An iron tyre is to be fitted onto a wooden wheel 1.0 m in diameter. The diameter of the tyre is 6 mm smaller than that of wheel. The tyre should be heated so that its temperature increases by a minimum of

To fit the iron tyre onto the wooden wheel, the tyre must be heated so that it expands to a diameter at least equal to the diameter of the wooden wheel.

1. Identify the given values:
Diameter of the wooden wheel, $D_w=1.0\text{m}$
Initial diameter of the iron tyre is 6 mm smaller than the wheel, so:
$D_t=1.0\text{m}-6\text{mm}=0.994\text{m}$
The required increase in diameter (expansion) is:
$\Delta D=6\text{mm}=6\times {10}^{-3}\text{m}$

2. Use the coefficient of linear expansion:
The coefficient of linear expansion for iron is:
$\alpha =1.2\times {10}^{-5}{\text{°C}}^{-1}$

3. Set up the linear thermal expansion formula:
The change in diameter due to temperature increase is given by:
$\Delta D=D_t\cdot \alpha \cdot \Delta T$

Rearranging the equation to solve for the temperature increase $\Delta T$:
$\Delta T=\frac{\Delta D}{D_t\cdot \alpha }$

4. Calculate the temperature change:
Approximating the initial tyre diameter $D_t\approx 1.0\text{m}$ for the calculation:
$\Delta T=\frac{6\times {10}^{-3}\text{m}}{1.0\text{m}\times 1.2\times {10}^{-5}{\text{°C}}^{-1}}$

Simplifying the division:
$\Delta T=\frac{6\times {10}^{-3}}{1.2\times {10}^{-5}}=500\text{°C}$

Therefore, the minimum temperature increase required to fit the iron tyre onto the wooden wheel is 500°C.