Standing Waves on a String

The fundamental frequency of a string stretched with a weight of $4 \mathrm{kg}$ is $256 \mathrm{Hz}$. The weight required to produce its octave is 

If vibrations of a string are to be increased by a factor of two, then tension in the string must be made 

When the stationary waves are formed, then

Choose the correct statement from the following options.

A wire having a linear mass density $5.0\times {10}^{-3}\mathrm{kg} {\mathrm{m}}^{-1}$ is stretched between two rigid supports with a tension of $450 \mathrm{N}$. The wire resonates at a frequency of $420 \mathrm{Hz}$. The next higher frequency at which the same wire resonates is $490 \mathrm{Hz}$. Find the length of the wire.

A stretched wire of length $260 \mathrm{cm}$ is set into vibrations. It is divided into three segments whose frequencies are in the ratio $2:3:4$ . Their lengths must be

A light string of length $\mathrm{L}$ is tied at one end to a fixed support and to a heavy string at the other end $\mathrm{A}$ as shown in the figure. The total length of the composite string between the fixed support and the pulley is $2\mathrm{L}$. A block of mass $\mathrm{M}$ is tied to the free end of heavy string. Mass per unit length of the strings are $\mathrm{\mu }$ and $16\mathrm{\mu }$ and tension is $\mathrm{T}$. The lowest positive value of frequency such that the junction point $\mathrm{A}$ is a node is

A string $2.0 \mathrm{m}$ long and fixed at its ends is driven by a $240 \mathrm{H}\mathrm{z}$ vibrator. The string vibrates in its third harmonic mode. The speed of the wave and its fundamental frequency is

A string is clamped at both the ends and it is vibrating in its $4^{\mathrm{t}\mathrm{h}}$ harmonic. The equation of the stationary wave is $y=0.3\sin \left(0.157x\right)\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\left(200\mathrm{\pi }t\right)$. The length of the string is
(All quantities are in $\mathrm{S}\mathrm{I}$ units.)

A wire of length $2\mathrm{L},$ is made by joining two wires $\mathrm{A}$ and $\mathrm{B}$ of same length but different radii $\mathrm{r}$ and $2\mathrm{r}$ and made of the same material. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire $\mathrm{A}$ is $\mathrm{p}$ and that in $\mathrm{B}$ is $\mathrm{q}$ then ratio $\mathrm{p}:\mathrm{q}$ is:

A granite rod of $60 \mathrm{cm}$ length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is $2.7\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ and its Young's modulus is $9.27\times {10}^{10} \mathrm{Pa}$. What will be the fundamental frequency of the longitudinal vibrations?

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

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 

An object of specific gravity $\mathrm{\rho }$ is hung from a thin steel wire. The fundamental frequency for transverse standing waves in the wire is $300 \mathrm{Hz}$. The object is immersed in water so that one half of its volume is submerged. The new fundamental frequency (in $\mathrm{Hz}$) is:

A sonometer wire of length $1.5\mathrm{m}$ is made of steel. The tension in it produces an elastic strain of $\text{1\%}$. What is the fundamental frequency of steel if density and elasticity of steel are $7.7\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ and $2.2\times {10}^{11} \mathrm{N} {\mathrm{m}}^{-2}$ respectively?