Advertisements
Advertisements
Question
A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
Advertisements
Solution 1
Frequency of the turning fork, ν = 340 Hz
Since the given pipe is attached with a piston at one end, it will behave as a pipe with one end closed and the other end open, as shown in the given figure.
Such a system produces odd harmonics. The fundamental note in a closed pipe is given by the relation:
`l_1 = lambda/4`
Where,
Length of the pipe, `l_1 = 25.5 cm = 0.255 m`
`:.lambda = 4l_1 = 4xx 0.255 = 1.02 m`
The speed of sound is given by the relation: `v = vlambda` = 340 × 1.02 = 346.8 m/s
Solution 2
Frequency of nth mode of vibration of the closed organ pipe of lenght
`l_1 = (2n - 1) v/(4l_1)`
Frequency of (n+1) th mode of vibration of closed pipe of length
`'l_2' = [2(n+1)-1] v/(4l_2) = (2n + 1) v/(4l_2)`
Both the modes are given to resonate with a frequency of 340 Hz.
`:. (2n - 1) v/(4l_1) = (2n+1) v/(4l_2)`
or `(2n -1)/(2n+2) = l_1/l_2 = 25.5/79.3 = 1/3`
[Approximation has been used because edge effect is being ignored. Moreover, we know that in the case of a closed organ pipe, the second resonance length is three times the first resonance length.]
On simplification, n = 1
Now, (2n-1)υ/4l1 = 340. Substituting values
(2 x 1 -1) υ x 100/4 x 25.5 = 340 or υ =346.8 ms-1
APPEARS IN
RELATED QUESTIONS
A stone dropped from the top of a tower of height 300 m high splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m s–1? (g= 9.8 m s–2)
A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 °C = 343 m s–1.
(i) For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?
A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy submarine moves towards the SONAR with a speed of 360 km h–1. What is the frequency of sound reflected by the submarine? Take the speed of sound in water to be 1450 m s–1.
Show that the particle speed can never be equal to the wave speed in a sine wave if the amplitude is less than wavelength divided by 2π.
Show that for a wave travelling on a string
\[\frac{y_{max}}{\nu_{max}} = \frac{\nu_{max}}{\alpha_{max}},\]
where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to write
\[\frac{y_{max} + \nu_{max}}{\nu_{max} - \nu_{max}} = \frac{\nu_{max} + \alpha_{max}}{\nu_{max} - \alpha_{max}}?\]
Two waves of equal amplitude A, and equal frequency travel in the same direction in a medium. The amplitude of the resultant wave is
Two sine waves travel in the same direction in a medium. The amplitude of each wave is A and the phase difference between the two waves is 120°. The resultant amplitude will be
The equation of a wave travelling on a string stretched along the X-axis is given by
\[y = A e {}^- \left( \frac{x}{a} + \frac{t}{T} \right)^2 .\]
(a) Write the dimensions of A, a and T. (b) Find the wave speed. (c) In which direction is the wave travelling? (d) Where is the maximum of the pulse located at t = T? At t = 2 T?
A sonometer wire of length l vibrates in fundamental mode when excited by a tuning fork of frequency 416. Hz. If the length is doubled keeping other things same, the string will ______.
A wave pulse is travelling on a string with a speed \[\nu\] towards the positive X-axis. The shape of the string at t = 0 is given by g(x) = Asin(x/a), where A and a are constants. (a) What are the dimensions of A and a ? (b) Write the equation of the wave for a general time t, if the wave speed is \[\nu\].
A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibration is 1⋅0 and the displacement becomes zero 200 times per second. The linear mass density of the string is 0⋅10 kg m−1 and it is kept under a tension of 90 N. (a) Find the speed and the wavelength of the wave. (b) Assume that the wave moves in the positive x-direction and at t = 0, the end x = 0 is at its positive extreme position. Write the wave equation. (c) Find the velocity and acceleration of the particle at x = 50 cm at time t = 10 ms.
A 200 Hz wave with amplitude 1 mm travels on a long string of linear mass density 6 g m−1 kept under a tension of 60 N. (a) Find the average power transmitted across a given point on the string. (b) Find the total energy associated with the wave in a 2⋅0 m long portion of the string.
A wire of length 2⋅00 m is stretched to a tension of 160 N. If the fundamental frequency of vibration is 100 Hz, find its linear mass density.
What is the interference of sound waves?
A sound wave is passing through air column in the form of compression and rarefaction. In consecutive compressions and rarefactions ______.
At what temperatures (in °C) will the speed of sound in air be 3 times its value at O°C?
A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this wire when a tension of 2.06 × 104N is applied?
A wave of frequency υ = 1000 Hz, propagates at a velocity v = 700 m/sec along x-axis. Phase difference at a given point x during a time interval M = 0.5 × 10-3 sec is ______.
