Establish relation between coefficient of linear expansion (α), coefficient of superficial expansion (β), and coefficient of volume expansion (γ).

**OR**

State and prove Bernoulli's theorem.

Question

Establish relation between coefficient of linear expansion (α), coefficient of superficial expansion (β), and coefficient of volume expansion (γ).

**OR**

State and prove Bernoulli's theorem.

Accepted Answer

>

> [!TLDR]
> Starting from linear expansion $L=L_0(1+\alpha\Delta T)$ and extending to area and volume, one finds $\beta=2\alpha$ and $\gamma=3\alpha$, i.e. $\alpha:\beta:\gamma = 1:2:3$.
>
> **Step 1 — Linear expansion.**
> For a rod of original length $L_0$ heated through a temperature rise $\Delta T$, the new length is
> $$L = L_0(1+\alpha\Delta T)$$
> where $\alpha$ is the coefficient of linear expansion.
>
> **Step 2 — Superficial (areal) expansion.**
> Consider a square lamina of side $L_0$, so its original area is $A_0 = L_0^2$. After heating, each side becomes $L=L_0(1+\alpha\Delta T)$, so the new area is
> $$A = L^2 = L_0^2(1+\alpha\Delta T)^2 = L_0^2\left(1+2\alpha\Delta T+\alpha^2\Delta T^2\right)$$
> Since $\alpha$ is very small, $\alpha^2\Delta T^2$ is negligible, so
> $$A \approx L_0^2(1+2\alpha\Delta T) = A_0(1+2\alpha\Delta T)$$
> By definition, $A = A_0(1+\beta\Delta T)$, so comparing:
> $$\beta = 2\alpha$$
>
> **Step 3 — Cubical (volume) expansion.**
> Similarly, for a cube of original volume $V_0=L_0^3$, the new volume after heating is
> $$V = L^3 = L_0^3(1+\alpha\Delta T)^3 = L_0^3\left(1+3\alpha\Delta T + 3\alpha^2\Delta T^2+\alpha^3\Delta T^3\right)$$ …

Q.Establish relation between coefficient of linear expansion (α), coefficient of superficial expansion (β), and coefficient of volume expansion (γ). OR State and prove Bernoulli's theorem.

Concept understanding — Thermal Expansion

Thermal Expansion: Why Things Grow When Heated

The Precise Physics

Why Liquids Are Different

Real-World Examples You Already Know …

Q.Establish relation between coefficient of linear expansion (α), coefficient of superficial expansion (β), and coefficient of volume expansion (γ). OR State and prove Bernoulli's theorem.

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