Two blocks of masses 3 kg and 5 kg are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is $${{24} \over \pi } \times {10^2}$$ Nm^-2. What is the minimum radius of the wire ? (Take g = 10 ms^-2)

Two waves are simultaneously passing through a string and their equations are :

y₁ = A₁ sin k(x $$-$$ vt), y₂ = A₂ sin k(x $$-$$ vt + x₀). Given amplitudes A₁ = 12 mm and A₂ = 5 mm, x₀ = 3.5 cm and wave number k = 6.28 cm^$$-$$1. The amplitude of resulting wave will be ................ mm.

A source of light is placed in front of a screen. Intensity of light on the screen is I. Two Polaroids P₁ and P₂ are so placed in between the source of light and screen that the intensity of light on screen is I/2. P₂ should be rotated by an angle of (degrees) so that the intensity of light on the screen becomes $${{3I} \over 8}$$.

If the maximum value of accelerating potential provided by a ratio frequency oscillator is 12 kV. The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is ...............

[m_p = 1.67 $$\times$$ 10^$$-$$27 kg, e = 1.6 $$\times$$ 10^$$-$$19C, Speed of light = 3 $$\times$$ 10⁸ m/s]