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 of determining the temperature to which the iron tire must be heated in order to fit onto the wooden wheel, we will follow these steps:

### Step 1: Identify the given values
- Diameter of the iron tire, \( D_{iron} = 2 \, \text{m} \)
- Diameter of the wooden wheel, \( D_{wood} = 2.01 \, \text{m} \)
- Coefficient of linear expansion, \( \alpha = 11 \times 10^{-6} \, \text{°C}^{-1} \)
- Room temperature, \( T_{room} = 20 \, \text{°C} \)

### Step 2: Calculate the change in diameter required
The change in diameter, \( \Delta D \), needed for the iron tire to fit onto the wooden wheel is:
\[ \Delta D = D_{wood} - D_{iron} = 2.01 \, \text{m} - 2 \, \text{m} = 0.01 \, \text{m} \]

### Step 3: Relate the change in diameter to temperature change
The change in diameter due to thermal expansion can be expressed as:
\[ \Delta D = \alpha \times D_{iron} \times \Delta T \]
where \( \Delta T = T - T_{room} \).

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 0.01 = 11 \times 10^{-6} \times 2 \times (T - 20) \]

### Step 5: Simplify the equation
First, calculate \( 11 \times 10^{-6} \times 2 \):
\[ 11 \times 10^{-6} \times 2 = 22 \times 10^{-6} \]
Now, substitute this back into the equation:
\[ 0.01 = 22 \times 10^{-6} \times (T - 20) \]

### Step 6: Solve for \( T \)
Rearranging the equation gives:
\[ T - 20 = \frac{0.01}{22 \times 10^{-6}} \]
Calculating the right side:
\[ T - 20 = \frac{0.01}{22 \times 10^{-6}} = \frac{0.01}{0.000022} \approx 454.545 \]
Now, add 20 to both sides:
\[ T = 454.545 + 20 \approx 474.545 \, \text{°C} \]

### Step 7: Final answer
Thus, the temperature to which the iron tire must be heated is approximately:
\[ T \approx 474.5 \, \text{°C} \]

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