Problems 7–9 are related to the following scenario:

We have a collection of 5 different mass/spring combinations that all have different
properties. All of them have some damping and all have the same mass. The properties are
summarized in the table below:
Name Natural Frequency
(cycles/second)
Time for oscillation
to stop
(seconds)
Notes
A 2.0 100 sec.
B 5.0 300 sec.
C 20.0 100 sec.
D — 10 sec. Does not oscillate.
E — 50 sec. Does not oscillate. The
mass and spring are the
same as in D.
7. If all the oscillators were driven by a device with a frequency of 4.6 Hz, which of the
oscillators would respond most strongly?
a. A
b. B
c. C
d. D
e. E

8. If one of the oscillators is critically damped, which one is it?
a. A
b. B
c. C
d. D
e. E

9. Which of the oscillators has the weakest damping coefficient?
a. A
b. B
c. C
d. D
e. E

Please help for 7. I found all the spring constants but do not know how to find the damping coeffiecent with the information give. I know 8,9 are opposites but need help.

Thanks

To determine which oscillator would respond most strongly to a device with a frequency of 4.6 Hz (question 7), we need to look at the natural frequencies of the oscillators and compare them to the driving frequency. The oscillator that has a natural frequency closest to 4.6 Hz will respond most strongly.

Given the information in the table, we can see that:

- Oscillator A has a natural frequency of 2.0 cycles/second (or Hz)
- Oscillator B has a natural frequency of 5.0 Hz
- Oscillator C has a natural frequency of 20.0 Hz
- Oscillators D and E do not oscillate, so they can be excluded from consideration.

Since the frequency of the driving device is 4.6 Hz, oscillator B has the closest natural frequency to the driving frequency. Therefore, the correct answer is:

7. b. B

Regarding the damping coefficient (questions 8 and 9), the table only provides information about the time it takes for each oscillator's oscillation to stop. Unfortunately, without further information, we cannot determine the precise values of the damping coefficients for each oscillator.

However, we can make some reasonable assumptions based on the given data. The oscillators with longer times for oscillation to stop (higher damping) are likely to have stronger damping coefficients, while the ones with shorter times (weaker damping) will have weaker damping coefficients.

Given this information, and based on the provided times for oscillation to stop, we can make reasonable estimations:

- Oscillator A has a longer time for oscillation to stop (100 seconds). Therefore, it is reasonable to assume that it has a stronger damping coefficient compared to the others.
- Oscillator B has a shorter time for oscillation to stop (300 seconds). Therefore, it is reasonable to assume that it has a weaker damping coefficient compared to the others.
- Oscillator C also has a longer time for oscillation to stop (100 seconds), suggesting a stronger damping coefficient.
- Oscillators D and E do not oscillate, so they can be excluded from consideration for questions 8 and 9.

Based on these estimations, we can determine the answers:

8. Since oscillator B has a shorter time for oscillation to stop, it is likely to be the least damped or critically damped. Therefore, the correct answer is:

8. b. B

9. Since oscillator A and oscillator C have longer times for oscillation to stop compared to oscillator B, they would have stronger damping coefficients. Therefore, oscillator B has the weakest damping coefficient. The correct answer is:

9. b. B