An iron tyre of diameter 2 m is to be fitted on to a wooden wheel of diameter 2.01 m. The temperature to which the tyre must be heated , if `alpha=11xx10^(-6).^@C^(-1)` and room temperature is `20^@C` , will be

Options

A. `474.5^@C`
B. `490.5^@C`
C. `440.5 ^@C`
D. `460.5^@C`

Answer

To solve the problem, we need to determine the temperature to which the iron tire must be heated so that its diameter expands to fit over the wooden wheel. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have an iron tire with an initial diameter of 2 m and a wooden wheel with a diameter of 2.01 m. We need to find the temperature increase required for the iron tire to expand enough to fit over the wooden wheel.

### Step 2: Identify Given Values
- Diameter of the iron tire, \( D_i = 2 \, \text{m} \)
- Diameter of the wooden wheel, \( D_w = 2.01 \, \text{m} \)
- Coefficient of linear expansion for iron, \( \alpha = 11 \times 10^{-6} \, \text{°C}^{-1} \)
- Initial temperature (room temperature), \( T_0 = 20 \, \text{°C} \)

### Step 3: Calculate the Change in Diameter
The change in diameter (\( \Delta D \)) needed for the tire to fit over the wheel is:
\[ \Delta D = D_w - D_i = 2.01 \, \text{m} - 2 \, \text{m} = 0.01 \, \text{m} \]

### Step 4: Use the Formula for Linear Expansion
The formula for linear expansion is:
\[ \Delta L = \alpha L_0 \Delta T \]
Where:
- \( \Delta L \) is the change in length (or diameter in this case),
- \( L_0 \) is the original length (or diameter),
- \( \Delta T \) is the change in temperature.

### Step 5: Set Up the Equation
For the iron tire:
\[ \Delta D = \alpha D_i \Delta T \]
Substituting the known values:
\[ 0.01 = (11 \times 10^{-6}) \times 2 \times \Delta T \]

### Step 6: Solve for \( \Delta T \)
Rearranging the equation to solve for \( \Delta T \):
\[ \Delta T = \frac{0.01}{(11 \times 10^{-6}) \times 2} \]
Calculating:
\[ \Delta T = \frac{0.01}{22 \times 10^{-6}} = \frac{0.01}{0.000022} \approx 454.545 \, \text{°C} \]

### Step 7: Calculate the Final Temperature
The final temperature \( T_f \) is given by:
\[ T_f = T_0 + \Delta T = 20 + 454.545 \approx 474.545 \, \text{°C} \]

### Final Answer
The temperature to which the tire must be heated is approximately **474.5 °C**.

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