Question

An iron tyre is to be fitted onto a wooden wheel $1.0$ metre in diameter. The diameter of the tyre is $6mm$, 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

Answer: 500.00

Diameter of wheel $=1m=\ell$
Diameter of tyre $=\left(1-6\times 10^{-3}\right)={\ell }_0$
Let the increment of temperature required be $\Delta T$, so
$\ell ={\ell }_0(1+\alpha \Delta T)$
$1=\left(1-6\times 10^{-}\right)\left(1+\frac{3.6\times 10^{-5}}{3}\Delta T\right)$
$\frac{1}{0.994}=1+1.2\times 10^{-5}\Delta T$
$\frac{1}{0.994}-1=1.2\times 10^{-5}\Delta T$
$\frac{0.006}{0.994\times 1.2\times 10^{-5}}=\Delta T$
$\Delta T=\frac{6\times 10^5}{12\times 0.994}\times \frac{10}{1000}$
$=500^{\circ }C$