A tuning of fork of frequency $392\, Hz$, resonates with $50\, cm$ length of a string under tension $(T)$. If length of the string is decreased by $2\%$, keeping the tension constant, the number of beats heard when the string and the tuning fork made to vibrate simultaneously is

Two vibrating strings of the same material but lengths $L$ and $2L$ have radii $2r$ and $r$ respectively. They are stretched under the same tension . Both the strings vibrate in their fundamental modes, the one of length $L$ with frequency $f_1$ and the other with frequency $f_2$. The ratio $\frac{f_1}{f_2}$ is given by

A string $1\,m$ long is drawn by a $300\,Hz$ vibrator attached to its end. The string vibrates in three segments. The speed of transverse waves in the string is equal to ..... $m/s$

Two identical flutes produce fundamental notes of frequency $300\,Hz$ at $27\,^oC$. If the temperature of air in one flute is increased to $31\,^oC$, the number of the beats heard per second will be

A uniform string suspended vertically. A transverse pulse is created at the top most of the string. Then

A wave travelling along the $x-$ axis is described by the equation $y\ (x, t )\ =\ 0.005\ cos\ (\alpha x - \beta t )$ . If the wavelength and the time period of the wave in $0.08\ m$ and $2.0\ s$ respectively then $\alpha $ and $\beta $ in appropriate units are