Advertisements
Advertisements
प्रश्न
A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 × 10–2 kg and its linear mass density is 4.0 × 10–2 kg m–1. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?
Advertisements
उत्तर १
a) Mass of the wire, m = 3.5 × 10–2 kg
Linear mass density, `mu = m/l = 4.0 xx 10^(-2) kg m^(-1)`
Frequency of vibration, ν = 45 Hz
:. Length of the wire, `l = m/mu = (3.5xx10^(-2))/(4.0xx10^(-2)) = 0.875 m`
The wavelength of the stationary wave (λ) is related to the length of the wire by the relation:
`lambda = (2l)/n`
Where n = number of nodes in the wire
For fundamental node, n = 1:
λ = 2l
λ = 2 × 0.875 = 1.75 m
The speed of the transverse wave in the string is given as:
v = νλ= 45 × 1.75 = 78.75 m/s
b) The tension produced in the string is given by the relation:
T = v2µ
= (78.75)2 × 4.0 × 10–2 = 248.06 N
उत्तर २
Here n= 45 Hz, M = `3.5 xx 10^(-2)` kg
Mass per unit length = `m = 4.0 xx 10^(-2) kg m^(-1)`
`:. l = M/m = (3.5xx10^(-2))/(4.0xx10^(-2)) = 7/8`
As `l/2 = lambda = 7/8 :. lambda = 7/4 m = 1.75 m`
a) The speed of the transverse wave,` v = vlambda = 45 xx 1.75 = 78.75 "m/s"`
b) As ` v = sqrt(T/m)`
`:. T = v^2xx m = (78.75)^2xx 4.0 xx 10^(-2) = 248.06 N`
APPEARS IN
संबंधित प्रश्न
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.
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air increases with humidity.
For the wave described in Exercise 15.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?
A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train (a) approaches the platform with a speed of 10 m s–1, (b) recedes from the platform with a speed of 10 m s–1? (ii) What is the speed of sound in each case? The speed of sound in still air can be taken as 340 m s–1.
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.
A sine wave is travelling in a medium. The minimum distance between the two particles, always having same speed, is
Choose the correct option:
Which of the following equations represents a wave travelling along Y-axis?
The equation of a wave travelling on a string is:
\[y = \left( 0 \cdot 10 \text{ mm } \right) \sin\left[ \left( 31 \cdot 4 m^{- 1} \right)x + \left( 314 s^{- 1} \right)t \right]\]
- In which direction does the wave travel?
- Find the wave speed, the wavelength and the frequency of the wave.
- What is the maximum displacement and the maximum speed of a portion of the string?
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.
Two long strings A and B, each having linear mass density
\[1 \cdot 2 \times {10}^{- 2} kg m^{- 1}\] , are stretched by different tensions 4⋅8 N and 7⋅5 N respectively and are kept parallel to each other with their left ends at x = 0. Wave pulses are produced on the strings at the left ends at t = 0 on string A and at t = 20 ms on string B. When and where will the pulse on B overtake that on A?
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?
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air is independent of pressure.
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 ______.
The displacement y of a particle in a medium can be expressed as, y = `10^-6sin(100t + 20x + pi/4)` m where t is in second and x in meter. The speed of the wave is ______.
