A thin uniform tube is bent into a circle of radius $$r$$ in the vertical plane. Equal volumes of two immiscible liquids, whose densities are $${\rho _1}$$ and $${\rho _2}$$ $$\left( {{\rho _1} > {\rho _2}} \right),$$ fill half the circle. The angle $$\theta $$ between the radius vector passing through the common interface and the vertical is :

Two tubes of radii r₁ and r₂, and lengths l₁ and l₂ , respectively, are connected in series and a liquid flows through each of them in stream line conditions. P₁ and P₂ are pressure differences across the two tubes.

If P₂ is 4P₁ and l₂ is $${{{1_1}} \over 4}$$, then the radius r₂ will be equal to :

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of :

A bottle has an opening of radius a and length b. A cork of length b and radius (a + $$\Delta $$a) where ($$\Delta $$a < < a) is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is $$\mu $$ then the force needed to push the cork into the bottle is :