2 HOMEWORK SOLUTIONS
PROBLEM THE HEARING OF A BAT: A moth of length 1.0 cm is flying about 1.0 m from a bat when the bat emits a sound wave at 80.0 kHz. The temperature of air is about 10.00C. To sense the presence of the moth using echolocation, the bat must emit a sound with a wavelength equal to or less than the length of the insect. The speed of sound that propagates in an ideal gas is given by
v=
where is the ratio of heat capacities ( = 1 . 4 for air), is the absolute temperature in kelvins (which is equal to the Celsius temperature plus 273.150C), M is the molar mass of the gas (for air, the average molar mass is M = 28.8 x 10-3 kg/mol), and R is the universal gas constant (R = 8.314 J/mol-K). (a) Find the wavelength of the 80.0-kHz wave emitted by the bat.
RT M
v = f v=
RT M
J 1 RT 1 (1.4 ) 8.314 molK (373.15K ) = = kg f M 28.8 10 -3 mol 80,000Hz
(
)
(b) Will the bat be able to locate the moth despite the darkness of the night? Yes, because the wavelength is smaller than the size of the moth. (c) How long after the bat emits the wave will it hear the echo from the moth?
= 4.85mm
t=
1. 0m = 2.58 10 -3 s m 388 s
ttotal = 2t = 5.15 10 -3 s
PROBLEM THE ELECTROMAGNETIC SPECTRUM (A) Which of the following statements correctly describe the various forms of EM radiation listed above? A. They have different wavelengths. B. They have different frequencies. C. They propagate at different speeds through a vacuum depending on their frequency. D. They propagate at different speeds through non-vacuum media depending on both their frequency and the material in which they travel. E. They require different media to propagate. (B) Which of the following statements correctly describe the various applications listed above? A. All these technologies use radio waves, including low-frequency microwaves. B. All these technologies use radio waves, including high-frequency microwaves. C. All these technologies use a combination of infrared waves and high-frequency microwaves. D. Microwave ovens emit in the same frequency band as some wireless Internet devices. E. The radiation emitted by wireless Internet devices has the shortest wavelength of all the technologies listed above. F. All these technologies emit waves with a wavelength in the range 0.10 to 10.0 m. G. All the technologies emit waves with a wavelength in the range 0.01 to 10.0 km.
(C) Based on this information, which of the following statements is correct? A. The earth absorbs visible light and emits radiation with a shorter wavelength. B. The earth absorbs visible light and emits radiation with a longer C. wavelength. The earth absorbs visible light and emits radiation with a lower frequency. D. The earth absorbs visible light and emits radiation with a higher frequency. (D) A large fraction of the ultraviolet (UV) radiation coming from the sun is absorbed by the atmosphere. The main UV absorber in our atmosphere is ozone, O3. In particular, ozone absorbs radiation with frequencies around 9.38 1014 Hz. What is the wavelength of the radiation absorbed by ozone?
c = f
=
3.00 10 8 m c s = = 320nm 14 f 9.38 10 Hz
STANDING WAVES ON GUITAR STRINGS Standing waves on a guitar string form when waves traveling down the string reflect off a point where the string is tied down or pressed against the fingerboard. The entire series of distortions may be superimposed on a single figure, like this, indicating different moments in time using traces of different colors or line styles.
(A) What is the wavelength of the standing wave shown on guitar string? The wave has two fixed points at the ends, and there are 3 antinodes, therefore n = 3 and
(B) What is the wavelength of the longest wavelength standing wave pattern that can fit on this guitar string?
2L n 2(60cm) = = 40cm 3 =
=
2L n n =1
=
(C)The frequency of the fundamental of the guitar string is 320 Hz. At what speed do waves move along that string?
2(60cm) = 120cm 1
v nv = 2L 2Lf 2(0.60m)(320Hz ) = v= n 1 m v = 384 s f=
(D) How does the overtone number relate to the standing wave pattern number, previously denoted with the variable n? Overtone Number = Pattern Number 1 The overtone number and the pattern number are easy to confuse but they differ by one. When referring to a standing wave pattern using a number, be explicit about which numbering scheme you are using. (E) A certain sound contains the following frequencies: 400 Hz, 1600 Hz, and 2400 Hz. Select the best description of this sound. This is a pure tone. This is a complex tone with a fundamental of 400 Hz, plus some of its overtones. This is a complex tone with a virtual pitch of 800 Hz. These frequencies are unrelated, so they are probably pure tones from three different sound sources. HARMONICS OF A PIANO WIRE A piano tuner stretches a steel piano wire with a tension of 800N. The steel wire has a length of 0.4m and a mass of 3.0g. (A) What is the frequency of the string's fundamental mode of vibration?
f= =
v nv = 2L
n T 1 800N = 2L 2(0.4m) 0.003kg
0.4 m
(B) What is the number n of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to 10,000 Hz?
f = 408Hz 1
f = nf n 1
n=
10,000Hz = 24 408Hz