## MCAT Physics and Math Review

**Chapter 7: Waves and Sound**

### Practice Questions

1. An opera singer has two precisely identical glasses. The singer produces as pure a tone as possible and shatters the first glass at a frequency of 808 Hz. She then sings a frequency of 838 Hz in the presence of the second glass. The second glass will likely:

1. shatter after a longer amount of time because the applied frequency is higher.

2. shatter after a shorter amount of time because the applied frequency is higher.

3. not shatter because the applied frequency is not equal to the natural frequency of the glass.

4. not shatter because higher-frequency sounds are associated with more attenuation.

2. A child is practicing the first overtone on his flute. If his brother covers one end of the flute for a brief second, how will the sound change, assuming that the new pitch represents the first overtone in the new setup?

1. The pitch of the sound will go up.

2. The pitch of the sound will go down.

3. The pitch of the sound will not change.

4. The change in the pitch depends on the starting pitch.

3. Which of the following is necessarily true regarding frequency, angular frequency, and period of a given wave?

1. The magnitude of the angular frequency is larger than the magnitude of the period.

2. The product of the frequency and period is equal to the angular frequency.

3. The magnitude of the angular frequency is larger than the magnitude of the frequency.

4. The product of the angular frequency and period is 1.

4. Ultrasound machines calculate distance based upon:

1. intensity of the reflected sound.

2. travel time of the reflected sound.

3. angle of incidence of the sound.

4. the detected frequency of the sound.

5. The period for a certain wave is 34 ms. If there is a Doppler shift that doubles the perceived frequency, which of the following must be true?

1. The detector is moving toward the source at a velocity equal to the speed of sound.

2. The source is moving toward the detector at a velocity equal to half the speed of sound.

3. The perceived period is 17 ms.

4. The perceived period is 68 ms.

1. III only

2. I and IV only

3. II and III only

4. I, II, and IV only

6. If the speed of a wave is and its wavelength is 10 cm, what is its period?

1. 0.01 s

2. 0.03 s

3. 0.1 s

4. 0.3 s

7. What is the angular frequency of the third harmonic in a pipe of length 0.6 m with one closed end? (Note: The speed of the sound is approximately )

1. 213 radians per second

2. 425π radians per second

3. 425 radians per second

4. 850** π** radians per second

8. A certain sound level is increased by 20 dB. By what factor does its intensity increase?

1. 2

2. 20

3. 100

4. log 2

9. In some forms of otosclerosis, the stapedial foot plate, which transmits vibrations from the bones of the middle ear to the fluid within the cochlea, can become fixed in position. This limits the displacement of the stapedial foot plate during vibration. Based on this mechanism, which of the following symptoms would most likely be seen in an individual with otosclerosis?

1. An increase in the perceived volume of sounds

2. A decrease in the perceived volume of sounds

3. An increase in the perceived pitch of sounds

4. A decrease in the perceived pitch of sounds

10.If two waves are 180° out of phase, what is the amplitude of the resultant wave if the amplitudes of the original waves are 5 cm and 3 cm?

1. 2 cm

2. 3 cm

3. 5 cm

4. 8 cm

11.A student is measuring sound frequencies from the side of a road while walking east. For which of the following situations could the student determine that the difference between the perceived frequency and the actual emitted frequency is zero?

1. A plane flying directly above him from east to west

2. A police car passing the student with its siren on

3. A person playing piano in a house on the street

4. A dog barking in a car that moves east

12.In which of the following media does sound travel the fastest?

1. Vacuum

2. Air

3. Water

4. Glass

13.Shock waves have the greatest impact when the source is traveling:

1. just below the speed of sound.

2. exactly at the speed of sound.

3. just above the speed of sound.

4. well above the speed of sound.

14.As an officer approaches a student who is studying with his radio playing loudly beside him, he experiences the Doppler effect. Which of the following statements remains true while the officer moves closer to the student?

1. The apparent frequency of the music is increased.

2. The same apparent frequency would be produced if the officer were stationary and the student approached him at the same speed.

3. The apparent velocity of the wave is decreased.

1. I only

2. II only

3. I and III only

4. I, II, and III

15.Ignoring attenuation, how does the intensity of a sound change as the distance from the source doubles?

1. It is four times as intense.

2. It is twice as intense.

3. It is half as intense.

4. It is one-quarter as intense.

PRACTICE QUESTIONS

### Answers and Explanations

1. **C**

If these two glasses are perfectly identical, then the fact that the first glass shattered at 808 Hz tells us that this is very close (if not identical) to the natural (resonant) frequency of the glass. If she produces a frequency that is not equal (or very close) to the natural frequency, then the applied frequency will not cause the glass to resonate, and there will not be the increase in wave amplitude associated with resonating objects. Attenuation will increase with increased frequency because there is more motion over which nonconservative forces can damp the sound wave; however, even if sound level was matched to that which shattered the first glass when accounting for attenuation, the glass would still not shatter for the reasons described above, eliminating **choice (D)**.

2. **B**

This question is testing our understanding of pipes open at one or both ends. To begin, remember that high-frequency sounds have a high pitch and low-frequency sounds have a low pitch. The pipe in this example begins as one that is open on both ends, and then one end is closed off. Our task, therefore, is to determine how the frequency of the second harmonic differs between a pipe that is open at both ends from one of equal length that is open at only one end. For a pipe of length *L* open at both ends, the wavelength for the second harmonic (first overtone) is equal to *L*:

In contrast, for a pipe open at one end and closed at the other, the wavelength is equal to :

Keep in mind that the first overtone for a closed pipe corresponds to the third harmonic, not the second. Thus, when the brother covers one end of the flute, the wavelength increases. Given that the wavelength and the frequency of a sound are inversely proportional, an increase in wavelength corresponds to a decrease in frequency. Therefore, when the brother covers one end of the flute, the sound produced by the instrument will be slightly lower in pitch than the original sound.

3. **C**

The angular frequency is related to the frequency through the equation *ω* = 2π*f*. Therefore, the magnitude of the angular frequency will always be larger than the magnitude of the frequency. The magnitude of the angular frequency may or may not be larger than the magnitude of period; these variables are inversely proportional, eliminating **choice (A)**. The product of the frequency and the period is always 1 because these two are inverses of each other, eliminating **choice (B)**. Finally, the product of the angular frequency and period will always be 2*π* because eliminating **choice (D)**.

4. **B**

While intensity, **choice (A),** could be used to measure distance, time of travel is an easier indication and most commonly used by ultrasound machines. Apparent frequency, **choice (D)**, is only used in Doppler ultrasound, and not to calculate distance. Angle of incidence, **choice (C)**, can be used to position various structures on the screen of an ultrasound, but is not used to calculate distance.

5. **A**

Period is inversely related to frequency. Because the perceived frequency is doubled, the perceived period must be halved, from 34 ms to 17 ms. While either condition I or II would cause a doubling of the perceived velocity, neither condition must necessarily be true because the opposite could be true instead.

6. **B**

This question is testing our understanding of traveling waves. We know that frequency and wavelength are related through the equation *ν* = *f*λ. Frequency and period are inverses of each other, so this equation could be rearranged to solve for period:

7. **D**

The angular frequency is related to the frequency of a wave through the formula *ω* = 2*πf*. Thus, our initial task is to calculate the frequency of the wave. Knowing its speed, we determine the frequency by first calculating its wavelength (*ν* = *f*λ). For the third harmonic of a standing wave in a pipe with one closed end, the wavelength is

The frequency of the wave is therefore

Finally, obtain the angular frequency simply by multiplying the frequency of the wave by 2** π**:

*ω* = 2*π**f* = 850** π** radians per second

8. **C**

Let *I*_{i} be the intensity before the increase and *I*_{f} be the intensity after the increase. Using the equation that relates sound level to intensity, obtain the ratio of *I*_{i} to *I*_{f}:

9. **B**

Saying that the stapedial footplate has limited displacement during vibration is another way of stating that the amplitude of the vibration has been decreased. Because amplitude is related to intensity, and intensity is related to sound level, the perceived sound level (volume) will be decreased as well. Pitch, described in **choices (C)** and **(D)**, is related to the frequency of a sound, not its amplitude.

10.**A**

When two waves are out of phase by 180°, the resultant amplitude is the difference between the two waves’ amplitudes. In this case, the resulting wave will have an amplitude of 5 cm – 3 cm = 2 cm.

11.**D**

This question is testing us on our understanding of the Doppler effect. A difference of zero between the perceived and the emitted frequencies implies that the source of the sound is not moving relative to the student. If the car in **choice (D)** is moving at the same speed as the student, then the relative motion between them could be 0. In all of the other cases, the student and the sound source are necessarily moving relative to each other.

12.**D**

Sound is a mechanical disturbance propagated through a deformable medium; it is transmitted by the oscillation of particles parallel to the direction of the sound wave’s propagation. As such, sound needs matter to travel through, eliminating **choice (A)**. The speed of propagation is fastest in solid materials, followed by liquids, and slowest in gases.

13.**B**

Shock waves are the buildup of wave fronts as the distance between those wave fronts decreases. This occurs maximally when an object is traveling at exactly the same speed as the wave is traveling (the speed of sound). Once an object moves faster than the speed of sound, some of the effects of the shock wave are mitigated because all of the wave fronts will trail behind the object, destructively interfering with each other.

14.**A**

Here, an observer is moving closer to a stationary source. The applicable version of the Doppler effect equation is where *ν* is the velocity of the sound. Because the numerator is greater than the denominator, *f′* will be greater than *f*; therefore, statement I is true. The scenario described in statement II will produce a similar, but not identical, frequency for the officer: the frequency formula would be The apparent frequency will increase, but the increase will not be exactly the same as if the officer had been moving. Statement III is false because we already know the frequency increases for the officer—a decrease in velocity would be associated with a decrease in frequency.

15.**D**

Intensity is equal to power divided by area. In this case, area refers to the surface area of concentric spheres emanating out from the source of the sound. This surface area is given by 4*πr*^{2}, so as distance (*r*) doubles, the intensity will decrease by a factor of four.