question_answer 20)

A brass boiler has a base area of \[m=100g=0\cdot 100kg\] and thickness \[AD=BD={{\left( {{l}^{2}}+{{x}^{2}} \right)}^{1/2}}\] cm. It boils water at the rate of \[\vartriangle l=\text{AD+DB-AB=2AD-AB}\]kg/min, when placed on a gas stove. Estimate the temperature of the part of the flame in contact with the boiler. Thermal conductivity of brass \[=2{{\left( {{l}^{2}}+{{x}^{2}} \right)}^{1/2}}-2l=2l{{\left( 1+\frac{{{x}^{2}}}{{{l}^{2}}} \right)}^{1/2}}\]\[-2l=2l\left[ 1+\frac{{{x}^{2}}}{2{{l}^{2}}} \right]-2l=\frac{{{x}^{2}}}{l}\]. Heat of vaporisation of water \[\therefore \]

Answer:

Here, \[=\frac{\vartriangle l}{2l}=\frac{{{x}^{2}}}{2{{l}^{2}}}\]\[\text{2 T cos }\!\!\theta\!\!\text{ = mg}\] \[\text{T=}\frac{\text{mg}}{\text{2 cos }\!\!\theta\!\!\text{ }}\] \[\text{cos }\!\!\theta\!\!\text{ =}\frac{\text{x}}{{{\left( {{\text{l}}^{\text{2}}}\text{+}{{\text{x}}^{\text{2}}} \right)}^{\text{1/2}}}}\text{=}\frac{\text{x}}{\text{l}{{\left( \text{1+}\frac{{{\text{x}}^{\text{2}}}}{{{\text{l}}^{\text{2}}}} \right)}^{\text{1/2}}}}\text{=}\frac{\text{x}}{\text{l}\left( \text{1+}\frac{\text{1}{{\text{x}}^{\text{2}}}}{\text{2}{{\text{l}}^{\text{2}}}} \right)}\]\[\text{As,xl,so1}\frac{\text{1}{{\text{x}}^{\text{2}}}}{\text{2}{{\text{l}}^{\text{2}}}}\] \[1+\frac{1{{x}^{2}}}{2{{l}^{2}}}\approx 1\] From, \[\therefore \text{cos }\!\!\theta\!\!\text{ =}\frac{\text{x}}{l}\] \[T=\frac{mg}{2\left( x/l \right)}=\frac{mgl}{2x}\] \[\text{=}\frac{\text{T}}{\text{A}}\text{=}\frac{\text{mgl}}{\text{2Ax}}\] \[\text{Y=}\frac{\text{Stress}}{\text{Strain}}\text{=}\frac{\text{mgl}}{\text{2Ax}}\text{ }\!\!\times\!\!\text{ }\frac{\text{2}{{\text{l}}^{\text{2}}}}{{{\text{x}}^{\text{2}}}}\text{=}\frac{\text{mg}{{\text{l}}^{\text{3}}}}{\text{A}{{\text{x}}^{\text{3}}}}\]