Advertisements
Advertisements
प्रश्न
The absolute temperature of air in a region linearly increases from T1 to T2 in a space of width d. Find the time taken by a sound wave to go through the region in terms of T1, T2, d and the speed v of sound at 273 K. Evaluate this time for T1 = 280 K, T2 = 310 K, d = 33 m and v = 330 m s−1.
Advertisements
उत्तर
Given:
The absolute temperature of air in a region increases linearly from T1 to T2 in a space of width d.
The speed of sound at 273 K is v.
vT is the velocity of the sound at temperature T.
Let us find the temperature variation at a distance x in the region.
Temperature variation is given by:
\[T = T_1 + \frac{\left( T_2 - T_1 \right)}{d}x \]
\[ v \propto \sqrt{T}\]
\[ \Rightarrow \frac{v_T}{v} = \sqrt{\left( \frac{T}{273} \right)}\]
\[ \Rightarrow v_T = v\sqrt{\left( \frac{T}{273} \right)}\]
\[ \Rightarrow dt = \frac{dx}{v_T} = \frac{du}{v} \times \sqrt{\left( \frac{273}{T} \right)}\]
\[ \Rightarrow t = \frac{\sqrt{273}}{v} \int\limits_0^d \frac{dx}{\left[ T_1 + \frac{\left( T_2 - T_1 \right)}{d}x \right]^\frac{1}{2}}\]
\[ \Rightarrow t = \frac{\sqrt{273}}{v} \times \frac{2d}{T_2 - T_1} \left[ T_1 + \frac{\left( T_2 - T_1 \right)}{d} \right]_0^d \]
\[ \Rightarrow t = \frac{\sqrt{273}}{v} \times \frac{2d}{T_2 - T_1}\left( \sqrt{T_2} - \sqrt{T_1} \right)\]
\[ \Rightarrow t = \left( \frac{2d}{v} \right)\left( \frac{\sqrt{273}}{T_2 - T_1} \right) \times \sqrt{T_2} - \sqrt{T_1} \left( \because A^2 - B^2 = \left( A - B \right)\left( A + B \right) \right)\]
\[ \Rightarrow T = \frac{2d}{v}\frac{\sqrt{273}}{\sqrt{T_2} + \sqrt{T_1}} . . . (i)\]
Evaluating this time:
Initial temperature T1 = 280 K
Final temperature T2 = 310 K
Space width d = 33 m
v = 330 m s−1
On substituting the respective values in the above equation, we get:
\[T = \frac{2 \times 33}{330}\frac{\sqrt{273}}{\sqrt{280} + \sqrt{310}} = 96 \text { ms }\]
APPEARS IN
संबंधित प्रश्न
A string clamped at both ends vibrates in its fundamental mode. Is there any position (except the ends) on the string which can be touched without disturbing the motion? What if the string vibrates in its first overtone?
The bulk modulus and the density of water are greater than those of air. With this much of information, we can say that velocity of sound in air
When sound wave is refracted from air to water, which of the following will remain unchanged?
A tuning fork of frequency 512 Hz is vibrated with a sonometer wire and 6 beats per second are heard. The beat frequency reduces if the tension in the string is slightly increased. The original frequency of vibration of the string is
An electrically maintained tuning fork vibrates with constant frequency and constant amplitude. If the temperature of the surrounding air increases but pressure remains constant, the produced will have
(a) larger wavelength
(b) larger frequency
(c) larger velocity
(d) larger time period.
A steel tube of length 1.00 m is struck at one end. A person with his ear closed to the other end hears the sound of the blow twice, one travelling through the body of the tube and the other through the air in the tube. Find the time gap between the two hearings. Use the table in the text for speeds of sound in various substances.
Find the minimum and maximum wavelengths of sound in water that is in the audible range (20−20000 Hz) for an average human ear. Speed of sound in water = 1450 m s−1.
The length of the wire shown in figure between the pulley is 1⋅5 m and its mass is 12⋅0 g. Find the frequency of vibration with which the wire vibrates in two loops leaving the middle point of the wire between the pulleys at rest.
Sound with intensity larger than 120 dB appears pain full to a person. A small speaker delivers 2.0 W of audio output. How close can the person get to the speaker without hurting his ears?
A particular guitar wire is 30⋅0 cm long and vibrates at a frequency of 196 Hz when no finger is placed on it. The next higher notes on the scale are 220 Hz, 247 Hz, 262 Hz and 294 Hz. How far from the end of the string must the finger be placed to play these notes?
The two sources of sound, S1 and S2, emitting waves of equal wavelength 20.0 cm, are placed with a separation of 20.0 cm between them. A detector can be moved on a line parallel to S1 S2 and at a distance of 20.0 cm from it. Initially, the detector is equidistant from the two sources. Assuming that the waves emitted by the sources are in detector should be shifted to detect a minimum of sound.
Figure shown two coherent sources S1 and S2 which emit sound of wavelength λ in phase. The separation between the sources is 3λ. A circular wire of large radius is placed in such way that S1,S2 is at the centre of the wire. Find the angular positions θ on the wire for which constructive interference takes place.
Two sources of sound S1 and S2 vibrate at same frequency and are in phase. The intensity of sound detected at a point P as shown in the figure is I0. (a) If θ equals 45°, what will be the intensity of sound detected at this point if one of the sources is switched off? (b) What will be the answer of the previous part if θ = 60°?
The first overtone frequency of a closed organ pipe P1 is equal to the fundamental frequency of a open organ pipe P2. If the length of the pipe P1 is 30 cm, what will be the length of P2?
The fundamental frequency of a closed pipe is 293 Hz when the air in it is a temperature of 20°C. What will be its fundamental frequency when the temperature changes to 22°C?
A small source of sound oscillates in simple harmonic motion with an amplitude of 17 cm. A detector is placed along the line of motion of the source. The source emits a sound of frequency 800 Hz which travels at a speed of 340 m s−1. If the width of the frequency band detected by the detector is 8 Hz, find the time period of the source.
A sound source, fixed at the origin, is continuously emitting sound at a frequency of 660 Hz. The sound travels in air at a speed of 330 m s−1. A listener is moving along the lien x= 336 m at a constant speed of 26 m s−1. Find the frequency of the sound as observed by the listener when he is (a) at y = − 140 m, (b) at y = 0 and (c) at y = 140 m.
The speed of a wave in a string is 20 m/s and the frequency is 50 Hz. The phase difference between two points on the string 10 cm apart will be ______.
