When tension $$T$$ is applied to a sonometer wire of length $$I$$, it vibrates with the fundamental frequency $$n$$. Keeping the experimental setup same, when the tension is increased by 8 N, the fundamental frequency becomes three times the earlier fundamental frequency $$n$$. The initial tension applied to the wire (in newton) was

The extension in a wire obeying Hooke's law is $$x$$. The speed of sound in the stretched wire is $$v$$. If the extension in the wire is increased to $$4 x$$, then the speed of sound in a wire is

Two waves $$Y_1=0.25 \sin 316 t$$ and $$Y_2=0.25 \sin 310 t$$ are propagating along the same direction. The number of beats produced per second are

Two identical strings of length $$l$$ and $$2l$$ vibrate with fundamental frequencies $$\mathrm{N} \mathrm{~Hz}$$ and $$1.5 N$$ Hz, respectively. The ratio of tensions for smaller length to large length is