1.A heavy uniform rope of mass $M$ and length $L$ hangs vertically from a ceiling. A transverse pulse is generated at the lower end. Find the time taken by the pulse to travel to the top end.
Sol: Tension at height $y$ from bottom $T(y) = \mu y g \implies v(y) = \sqrt{\frac{T}{\mu}} = \sqrt{g y}$. $t = \int_0^L \frac{dy}{\sqrt{g y}} = \frac{2\sqrt{L}}{\sqrt{g}} = 2\sqrt{\frac{L}{g}}$.
2.Two tuning forks A and B produce 5 beats/sec when sounded together. When fork A is loaded with a little wax, beat frequency becomes 2 beats/sec. If frequency of B is $256\text{ Hz}$, find frequency of A.
Sol: Initial $f_A = 256 \pm 5 = 261\text{ Hz}$ or $251\text{ Hz}$. Loading A with wax decreases $f_A$. If $f_A = 261\text{ Hz}$, decreasing it brings it closer to $256\text{ Hz}$, reducing beat frequency to $2$ ($258 - 256 = 2$). Therefore $f_A = 261\text{ Hz}$.
