Write the relation between the coefficient of linear expansion and the coefficient of volume expansion.
(N/A) Let a solid cube have a side length $L$ at temperature $T$. The initial volume is $V = L^3$.
When the temperature increases by $\Delta T$,the new length becomes $L' = L(1 + \alpha \Delta T)$,where $\alpha$ is the coefficient of linear expansion.
The new volume $V'$ is given by $V' = (L')^3 = [L(1 + \alpha \Delta T)]^3$.
Using the binomial expansion $(1 + x)^n \approx 1 + nx$ for small $x$,we get $V' \approx L^3(1 + 3\alpha \Delta T) = V(1 + 3\alpha \Delta T)$.
The coefficient of volume expansion $\gamma$ is defined by the relation $V' = V(1 + \gamma \Delta T)$.
Comparing the two expressions,we find that $\gamma = 3\alpha$.
When the temperature increases by $\Delta T$,the new length becomes $L' = L(1 + \alpha \Delta T)$,where $\alpha$ is the coefficient of linear expansion.
The new volume $V'$ is given by $V' = (L')^3 = [L(1 + \alpha \Delta T)]^3$.
Using the binomial expansion $(1 + x)^n \approx 1 + nx$ for small $x$,we get $V' \approx L^3(1 + 3\alpha \Delta T) = V(1 + 3\alpha \Delta T)$.
The coefficient of volume expansion $\gamma$ is defined by the relation $V' = V(1 + \gamma \Delta T)$.
Comparing the two expressions,we find that $\gamma = 3\alpha$.
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