Wave Speed On A String

The speed of a transverse wave on a stretched string depends on the tension $T$ that provides the restoring force and the linear mass density $\mu$ that provides the inertia, through $v=\sqrt{T/\mu }$. A tighter or lighter string carries waves faster. Substituting $T=80\text{ N}$ and $\mu =0.005\text{ kg/m}$ gives $v=\sqrt{80/0.005}=\sqrt{16000}\approx 126\text{ m/s}$. The value $16\text{ m/s}$ takes the ratio $T/\mu$ but forgets the square root applied to the wrong quantity. The value $400\text{ m/s}$ would arise from a tension of 800 N, an order-of-magnitude error. The value $63\text{ m/s}$ halves the correct answer, likely from doubling $\mu$. This is the NCERT formula for wave speed on a string, derived from a force balance on a small curved element. As a plausibility check, dividing a large tension by a small mass density gives a large number under the root, so a speed greater than 100 m/s is expected, confirming that $126\text{ m/s}$ is reasonable for a taut light string.