A copper rod of $88 \,cm$ and an aluminium rod of unknown length have their increase in length independent of increase in temperature. The length of aluminium rod is $(a_{C_{u}}=1.7 \times 10^{-5}\,K^{-1}$ and $\left.a_{A_{l}}=2.2 \times 10^{-5} K^{-1}\right)$

Question

A 200 g sample of water at 80°C is mixed with 100 g of water at 20°C. Assuming no heat loss to the surroundings, what is the final temperature of the mixture?

Answer 1

To solve this problem, we need to ensure that the increase in length of both the copper rod and the aluminium rod is independent of the increase in temperature. This implies that the fractional change in length per degree change in temperature should be the same for both rods.

The formula for the change in length due to thermal expansion is given by:

\Delta L = a \cdot L \cdot \Delta T

where \Delta L is the change in length, a is the coefficient of linear expansion, L is the original length, and \Delta T is the change in temperature.

Since the increase in lengths of the two rods is independent of temperature, we have:

a_{C_{u}} \cdot L_{C_{u}} = a_{A_{l}} \cdot L_{A_{l}}

Substituting the given values:

1.7 \times 10^{-5} \cdot 88 = 2.2 \times 10^{-5} \cdot L_{A_{l}}

Solving for L_{A_{l}}:

L_{A_{l}} = \frac{1.7 \times 10^{-5} \cdot 88}{2.2 \times 10^{-5}}

L_{A_{l}} = \frac{149.6}{2.2}

L_{A_{l}} = 68 \, cm

Therefore, the length of the aluminium rod is 68 cm. This matches the given correct answer.

Answer 2