Advertisements
Advertisements
Question
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
Solution 1
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
Solution 2
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`
RELATED QUESTIONS
You have learnt that a travelling wave in one dimension is represented by a function y= f (x, t)where x and t must appear in the combination x – v t or x + v t, i.e. y = f (x ± v t). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave:
(a) `(x – vt )^2`
(b) `log [(x + vt)/x_0]`
(c) `1/(x + vt)`
A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s–1? The operating frequency of the scanner is 4.2 MHz.
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 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.
Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse (S) and longitudinal (P) sound waves. Typically the speed of S wave is about 4.0 km s–1, and that of P wave is 8.0 km s–1. A seismograph records P and S waves from an earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, at what distance does the earthquake occur?
Choose the correct option:
Which of the following equations represents a wave travelling along Y-axis?
Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double of the radius of B. A transverse wave travels on A with speed `v_A` and on B with speed `v_B`. The ratio `v_A/v_B` is ______.
A wave pulse, travelling on a two-piece string, gets partially reflected and partially transmitted at the junction. The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength λ and the transmitted wave λ'
Two wires A and B, having identical geometrical construction, are stretched from their natural length by small but equal amount. The Young modules of the wires are YA and YB whereas the densities are \[\rho_A \text{ and } \rho_B\]. It is given that YA > YB and \[\rho_A > \rho_B\]. A transverse signal started at one end takes a time t1 to reach the other end for A and t2 for B.
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.
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 wave travelling on a string at a speed of 10 m s−1 causes each particle of the string to oscillate with a time period of 20 ms. (a) What is the wavelength of the wave? (b) If the displacement of a particle of 1⋅5 mm at a certain instant, what will be the displacement of a particle 10 cm away from it at the same instant?
Two waves, travelling in the same direction through the same region, have equal frequencies, wavelengths and amplitudes. If the amplitude of each wave is 4 mm and the phase difference between the waves is 90°, what is the resultant amplitude?
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.
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 4 m.
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 ______.
Two perfectly identical wires kept under tension are in unison. When the tension in the wire is increased by 1% then on sounding them together 3 beats are heard in 2 seconds. What is the frequency of each wire?
