$A$ transverse wave is propagating on a string. The linear mass density of the vibrating string is $10^{-3} \ kg/m$. The equation of the wave is $Y = 0.05 \sin(x + 15t)$,where $x$ and $Y$ are in meters and time $t$ is in seconds. The tension in the string is: (in $N$)

$A$ wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $50 \, Hz$. The mass of the wire is $30 \, g$ and its linear density is $4 \times 10^{-2} \, kg/m$. The speed of the transverse wave on the string is ...... $m/s$.

$Assertion :$ Two waves moving in a uniform string having uniform tension cannot have different velocities.
$Reason :$ Elastic and inertial properties of string are same for all waves in same string. Moreover,the speed of a wave in a string depends on its elastic and inertial properties only.

$A$ string of mass $m$ and length $l$ hangs from a ceiling as shown in the figure. $A$ wave in the string moves upward. $v_A$ and $v_B$ are the speeds of the wave at points $A$ and $B$ respectively. Then $v_B$ is

$A$ string of length $0.4\, m$ and mass $10^{-2}\, kg$ is tightly clamped at its ends. The tension in the string is $1.6\, N$. Identical wave pulses are produced at one end at equal intervals of time $\Delta t$. The minimum value of $\Delta t$ which allows constructive interference between successive pulses is .... $s$

$A$ string of length $1 \,m$ and mass $490 \,g$ is put under a tension of $25 \,N$. $A$ wave of frequency $120 \,Hz$ is sent along it. The speed of this wave is (in $\,m/s$)