A bucket containing water is revolved in a vertical circle of radius $$r$$. To prevent the water from falling down, the minimum frequency of revolution required is

($$\mathrm{g}=$$ acceleration due to gravity)

Two monatomic ideal gases A and B of molecular masses '$$m_1$$' and '$$m_2$$' respectively are enclosed in separate containers kept at the same temperature. The ratio of the speed of sound in gas A to that in gas B is given by

A particle starts oscillating simple harmonically from its mean position with time period '$$T$$'. At time $$t=\frac{T}{12}$$, the ratio of the potential energy to kinetic energy of the particle is $$\left(\sin 30^{\circ}=\cos 60^{\circ}=0.5, \cos 30^{\circ}=\sin 60^{\circ}=\sqrt{3} / 2\right)$$

A hollow pipe of length $$0.8 \mathrm{~m}$$ is closed at one end. At its open end, a $$0.5 \mathrm{~m}$$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of pipe. If the tension in the string is $$50 \mathrm{~N}$$ and speed of sound in air is $$320 \mathrm{~m} / \mathrm{s}$$, the mass of the string is