One end of two wires of the same metal and of the same length (with radius, $r$ and $2r$) are joined together. The wire is used as a sonometer wire and the junction is placed in between two bridges. The tension $T$ is applied to the wire. If at a junction a node is formed then the ratio of a number of loops formed in the wires will be

A circular loop of rope of length $L$ rotates with uniform angular velocity $\omega$ about axis through its centre on a horizontal smooth platform. Velocity of pulse (with respect to rope) produced due to sight radial displacement is given by

Three waves of equal frequency having amplitudes $10 \mu \mathrm{m}$, $4 \mu \mathrm{m}$ and $7 \mu \mathrm{m}$ arrive at a given point with a successive phase difference of $\frac{\mathrm{\pi }}{2}$. The amplitude of the resulting wave in $\mu \mathrm{m}$ is given by

A steel wire of length $1 \mathrm{m}$ and mass $0.1 \mathrm{kg}$ and having a uniform cross-sectional area of ${10}^{-6} {\mathrm{m}}^2$ is rigidly fixed at both ends. The temperature of the wire is lowered by $20º\mathrm{C}$ if the transverse waves are set up plucking the string in the middle the frequency of the fundamental note of vibration is

$\left({\mathrm{Y}}_{\text{steel }}=2\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2 {\mathrm{\alpha }}_{\mathrm{steel}}=1.21\times {10}^{-5}/º\mathrm{C}\right)$

A certain transverse sinusoidal wave of wavelength $20 \mathrm{cm}$ is moving in the positive $X$ direction. The transverse velocity of the particle at $x=0$ as a function of time is shown. The amplitude of the motion is