A brass rod of density \(8.3\frac{g}{cm^{3}}\) length \(3m\) is clamped at the cemtre. It is expected to give a longitudinal vibrations and the frequency of the fundamental note is \(600Hz\) . Calculate: the velocity of the sound in the rod and its Young’s Modulus.
Ans: \(V_{sound}=\left(3.6\times10^{3}\right)\frac{m}{s}\) and \(Y=\left(10.76\times10^10\right)\frac{N}{m^{2}}\).
The ends of a stretched wire of length are fixed at \(X=0\) and \(X=L\). In one experiment, the displacement of the wire is \(y_{1}=rSin\left(\frac{\pi x}{L}\right)Sin\omega t\) and energy \(E_{1}\), and in another experiment its displacement is \(y_{2}=rSin\left(\frac{2\pi x}{L}\right)Sin2\omega t\) and energy \(E_{2}\). Then how are \(E_{1}\) and \(E_{2}\) are related?
Ans: \(\frac{E_{1}}{E_{2}}=\frac{1}{4}\).
The equation of transverse wave in a stretched string is \(y=5Sin2\pi\left(\frac{t}{0.04}-\frac{x}{50}\right)\) with length expressed in \(cm\) and time time taken in \(sec\). Find: (i) Wave Length. (ii) Amplitude. (iii) Frequency. (iv) Velocity of wave. Also, calculate the maximum velocity and acceleration of string.
Ans:(i) \(\lambda=0.5m\) (ii) \(r=0.05m\) (iii) \(\nu=25Hz\) (iv) \(V=5\times\frac{2\pi}{0.04}Cos2\pi\left(\frac{1}{0.04}-\frac{x}{50}\right)\). and \(V_{max}=7.85\frac{m}{s}\) & \(a_{max}=-125\pi^{2}\frac{m}{s^{2}}\).
An open pipe is in seconds harmonic with frequency \(f_{1}\). Now one end of the tube is closed and frequency increased to \(f_{2}\) such that resonance again occurs in \(n^{th}\) harmonic. Find: the value of \(n\). How are \(f_{1}\) and \(f_{2}\) related to each other?
Ans: \(\frac{f_{1}}{f_{2}}=\frac{4}{5}\).
A string is stretched between two fixed points separated by \(75cm\). It is observed to have resonant frequencies \(420Hz\) and \(315Hz\). There are no other resonant frequencies between these two. What is the lowest resonant frequency for string?
Ans: \(105Hz\).
(a) If, the successive overtones of a vibrating string are \(280Hz\) and \(315Hz\). What is the frequency of the fundamental note? (b) If, the amplitude of a sound wave is tripled, by how many \(dB\) will the intensity level increases?
Ans:(a) \(\nu=70Hz\) and (b) \(I_{2}=3^{20}I_{1}\).
Two vibrating strings of the same materials but lengths \(l\) and \(2l\) have radii \(2r\) and \(r\) respectively. They are stretched under the same tension. Both the string vibrates in their fundamental modes, the one of length \(l\) with frequency \(\nu_{1}\) and the other with frequency \(\nu_{2}\). What is the ratio \(\frac{\nu_{1}}{\nu_{2}}\)?
Ans: \(\frac{\nu_{1}}{\nu_{2}}=1\).
The amplitude of a wave disturbance propagating in the positive x-direction is given by \(y=\frac{1}{\left(1+x^{2}\right)}\) at \(t=0\) and \(y=\frac{1}{\left[1+\left(x-1\right)^{2}\right]}\) at \(t=2s\). Where \(x\) and \(y\) are in \(metre\). The shape of distribution does not change during the propagation. What is the velocity of wave?
Ans: \(v=0.5\frac{m}{s}\).
The first overtone of an open organ pipe beats with the first overtone of a closed organ pipe with a beat frequency \(2.2Hz\). The fundamental frequency of closed organ pipe is \(110Hz\). Find: the length of the organ pipes. Take: Velocity of sound in air \(=330\frac{m}{s}\).
Ans: \(L_{closed}=0.75m\) and \(L_{open}=0.99m\).
A police car moving at \(22\frac{m}{s}\), chase a motorcyclist. The policeman sounds his horn at \(176Hz\) while both of them move towards a stationary siren of frequency \(165Hz\). Calculate: The speed of the motorcycle, if it is given that he does not observe any beats.
Ans: \(v=22\frac{m}{s}\).
Calculate: The ratio of the speed of sound in neon to that in water vapours at any temperature. Molecular weight of neon is \(\left(2.02\times10^{-2}\right)\frac{kg}{mole}\) and for water vapours molecular weight is \(\left(1.8\times10^{-2}\right)\frac{kg}{mole}\).
Ans: \(\frac{v_{neon}}{v_{vapour}}=1.055\).
A load of \(20kg\) is suspended by a steel wire in a sonometer experiment. Velocity of waves when the wire is rubbed with the resined cloth along the length is \(20\) times the velocity of the wave in the same string, when it is plucked. Find: the area of cross-section of the wire, If Young’s Modulus \((Y)\) for steel is \(\left(19.6\times10^{10}\right)\frac{N}{m^{2}}\) and \(g=9.8\frac{m}{s^{2}}\).
Ans: \(A=\left(4\times10^{-7}\right)m^{2}\).
An open pipe is suddenly closed at one end with the result that the frequency of the third harmonic of the closed pipe is found to be higher by \(400Hz\) than the fundamental frequency of the pipe. What is the fundamental frequency of the open pipe.
