In order to double the frequency of the fundamental note emitted by a stretched string, the length is reduced to $\frac{3}{4}$$^{th}$ of the original length and the tension is changed. The factor by which the tension is to be changed, is

A string is fixed at both ends vibrates in a resonant mode with a separation $2.0 \,\,cm$ between the consecutive nodes. For the next higher resonant frequency, this separation is reduced to $1.6\,\, cm$. The length of the string is …. $cm$

The given diagram shows three light pieces of paper placed on a wire that is stretched over two supports, $Q$ and $R$ , a distance $4x$ apart. When the wire is made to vibrate at a particular frequency, all the pieces of paper, except the middle one, fall off the wire. Which of the following could be the wavelength of the vibration?

A tuning fork vibrating with a sonometer having a wire of length $20 \,cm$ produces $5$ beats per second. The beats frequency does not change if the length of the wire is changed to $21 \,cm$. The frequency of the tuning fork must be ………… $Hz$

Two uniform strings of mass per unit length $\mu$ and $4 \mu$, and length $L$ and $2 L$, respectively, are joined at point $O$, and tied at two fixed ends $P$ and $Q$, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$, which of the following statement($s$) is(are) correct?

$(A)$ With a node at $O$, the minimum frequency of vibration of the composite string is $v_0$

$(B)$ With an antinode at $O$, the minimum frequency of vibration of the composite string is $2 v_0$

$(C)$ When the composite string vibrates at the minimum frequency with a node at $O$, it has $6$ nodes, including the end nodes

$(D)$ No vibrational mode with an antinode at $O$ is possible for the composite string