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Step-by-Step Solution
Step 1: Identify the physical principle
The problem is based on the concept of linear thermal expansion, which states that the change in length of a rod due to temperature change is proportional to its initial length and the coefficient of linear expansion.
Step 2: Write down the formula for linear expansion
The increase in length of each rod can be given by:
$$\Delta l = l \,\alpha \,\Delta T$$
where
$l$ is the original length of the rod,
$\alpha$ is the coefficient of linear expansion of the material, and
$\Delta T$ is the change in temperature.
Step 3: Equate the increase in lengths
Since the copper rod and aluminium rod have the same increase in length (independent of the increase in temperature),
$$l_{Cu}\,\alpha_{Cu}\,\Delta T = l_{Al}\,\alpha_{Al}\,\Delta T$$
We can cancel out the common factor $\Delta T$ on both sides, yielding:
$$l_{Cu}\,\alpha_{Cu} = l_{Al}\,\alpha_{Al}$$
Step 4: Substitute the known values
Substitute the given numerical values into the equation:
$$88 \times 1.7 \times 10^{-5} = l_{Al} \times 2.2 \times 10^{-5}$$
Step 5: Solve for the length of the aluminium rod
Rearrange the equation to find $l_{Al}$:
$$l_{Al} = \frac{88 \times 1.7 \times 10^{-5}}{2.2 \times 10^{-5}}$$
Cancel out the factor of $10^{-5}$:
$$l_{Al} = \frac{88 \times 1.7}{2.2} = 68 \text{ cm}$$
Final Answer
The length of the aluminium rod is 68 cm.
