy is stored in 1.0 m of the rope when a driving frequency of 6.0 Hz produces a traveling wave with an amplitude of 0.10 m?

Question 59

A string of length 2.5 m is fixed at both ends. When the string vibrates at a frequency of 85 Hz, a standing wave with five loops is formed.
(a) Determine the distance between two adjacent nodes.
(b) Determine the wavelength of the waves that travel on the string.
(c) Determine the velocity of waves.
(d) Determine the fundamental frequency of this string.

Question 60

A 2.0 kg mass is attached to the end of a horizontal spring (k = 50. N/m) and set into simple harmonic motion with an amplitude of 10. cm. What is the total mechanical energy of this system?

A) 2.5 J
B) 0.02 J
C) 0.25 J
D) 1.00 J
E) 25 J

Question 61

A mass vibrates back and forth from the free end of an ideal spring (k = 20. N/m) with an amplitude of 0.25 m. What is the maximum kinetic energy of this vibrating mass?

A) 1.3 J
B) 2.5 J
C) 0.63 J
D) 5.0 J
E) It is impossible to give an answer since kinetic energy cannot be determined without knowing the object's mass.

Question 62

A mass vibrates back and forth from the free end of an ideal spring (k = 20. N/m) with an amplitude of 0.30m. What is the kinetic energy of this vibrating mass when it is 0.30 m from its equilibrium position?

A) 0.45 J
B) 1.4 J
C) 0.90 J
D) zero
E) It is impossible to give an answer without knowing the object's mass.

Question 63

A mass is attached to a spring. It oscillates at a frequency of 4.0 Hz when displaced a distance of 2.0 cm from equilibrium and released. What is the maximum velocity attained by the mass?

A) 0.40 m/s
B) 0.50 m/s
C) 0.32 m/s
D) 0.08 m/s
E) 0.02 m/s

Question 64

The mass of a mass-and-spring system is displaced 10 cm from its equilibrium position and released. A frequency of 4 Hz is observed. What frequency would be observed if the mass had been displaced only 5 cm and then released?

A) 8 Hz
B) 6 Hz
C) 4 Hz
D) 2 Hz

Question 65

A 2-kg mass is hung from a spring (k = 18 N/m), displaced slightly from its equilibrium position, and released. What is the frequency of its vibration?

A) 1.5 $\pi$ Hz
B) 1.5 Hz
C) 3.0/ $\pi$ Hz
D) 1.5/ $\pi$ Hz
E) 3.0 Hz

Question 66

A mass attached to the free end of an ideal spring executes SHM according to the equation x = 0.5 sin(20 t) where x is in meters and t is in seconds. What is the magnitude of the maximum acceleration for this mass?

A) 200 m/s²
B) 10 m/s²
C) 100 m/s²
D) 20 m/s²
E) (20/ $\pi$ ) m/s²

Question 67

A mass attached to the free end of an ideal spring executes SHM according to the equation x = 0.50 sin(18.t) where x is in meters and t is in seconds. What is the maximum velocity of the mass?

A) 36 m/s
B) 3 m/s
C) 3 m/s
D) 9 m/s
E) 3 m/s

Question 68

A simple pendulum consists of a 0.25-kg spherical mass attached to a massless string. When the mass is displaced slightly from its equilibrium position and released, the pendulum swings back and forth with a frequency of 2. Hz. What frequency would have resulted if a 0.50 kg mass (same diameter sphere) had been attached to the string instead?

A) 4. Hz
B) 1.41 Hz
C) 1. Hz
D) 2 $\pi$ Hz
E) 2. Hz

Question 69

What mass should be attached to a vertical spring (k = 39.5 N/m) so that the natural vibration frequency of the system will be 1.00 Hz?

A) 39.5 kg
B) 2.00 kg
C) 1.00 kg
D) 1.56 kg
E) 6.29 kg

Question 70

Grandfather clocks often are built so that each one-way swing of the pendulum is a second. How long is the length of a simple pendulum for a 2.00 second period?

A) 101.cm
B) 0.993 m
C) 0.500 m
D) 500. cm
E) 24.8 cm

Question 71

Waves on a lake pass under a floating bird causing the bird to bob up and down with a period of 2.5 seconds. If the distance from wave trough to wave trough is 3.0 meters, what is the speed of the wave?

A) 1.5 m/s
B) 2.1 m/s
C) 1.2 m/s
D) 1.9 m/s
E) 0.21 m/s

Question 72

What is the velocity of propagation if a wave has a frequency of 12. Hz and a wavelength of 3.0 m?

A) 28 m/s
B) 4.0 m/s
C) 6.0 m/s
D) 0.25 m/s
E) 36 m/s

Question 73

The equation of motion of the wave shown in Figure 13-6 is

A) y = 1.0 cos (2 $\pi$ t).
B) y = 1.0 sin ( $\pi$ t).
C) y = 0.5 cos (40t).
D) y = 0.5 cos ( $\pi$ t).
E) y = 0.5 sin (40t).

Question 74

What is the wavelength (in meters) of a wave whose displacement is given by y = 0.5 sin(0.20x + 120t), where x and y are in meters and t in seconds?

A) 10. m
B) 0.20 m
C) 19. m
D) 31. m
E) 5.0 m

Question 75

What is the frequency (in Hz) of a wave whose displacement is given y = 50. sin(0.20x + 120t), where x and y are in cm and t in seconds?

A) 0.5 Hz
B) 19. Hz
C) 0.120 kHz
D) 120 Hz
E) 10 Hz

Question 76

A spring, fixed at both ends, vibrates at a frequency of 12. Hz with a standing transverse wave pattern as shown in Figure 13-7.

What is this spring's fundamental frequency?

A) 36 Hz
B) 24. Hz
C) 16 Hz
D) 4.0 Hz
E) 8.0 Hz

Question 77

A string of linear density 1.5 g/m is under a tension of 20. N. What should its length be if its fundamental resonance frequency is 220. Hz?

A) 0.26 m
B) 0.96 m
C) 43. cm
D) 2.1 m
E) 1.1 m

Question 78

A string, fixed at both ends, vibrates at a frequency of 12 Hz with a standing transverse wave pattern containing 3 loops. What frequency is needed if the standing wave pattern is to contain 4 loops?

A) 48 Hz
B) 16 Hz
C) 36 Hz
D) 18 Hz
E) 12 Hz

Question 79

Find the first three harmonics of a string of linear mass density 2.00 g/m and length 0.600 m when it is subjected to tension of 50.0 N.

A) 264 Hz, 528 Hz, 792 Hz
B) 66 Hz, 132 Hz, 198 Hz
C) 132 Hz, 264 Hz, 396 Hz
D) none of the above