The frequencies of three tuning forks $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ are related as $$\mathrm{n}_{\mathrm{A}}>\mathrm{n}_{\mathrm{B}}>\mathrm{n}_{\mathrm{C}}$$. When the forks $$\mathrm{A}$$ and $$\mathrm{B}$$ are sounded together, the number of beats produced per second is '$$n_1$$'. When forks $$\mathrm{A}$$ and $$\mathrm{C}$$ are sounded together the number of beats produced per second is '$$n_2$$'. How may beats are produced per second when forks $$\mathrm{B}$$ and $$\mathrm{C}$$ are sounded together?

The equation of wave is given by $$\mathrm{y}=10 \sin \left(\frac{2 \pi \mathrm{t}}{30}+\alpha\right)$$. If the displacement is $$5 \mathrm{~cm}$$ at $$\mathrm{t}=0$$, then the total phase at $$\mathrm{t}=7.5 \mathrm{~s}$$ will be

$$\left[\sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}, \cos 30^{\circ}=\sin 60^{\circ}=\frac{\sqrt{3}}{2}\right] $$

A sonometer wire resonates with 4 antinodes between two bridges for a given tuning fork, when 1 kg mass is suspended from the wire. Using same fork, when mass M is suspended, the wire resonates producing 2 antinodes between the two bridges (distance between two bridges is as before). The value of M is

Two wires of same material of radius 'r' and '2r' respectively are welded together end to end. The combination is then used as a sonometer wire under tension 'T'. The joint is kept midway between the two bridges. The ratio of the number of loops formed in the wires such that the joint is a node is