Standing waves in strings

Two uniform strings of mass per unit length $\mu$ and $4\mu$, and length $L$ and $2L$, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $\nu_0 = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$, which of the following statement(s) is(are) correct?

A string of length 1 m and mass $2\times10^{-5}$ kg is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies 750 Hz and 1000 Hz. The value of tension $T$ is _____ Newton.

Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by
$$y=\left(10\cos\pi x\sin\frac{2\pi t}T\right)\text{cm}$$
The amplitude of the particle at $x=\dfrac43$ cm will be ______ cm.

A wire of length 30 cm, stretched between rigid supports, has it's n$^{\text{th}}$ and (n+1)$^{\text{th}}$ harmonics at 400 Hz and 450 Hz, respectively. If tension in the string is 2700 N, it's linear mass density is ______ kg/m.

If the tension in each string is $T_0$, the correct match for the highest fundamental frequency in $f_0$ units will be,

The length of the strings 1, 2, 3 and 4 are kept fixed at $L_0$, $\dfrac{3L_0}2$, $\dfrac{5L_0}4$, and $\dfrac{7L_0}4$, respectively. Strings 1, 2, 3, and 4 are vibrated at their 1st, 3rd, 5th, and 14th harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of $T_0$ will be,

Two wires $W_1$ and $W_2$ have the same radius r and respective densities $\rho_1$ and $\rho_2$ such that $\rho_2=4\rho_1$. They are joined together at the point O, as shown in the figure. The combination is used as a sonometer wire and kept under tension T. The point O is midway between the two bridges. When a stationary wave is set up in the composite wire, the joint is found to be a node. The ratio of the number of antinodes formed in $W_1$ to $W_2$ is :

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by
$$y(x,t)=0.5\sin\left(\frac{5\pi}4x\right)\cos(200\pi t).$$
What is the speed of the travelling wave moving in the positive $x$ direction ?

($x$ and t are in meter and second, respectively.)

One end of a taut string of length 3 m along the $x$ axis is fixed at $x=0$. The speed of the waves in the string is $100\ \mathrm{m\,s^{-1}}$. The other end of the string is vibrating in the $y$ direction so that stationary waves are set up in the string. The possible waveform(s) of these stationary waves is(are)

The total length of a sonometer wire between fixed ends is 110 cm. Two bridges are placed to divide the length of wire in ratio 6 : 3 : 2. The tension in the wire is 400 N and the mass per unit length is 0.01 kg/m. What is the minimum common frequency with which three parts can vibrate ?

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, $y(x,t)=(0.01\ \mathrm m)\sin\left[\left(62.8\ \mathrm m^{-1}\right)x\right]\cos\left[\left(628\ \mathrm s^{-1}\right)t\right]$. Assuming $\pi=3.14$, the correct statement(s) is (are)

Column I shows four systems, each of the same length L, for producing standing waves. The lowest possible natural frequency of a system is called its fundamental frequency, whose wavelength is denoted as $\lambda_f$. Match each system with statements given in Column II describing the nature and wavelength of the standing waves.

A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is 50 N and the speed of sound is 320 m s⁻¹, the mass of the string is

A 20 cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in the string is 0.5 N. The string is set into vibrations using an external vibrator of frequency 100 Hz. Find the separation (in cm) between the successive nodes on the string.