Question 1

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.

Accepted Answer

The maximum kinetic energy of the vibrating mass can be found using the formula for the potential energy stored in a spring at maximum compression or extension, which is $KE_{max} = PE_{max} = \frac{1}{2}kA^2$, where $k$ is the spring constant and $A$ is the amplitude. Substituting the given values, $KE_{max} = \frac{1}{2} \times 20 \, \text{N/m} \times (0.25 \, \text{m})^2 = 0.625 \, \text{J}$, which rounds to $0.63 \, \text{J}$. This calculation does not require the mass of the object, as the energy is solely dependent on the spring constant and the amplitude of the motion.

Question 2

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.

Accepted Answer

At the amplitude (0.30 m from equilibrium), the mass is momentarily at rest, so its kinetic energy is zero.

Question 3

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

Accepted Answer

The answer of A mass is attached to a spring....

Question 4

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

Accepted Answer

The frequency of a mass-and-spring system is determined by the mass and the stiffness of the spring, not by the amplitude of the displacement. Therefore, halving the displacement does not change the frequency.

Question 5

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

Accepted Answer

The frequency of vibration for a mass-spring system is given by $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$, where $k$ is the spring constant and $m$ is the mass. Substituting $k = 18 \, \text{N/m}$ and $m = 2 \, \text{kg}$, we get $f = \frac{1}{2\pi}\sqrt{\frac{18}{2}} = \frac{1.5}{\pi} \, \text{Hz}$.

Question 6

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

Accepted Answer

The maximum acceleration in SHM is given by $a_{max} = \omega^2 \times A$, where $\omega$ is the angular frequency and $A$ is the amplitude. Here, $\omega = 20$ rad/s and $A = 0.5$ m. Thus, $a_{max} = 20^2 \times 0.5 = 400 \times 0.5 = 200$ m/s$^2$.

Question 7

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

Accepted Answer

The answer of A mass attached to the free end...

Question 8

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

Accepted Answer

The answer of A simple pendulum consists of a 0.25-kg...

Question 9

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

Accepted Answer

The natural vibration frequency $f$ of a mass-spring system is given by $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$, where $k$ is the spring constant and $m$ is the mass attached to the spring. Rearranging this formula to solve for $m$ gives $m = \frac{k}{(2\pi f)^2}$. Substituting $k = 39.5 \, \text{N/m}$ and $f = 1.00 \, \text{Hz}$ into this formula yields $m = 1.00 \, \text{kg}$.

Question 10

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

Accepted Answer

The period $T$ of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity ($9.8 \, \text{m/s}^2$). For a 2.00 second period ($T = 2.00 \, \text{s}$), solving for $L$ gives $L \approx 0.993 \, \text{m}$.

Question 11

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

Accepted Answer

The answer of Waves on a lake pass under a...

Question 12

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

Accepted Answer

The velocity of propagation of a wave is calculated by multiplying its frequency by its wavelength. Here, velocity = frequency * wavelength = 12 Hz * 3.0 m = 36 m/s.

Question 13

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).

Accepted Answer

The answer of The equation of motion of the wave...

Question 14

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

Accepted Answer

The answer of What is the wavelength (in meters) of...

Question 15

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

Accepted Answer

The answer of What is the frequency (in Hz) of...

Question 16

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

Accepted Answer

The answer of A spring, fixed at both ends, vibrates...

Question 17

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

Accepted Answer

The fundamental resonance frequency of a string is given by:f = (1/2L) * sqrt(T/μ)where:f is the frequencyL is the length of the stringT is the tension in the stringμ is the linear density of the stringPlugging in the given values, we get:220 Hz = (1/2L) * sqrt(20 N / 1.5 g/m)Solving for L, we get:L = 0.26 m

Question 18

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

Accepted Answer

The frequency of a standing wave in a string fixed at both ends is directly proportional to the number of loops (or harmonics). If 3 loops correspond to 12 Hz, then 4 loops (which is 4/3 times more) will correspond to a frequency that is 4/3 times 12 Hz, which equals 16 Hz.

Question 19

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

Accepted Answer

The fundamental frequency (first harmonic) is calculated using the formula $ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} $, where $ L = 0.600 \, \text{m} $, $ T = 50.0 \, \text{N} $, and $ \mu = 2.00 \, \text{g/m} = 0.002 \, \text{kg/m} $. This gives $ f_1 = 132 \, \text{Hz} $. The second harmonic is $ 2 \times 132 = 264 \, \text{Hz} $, and the third harmonic is $ 3 \times 132 = 396 \, \text{Hz} $.

Quiz 13: Vibrations and Waves

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 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 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?

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 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 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 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?

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?

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?

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?

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

The equation of motion of the wave shown in Figure 13-6 is $The equation of motion of the wave shown in Figure 13-6 is A) y = 1.0 cos (2 \pi t) . B) y = 1.0 sin ( \pi t) . C) y = 0.5 cos (40t) . D) y = 0.5 cos ( \pi t) . E) y = 0.5 sin (40t) .$

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?

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 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 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 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?

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.