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 `(alpha cu=1.7xx10^(-5) K^(-1) " and " alpha Al=2.2xx10^(-5) K^(-1))`

Options

A. 68 cm
B. 6.8cm
C. 113.9cm
D. 88 cm

Explanation

To solve the problem, we need to understand the relationship between the change in length of the rods and their coefficients of linear expansion. The increase in length of a rod due to temperature change can be expressed using the formula:

\[ \Delta L = \alpha \cdot L_0 \cdot \Delta T \]

Where:
- \(\Delta L\) is the change in length,
- \(\alpha\) is the coefficient of linear expansion,
- \(L_0\) is the original length of the rod,
- \(\Delta T\) is the change in temperature.

Given:
- Length of copper rod, \(L_{Cu} = 88 \, \text{cm} = 0.88 \, \text{m}\)
- Coefficient of linear expansion for copper, \(\alpha_{Cu} = 1.7 \times 10^{-5} \, \text{K}^{-1}\)
- Coefficient of linear expansion for aluminium, \(\alpha_{Al} = 2.2 \times 10^{-5} \, \text{K}^{-1}\)
- The increase in length is independent of the increase in temperature.

Since the increase in length is independent of temperature, we can set the change in length of both rods equal to each other:

\[ \Delta L_{Cu} = \Delta L_{Al} \]

Substituting the formula for change in length:

\[ \alpha_{Cu} \cdot L_{Cu} \cdot \Delta T = \alpha_{Al} \cdot L_{Al} \cdot \Delta T \]

Since \(\Delta T\) is common on both sides, we can cancel it out (assuming \(\Delta T \neq 0\)):

\[ \alpha_{Cu} \cdot L_{Cu} = \alpha_{Al} \cdot L_{Al} \]

Now, we can rearrange the equation to find the unknown length of the aluminium rod:

\[ L_{Al} = \frac{\alpha_{Cu} \cdot L_{Cu}}{\alpha_{Al}} \]

Substituting the known values:

\[ L_{Al} = \frac{(1.7 \times 10^{-5}) \cdot (0.88)}{2.2 \times 10^{-5}} \]

Calculating \(L_{Al}\):

1. Calculate the numerator:
\[ 1.7 \times 10^{-5} \cdot 0.88 = 1.496 \times 10^{-5} \]

2. Now divide by \(\alpha_{Al}\):
\[ L_{Al} = \frac{1.496 \times 10^{-5}}{2.2 \times 10^{-5}} \approx 0.679 \, \text{m} \]

Thus, the length of the aluminium rod is approximately \(0.679 \, \text{m}\) or \(67.9 \, \text{cm}\).