Speed Of Waves On Strings

Wave speed on a stretched string, derived in NCERT Class 11, Chapter 15 (Waves), is $v=\sqrt{\frac{T}{\mu }}$, where $T$ is the tension and $\mu$ is the mass per unit length. Substituting $T=80$ N and $\mu =0.005$ kg/m gives $\frac{T}{\mu }=\frac{80}{0.005}=16000$ m${}^2$/s${}^2$, and taking the square root, $v=\sqrt{16000}\approx 126.5$ m/s. A tighter string or a lighter string transmits waves faster. The option 16 m/s mistakenly reports $T/\mu$ divided by 1000 or forgets the square root scale. The option 400 m/s wrongly uses $T/\mu =160000$ from a decimal error in $\mu$. The option 63 m/s halves the correct answer, as if the square root were mishandled. The physical origin of this formula is that tension supplies the restoring force that snaps a displaced element back, while the linear mass density represents the inertia resisting that motion, so their ratio under a square root sets the propagation speed, exactly analogous to how stiffness and mass set an oscillator's frequency. This is why a guitarist raises a string's pitch by tightening it, since greater tension speeds up the waves and raises the resonant frequency. A plausibility check: musical string wave speeds of the order of a hundred metres per second are typical, and $\sqrt{\text{N}/(\text{kg/m})}$ works out to $\sqrt{{\text{m}}^2/{\text{s}}^2}$, which reduces to metres per second, so the magnitude and units both confirm the result.