The design of some physical instrument requires that there be a constant difference in length of 10 cm between an iron rod and a copper cylinder laid side by side at all temperature find their lengths
`(alpha_(Fe)=11 xx 10^(6).^(@)C^(-1),alpha_(Cu)=17 xx 10^(-6) .^(@)C^(-1))`

Options

A. 28.3 cm, and 18.3 cm
B. 23.8 cm, 13.8 cm
C. 28.9 cm, 10.9
D. 27.5 cm, 14.5 cm

Explanation

To solve the problem of maintaining a constant length difference of 10 cm between an iron rod and a copper cylinder at all temperatures, we will follow these steps:

### Step 1: Understand the relationship between lengths and temperature
The change in length of a rod due to temperature change is given by the formula:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]
where:
- \( \Delta L \) is the change in length,
- \( L_0 \) is the original length,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta T \) is the change in temperature.

### Step 2: Set up the initial condition
Let:
- \( L_{Fe} \) = length of the iron rod,
- \( L_{Cu} \) = length of the copper cylinder.

According to the problem, the initial condition is:
\[ L_{Fe} - L_{Cu} = 10 \text{ cm} \]

### Step 3: Express the change in lengths
The change in lengths for both materials when the temperature changes can be expressed as:
\[ \Delta L_{Fe} = L_{Fe} \cdot \alpha_{Fe} \cdot \Delta T \]
\[ \Delta L_{Cu} = L_{Cu} \cdot \alpha_{Cu} \cdot \Delta T \]

### Step 4: Set up the equation for constant length difference
For the length difference to remain constant at 10 cm, the change in length of the iron rod must equal the change in length of the copper cylinder:
\[ \Delta L_{Fe} - \Delta L_{Cu} = 0 \]
This implies:
\[ L_{Fe} \cdot \alpha_{Fe} \cdot \Delta T = L_{Cu} \cdot \alpha_{Cu} \cdot \Delta T \]
Since \( \Delta T \) is common and non-zero, we can cancel it out:
\[ L_{Fe} \cdot \alpha_{Fe} = L_{Cu} \cdot \alpha_{Cu} \]

### Step 5: Relate the lengths of the rods
From the above equation, we can express the lengths in terms of each other:
\[ \frac{L_{Fe}}{L_{Cu}} = \frac{\alpha_{Cu}}{\alpha_{Fe}} \]
Substituting the values of the coefficients of linear expansion:
\[ \alpha_{Fe} = 11 \times 10^{-6} \, ^\circ C^{-1}, \quad \alpha_{Cu} = 17 \times 10^{-6} \, ^\circ C^{-1} \]
Thus:
\[ \frac{L_{Fe}}{L_{Cu}} = \frac{17 \times 10^{-6}}{11 \times 10^{-6}} = \frac{17}{11} \]
This implies:
\[ L_{Fe} = \frac{17}{11} L_{Cu} \]

### Step 6: Substitute into the initial condition
Now substitute \( L_{Fe} \) into the initial condition:
\[ \frac{17}{11} L_{Cu} - L_{Cu} = 10 \]
This simplifies to:
\[ \frac{17L_{Cu} - 11L_{Cu}}{11} = 10 \]
\[ \frac{6L_{Cu}}{11} = 10 \]

### Step 7: Solve for \( L_{Cu} \)
Multiplying both sides by 11:
\[ 6L_{Cu} = 110 \]
Dividing by 6:
\[ L_{Cu} = \frac{110}{6} \approx 18.33 \text{ cm} \]

### Step 8: Find \( L_{Fe} \)
Now substitute \( L_{Cu} \) back to find \( L_{Fe} \):
\[ L_{Fe} = \frac{17}{11} \cdot 18.33 \approx 28.33 \text{ cm} \]

### Final Answer
Thus, the lengths are:
- Length of the iron rod \( L_{Fe} \approx 28.33 \text{ cm} \)
- Length of the copper cylinder \( L_{Cu} \approx 18.33 \text{ cm} \)