Question

An iron tyre is to be fitted onto a wooden wheel $1.0$ metre in diameter. The diameter of the tyre is $6 \,mm$, smaller than that of the wheel. The tyre should be heated so that its temperature increases by a minimum of (given coefficient of volume expansion of iron is $3.6 \times 10^{-5} /{ }^{\circ} C$ ) (in ${ }^{\circ} C$ )

Answer 1

Diameter of wheel $=1\, m =\ell$
Diameter of tyre $=\left(1-6 \times 10^{-3}\right)=\ell_{0}$
Let the increment of temperature required be $\Delta T$, so
$\ell=\ell_{0}(1+\alpha \Delta T )$
$1=\left(1-6 \times 10^{-}\right)\left(1+\frac{3.6 \times 10^{-5}}{3} \Delta T \right)$
$\frac{1}{0.994}=1+1.2 \times 10^{-5} \Delta T$
$\frac{1}{0.994}-1=1.2 \times 10^{-5} \Delta T$
$\frac{0.006}{0.994 \times 1.2 \times 10^{-5}}=\Delta T$
$\Delta T =\frac{6 \times 10^{5}}{12 \times 0.994} \times \frac{10}{1000}$
$=500^{\circ} C$