Advertisements
Advertisements
Question
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.
Advertisements
Solution
Given:
Amplitude r = 17 cm = \[\frac{17}{100}\]= 0.17 m
Frequency of sound emitted by source f = 800 Hz
Velocity of sound \[v\]= 340 m/s
Frequency band = f2\[-\]f1= 8 Hz
Here,
\[f_2\] and \[f_1\]correspond to the maximum and minimum apparent frequencies (Both will be at the mean position because the velocity is maximum).
\[\text { Now }, f_1 = \left( \frac{340}{340 + v_s} \right)f\]
\[ \text { and } f_2 = \left( \frac{340}{340 - v_s} \right)f\]
\[ \therefore f_2 - f_1 = 8\]
\[ \Rightarrow 8 = \left( \frac{340}{340 - v_s} \right)f - \left( \frac{340}{340 + v_s} \right)f\]
\[ \Rightarrow 8 = 340f\left[ \frac{1}{340 - v_s} - \frac{1}{340 + v_s} \right]\]
\[ \Rightarrow 8 = 340 \times 800 \times \left[ \frac{2 v_s}{{340}^2 - {v_s}^2} \right]\]
\[ \Rightarrow \frac{2 v_s}{{340}^2 - v_s^2} = \frac{8}{340 \times 800}\]
\[ \Rightarrow {340}^2 - {v_s}^2 = 68000 v_s \]
Solving for vs, we get:
\[v_s\]= 1.695 m/s
For SHM:
\[v_s = r\omega\]
\[ \Rightarrow \omega = \left( \frac{1 . 695}{0 . 17} \right) = 10\]
\[ \therefore T = \frac{2\pi}{w} = \frac{\pi}{5} = 0 . 63 \sec\]
APPEARS IN
RELATED QUESTIONS
A wave is represented by an equation \[y = c_1 \sin \left( c_2 x + c_3 t \right)\] In which direction is the wave going? Assume that \[c_1 , c_2\] \[c_3\] are all positive.
The equation \[y = A \sin^2 \left( kx - \omega t \right)\]
represents a wave motion with
If you are walking on the moon, can you hear the sound of stones cracking behind you? Can you hear the sound of your own footsteps?
When sound wave is refracted from air to water, which of the following will remain unchanged?
When two waves with same frequency and constant phase difference interfere,
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.
A man stands before a large wall at a distance of 50.0 m and claps his hands at regular intervals. Initially, the interval is large. He gradually reduces the interval and fixes it at a value when the echo of a clap merges every 3 seconds, find the velocity of sound in air.
Two point sources of sound are kept at a separation of 10 cm. They vibrate in phase to produce waves of wavelength 5.0 cm. What would be the phase difference between the two waves arriving at a point 20 cm from one source (a) on the line joining the sources and (b) on the perpendicular bisector of the line joining the sources?
A string, fixed at both ends, vibrates in a resonant mode with a separation of 2⋅0 cm between the consecutive nodes. For the next higher resonant frequency, this separation is reduced to 1⋅6 cm. Find the length of the string.
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?
A source S and a detector D are placed at a distance d apart. A big cardboard is placed at a distance \[\sqrt{2}d\] from the source and the detector as shown in figure. The source emits a wave of wavelength = d/2 which is received by the detector after reflection from the cardboard. It is found to be in phase with the direct wave received from the source. By what minimum distance should the cardboard be shifted away so that the reflected wave becomes out of phase with the direct wave?
In a standing wave pattern in a vibrating air column, nodes are formed at a distance of 4.0 cm. If the speed of sound in air is 328 m s−1, what is the frequency of the source?
The separation between a node and the next antinode in a vibrating air column is 25 cm. If the speed of sound in air is 340 m s−1, find the frequency of vibration of the air column.
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?
Show that if the room temperature changes by a small amount from T to T + ∆T, the fundamental frequency of an organ pipe changes from v to v + ∆v, where \[\frac{∆ v}{v} = \frac{1}{2}\frac{∆ T}{T} .\]
A piano wire A vibrates at a fundamental frequency of 600 Hz. A second identical wire Bproduces 6 beats per second with it when the tension in A is slightly increased. Find the the ratio of the tension in A to the tension in B.
A tuning fork of frequency 256 Hz produces 4 beats per second with a wire of length 25 cm vibrating in its fundamental mode. The beat frequency decreases when the length is slightly shortened. What could be the minimum length by which the wire we shortened so that it produces no beats with the tuning fork?
Two electric trains run at the same speed of 72 km h−1 along the same track and in the same direction with separation of 2.4 km between them. The two trains simultaneously sound brief whistles. A person is situated at a perpendicular distance of 500 m from the track and is equidistant from the two trains at the instant of the whistling. If both the whistles were at 500 Hz and the speed of sound in air is 340 m s−1, find the frequencies heard by the person.
In an experiment to determine the velocity of sound in air at room temperature using a resonance tube, the first resonance is observed when the air column has a length of 20.0 cm for a tuning fork of frequency 400 Hz is used. The velocity of the sound at room temperature is 336 ms-1. The third resonance is observed when the air column has a length of ______ cm.
