question_answer 11)

The coefficient of volume expansion of glycerine is \[-2l=2l\left[ 1+\frac{{{x}^{2}}}{2{{l}^{2}}} \right]-2l=\frac{{{x}^{2}}}{l}\]\[\therefore \]. What is the fractional 1 change in density for a \[{{30}^{o}}C\]rise in temperature?

Answer:

Here, \[=\frac{\vartriangle l}{2l}=\frac{{{x}^{2}}}{2{{l}^{2}}}\]\[\text{2 T cos }\!\!\theta\!\!\text{ = mg}\]\[\text{T=}\frac{\text{mg}}{\text{2 cos }\!\!\theta\!\!\text{ }}\] \[\text{cos }\!\!\theta\!\!\text{ =}\frac{\text{x}}{{{\left( {{\text{l}}^{\text{2}}}\text{+}{{\text{x}}^{\text{2}}} \right)}^{\text{1/2}}}}\text{=}\frac{\text{x}}{\text{l}{{\left( \text{1+}\frac{{{\text{x}}^{\text{2}}}}{{{\text{l}}^{\text{2}}}} \right)}^{\text{1/2}}}}\text{=}\frac{\text{x}}{\text{l}\left( \text{1+}\frac{\text{1}{{\text{x}}^{\text{2}}}}{\text{2}{{\text{l}}^{\text{2}}}} \right)}\] As, \[\text{As,xl,so1}\frac{\text{1}{{\text{x}}^{\text{2}}}}{\text{2}{{\text{l}}^{\text{2}}}}\] \[1+\frac{1{{x}^{2}}}{2{{l}^{2}}}\approx 1\]\[\therefore \text{cos }\!\!\theta\!\!\text{ =}\frac{\text{x}}{l}\] As, \[T=\frac{mg}{2\left( x/l \right)}=\frac{mgl}{2x}\] Fractional change in density \[\text{=}\frac{\text{T}}{\text{A}}\text{=}\frac{\text{mgl}}{\text{2Ax}}\]