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MCAT Sample Questions : Physical Sciences

Passage VI

The timbre, or quality, of a musical tone depends on the number and relative strengths of the harmonics including the fundamental frequency of the note. Figure 1a illustrates the first three harmonics of a tone. The addition of the first two harmonics is pictured in Figure 1b, and the addition of the first 3 harmonics is shown in Figure 1c.

Figure 1 Elements of a complex tone

The graphs in Figure 2 illustrate the characteristics of two adjacent tones from a bassoon. Figure 2a shows the pressure variations and the amplitudes of the harmonics for one of the tones, and Figure 2b shows the same information for the other tone

Figure 2 Pressure variations and amplitudes of harmonics for adjacent bassoon tones.

Following are some sample questions on this passage:

  1. Which of the waveforms shown in Figure 1 has the shortest period?

    1. First harmonic
    2. Second harmonic
    3. Third harmonic
    4. The waveform in Figure 1c

    Answer: C

    Explanation: The tone with the shortest period has the shortest wavelength. In Figure 1a, the period of the third harmonic (the curve with the smaller dashes) is seen to be shorter than the other two harmonics. Thus, answer choice C is the best answer.

  2. At the second position where the three curves intersect in Figure 1a, the curves are all:

    1. in phase.
    2. out of phase.
    3. at zero displacement.
    4. at maximum displacement.

    Answer: C

    Explanation: The three curves in Figure 1a intersect at three points in time. The second intersection occurs in the middle of the time axis. At that point all three curves have zero displacement. Therefore, answer choice C is the best answer.

  3. If the frequency of the first harmonic in Figure 2a is 100 Hz, what is the period of the second harmonic?

    1. 0.005 sec
    2. 0.01 sec
    3. 50.0 sec
    4. 200.0 sec

    Answer: A

    Explanation: The period T and frequency f of a tone are related by T = 1/f. If the first harmonic has a frequency of 100 Hz, then the second harmonic has a frequency of 200 Hz. The period corresponding to 200 Hz is 1/200 s-1 = 0.005 s. Thus, answer choice A is the best answer.

  4. Which of the following graphs best illustrates the relative amplitudes of the harmonics in Figure 1?

    Answer: A

    Explanation: The amplitudes of the three harmonics can be compared in Figure 1a. The first harmonic is seen to be largest while the other two have equal amplitudes. Answer choice A best represents these observations.

  5. If a fourth harmonic exists for the tone graphed in Figure 1, then, compared to the third harmonic, the fourth harmonic will have:

    1. lower amplitude.
    2. higher amplitude.
    3. lower frequency.
    4. higher frequency.

    Answer: D

    Explanation: A fourth harmonic would have a shorter period than the other three. Since T = 1/f, the fourth harmonic would have a higher frequency than the third harmonic. Therefore, answer choice D is the best answer.

  6. The period of the waveform shown in Figure 1c is the:

    1. same as the period of the first harmonic.
    2. same as the period of the second harmonic.
    3. same as the period of the third harmonic.
    4. sum of the periods of the first, second, and third harmonics.

    Answer: A

    Explanation: The waveform in Figure 1c begins to repeat at the zero displacement point near the end of the time axis. This is the same time period as the first harmonic as seen in Figure 1a. Thus, answer choice A is the best answer.

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