A particle executes a simple harmonic motion with a periodic time $8$ second. At time $t = 0$, it is at a mean position. The ratio of the distance traveled by a particle in the 2nd and that in the 1st second of its motion is
($\sin 45^\circ = \cos 45^\circ = \dfrac{1}{\sqrt{2}}$, $\sin 90^\circ = \cos 0^\circ = 1$)

The third overtone of a closed pipe of length '$L_c$' has the same frequency as the third overtone of the open pipe of length '$L_0$'. Both the pipes have same diameters. The ratio $L_c : L_0$ is equal to

A tuning fork gives $5$ beats per second with a $33$ cm length of sonometer wire. If the length of the wire is shortened by $1$ cm, the number of beats is still the same. The frequency of the fork is

A set of $14$ tuning forks is arranged in a series of increasing frequencies. Each fork produces '$x$' beats per second with the preceding fork and the last fork is an octave of the first fork. If the seventh fork has the frequency of $114$ Hz, the value of $x$ is