A light cylindrical vessel is kept on a horizontal surface. Area of base is A. A hole of cross-sectional area 'a' is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is (a < < A) :

A body takes 4 min. to cool from 61$$^\circ$$ C to 59$$^\circ$$ C. If the temperature of the surroundings is 30$$^\circ$$ C, the time taken by the body to cool from 51$$^\circ$$ C to 49$$^\circ$$ C is :

Two spherical soap bubbles of radii r₁ and r₂ in vacuum combine under isothermal conditions. The resulting bubble has a radius equal to :

Two wires of same length and radius are joined end to end and loaded. The Young's modulii of the materials of the two wires are Y₁ and Y₂. The combination behaves as a single wire then its Young's modulus is :