Two rods of lengths $2m$ and $1m$ have coefficients of linear expansion $2\alpha$ and $\alpha$ respectively. The rods are joined to form a composite rod of total length $3m$. We need to find the average coefficient of linear expansion of the composite rod.

1. Problem Description

Two rods of lengths $2m$ and $1m$ have coefficients of linear expansion $2\alpha$ and $\alpha$ respectively. The rods are joined to form a composite rod of total length $3m$. We need to find the average coefficient of linear expansion of the composite rod.

2. Solution Steps

Let $L_1$ be the length of the first rod and $\alpha_1$ be its coefficient of linear expansion.

Let $L_2$ be the length of the second rod and $\alpha_2$ be its coefficient of linear expansion.

We are given that $L_1 = 2m$, $\alpha_1 = 2\alpha$, $L_2 = 1m$, and $\alpha_2 = \alpha$.

The total length of the composite rod is $L = L_1 + L_2 = 2m + 1m = 3m$.

Let $\alpha$ be the average coefficient of linear expansion of the composite rod.

The total expansion of the composite rod is the sum of the expansions of the individual rods.

Therefore, $L\alpha = L_1\alpha_1 + L_2\alpha_2$.

Substituting the given values, we have:

$3\alpha = 2(2\alpha) + 1(\alpha)$

$3\alpha = 4\alpha + \alpha$

$3\alpha = 5\alpha$

$\alpha = \frac{5}{3}\alpha$

3. Final Answer

The average coefficient of linear expansion of the composite rod will be $\frac{5}{3}\alpha$.