On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius $$R$$ and making a circular contact of radius $$r$$ with the bottom $$R$$ and making a circular contact of radius $$r$$ with the bottom of the vessel. If $$r < < R$$ and the surface tension of water is $$T,$$ value of $$r$$ just before bubbles detach is: (density of water is $${\rho _w}$$)

Three rods of Copper, Brass and Steel are welded together to form a $$Y$$ shaped structure. Area of cross - section of each rod $$ = 4c{m^2}.$$ End of copper rod is maintained at $${100^ \circ }C$$ where as ends of brass and steel are kept at $${0^ \circ }C$$. Lengths of the copper, brass and steel rods are $$46,$$ $$13$$ and $$12$$ $$cms$$ respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are $$0.92, 0.26$$ and $$0.12$$ $$CGS$$ units respectively. Rate of heat flow through copper rod is:

The pressure that has to be applied to the ends of a steel wire of length $$10$$ $$cm$$ to keep its length constant when its temperature is raised by $${100^ \circ }C$$ is: (For steel Young's modulus is $$2 \times {10^{11}}\,\,N{m^{ - 2}}$$ and coefficient of thermal expansion is $$1.1 \times {10^{ - 5}}\,{K^{ - 1}}$$ )

A mass $$'m'$$ is supported by a massless string wound around a uniform hollow cylinder of mass $$m$$ and radius $$R.$$ If the string does not slip on the cylinder, with what acceleration will the mass fall or release?