First slide

Thermal expansion

Question

The coefficient of linear expansion of homogeneous rod changes linearly from α₁ to α₂ from one end to the other end. The effective coefficient of linear expansion is

Moderate

A

$α_{1} + α_{2}$
B

$\frac{α_{1} + {α_{2}}_{}}{2}$
C

$\frac{α_{1} - α_{2}}{2}$
D

$α_{1} - α_{2}$

Solution

At a distance x, $α_{x} = α_{1} + \frac{α_{2} - α_{1}}{l} x$

total expansion of the rod for a rise in temperature Δθ.
$∆ l = \int_{0}^{l} d x (α_{1} + (\frac{α_{2} - α_{1}}{l} x)) ∆ θ = (\frac{α_{1} + α_{2}}{2}) l ∆ θ$
It should be noted that, if a quantity changes line only from a minimum value to a maximum value, then its average value is the arithmetic mean of the two.

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