(i) A wire is stretched between two rigid supports. It is observed that the wire resonates at a frequency of `420` Hz. If a wooden bridge is placed at the midpoint of the wire (so that the midpoint becomes a node), it was observed that the smallest frequency at which the wire resonates is `420` Hz. Find the smallest frequency at which the wire will resonate when there is no wooden bridge.
(ii) A string of length `L` is fixed at one end and is under tension due to a weight hanging from the other end, as shown in the figure. The point of the string on the pulley behaves as a fixed point. Coordinate axes are chosen so that the horizontal segment of the string runs from `x = – L // 2` to `x = L // 2`. The string is vibrating at one of its resonant frequencies with transverse displacement `(y)` given by
`y(x,t) = 0.05 cos(12.0x) sin(360t)`
with x, y in meter and t in second. Write two smallest possible values of L consistent with the given equation?

Answer

(i) 210 Hz
(ii) `pi/12, pi/4`