Advertisements
Advertisements
प्रश्न
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air increases with temperature.
Advertisements
उत्तर १
Take the relation:
`v = sqrt((gammaP)/rho)` ...(i)
For one mole of an ideal gas, the gas equation can be written as:
PV = RT
`"P" = ("RT")/"V"` .....(ii)
Substituting equation (ii) in equation (i), we get:
`"v" = sqrt((gamma"RT")/("V"rho)) = sqrt((gamma"RT")/"M")` ....(iv)
Where,
Mass, M = ρV is a constant
γ and R are also constants
We conclude from equation (iv) that `v prop sqrt"T"`.
Hence, the speed of sound in a gas is directly proportional to the square root of the temperature of the gaseous medium, i.e., the speed of the sound increases with an increase in the temperature of the gaseous medium and vice versa.
उत्तर २
We are given that `v = sqrt((gamma p)/rho)`
We know PV = nRT (For n moles of ideal gas)
`=> "PV" = "m"/"M" "RT"`
where m is the total mass and M is the molecular mass of the gas
`:. "P" = "m"/"M" * "RT"/M`
`= (rho"RT")/"M"`
`=> "P"/rho = "RT"/"M"`
Since `"P"/rho = "RT"/"M"`, therefore, `v = sqrt((gamma "P")/rho)`
= `sqrt((gamma "RT")/"M")`
संबंधित प्रश्न
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air increases with humidity.
A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with at a speed of 10 m s–1. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of 10 m s–1? The speed of sound in still air can be taken as 340 m s–1.
Two wave pulses travel in opposite directions on a string and approach each other. The shape of one pulse is inverted with respect to the other.
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 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.
A steel wire fixed at both ends has a fundamental frequency of 200 Hz. A person can hear sound of maximum frequency 14 kHz. What is the highest harmonic that can be played on this string which is audible to the person?
The equation for the vibration of a string, fixed at both ends vibrating in its third harmonic, is given by
\[y = \left( 0 \cdot 4 cm \right) \sin\left[ \left( 0 \cdot 314 {cm}^{- 1} \right) x \right] \cos \left[ \left( 600\pi s^{- 1} \right) t \right]\]
(a) What is the frequency of vibration? (b) What are the positions of the nodes? (c) What is the length of the string? (d) What is the wavelength and the speed of two travelling waves that can interfere to give this vibration?
A 40 cm wire having a mass of 3⋅2 g is stretched between two fixed supports 40⋅05 cm apart. In its fundamental mode, the wire vibrates at 220 Hz. If the area of cross section of the wire is 1⋅0 mm2, find its Young modulus.
Following figure shows a string stretched by a block going over a pulley. The string vibrates in its tenth harmonic in unison with a particular tuning for. When a beaker containing water is brought under the block so that the block is completely dipped into the beaker, the string vibrates in its eleventh harmonic. Find the density of the material of the block.
The string of a guitar is 80 cm long and has a fundamental frequency of 112 Hz. If a guitarist wishes to produce a frequency of 160 Hz, where should the person press the string?
For the travelling harmonic wave
y (x, t) = 2.0 cos 2π (10t – 0.0080x + 0.35)
Where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of 0.5 m.
Speed of sound wave in air ______.
A sound wave is passing through air column in the form of compression and rarefaction. In consecutive compressions and rarefactions ______.
A string of mass 2.5 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, the disturbance will reach the other end in ______.
A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36t + 0.018x + π/4) where x and y are in cm and t is in s. The positive direction of x is from left to right.
- The wave is travelling from right to left.
- The speed of the wave is 20 m/s.
- Frequency of the wave is 5.7 Hz.
- The least distance between two successive crests in the wave is 2.5 cm.
Given below are some functions of x and t to represent the displacement of an elastic wave.
- y = 5 cos (4x) sin (20t)
- y = 4 sin (5x – t/2) + 3 cos (5x – t/2)
- y = 10 cos [(252 – 250) πt] cos [(252 + 250)πt]
- y = 100 cos (100πt + 0.5x)
State which of these represent
- a travelling wave along –x direction
- a stationary wave
- beats
- a travelling wave along +x direction.
Given reasons for your answers.
The amplitude of wave disturbance propagating in the positive x-direction given is by `1/(1 + x)^2` at time t = 0 and `1/(1 + (x - 2)^2)` at t = 1 s, where x and y are in 2 metres. The shape of wave does not change during the propagation. The velocity of the wave will be ______ m/s.
