Question

An iron tyre is to be fitted onto a wooden wheel $100\, cm$ in diameter. The diameter of the tyre is $5\, mm$ smaller than that of the wheel. How much should the temperature (in ${ }^{\circ} C$ ) of the tyre be increased for this purpose ? $\alpha$ for iron $=12 \times 10^{-6}$ per $^{\circ} C$

Answer 1

To fit the tyre, its diameter has to be increased from $99.5$ to $100$.
So let temperature to which tyre must be heated be $T$.
$\ell=\ell_{0}[1+\alpha \Delta \theta]$
$100=99.5\left[1+12 \times 10^{-6} T \right]$
$1.0050251=1+12 \times 10^{-6} T$
$T =\frac{0.0050251}{12} \times 10^{6}=418.8^{\circ} C$