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An iron tyre is to be fitted on to a wooden wheel 1m in diameter. The diameter of tyre is $6\, mm$ smaller than that of wheel. The tyre should be heated so that its temperature increases by a minimum of ........ $^oC$ (the coefficient of cubical expansion of iron is ${3.6 \times 10^{-5} } °C^{-1}$)
$167$
$334$
$500$
$1000$
Solution
(c) Initial diameter of tyre $= (1000 -6) \,mm = 994\,mm,$ so initial radius of tyre $R = \frac{{994}}{2} = 497\,mm$
and change in diameter $\Delta$$D = 6\, mm$ so $\Delta R = \frac{6}{2} = 3\,mm$
After increasing temperature by $\Delta \theta$ tyre will fit onto wheel
Increment in the length (circumference) of the iron tyre
$\Delta L = L \times \alpha \Delta \theta$ $ = L \times \frac{\gamma }{3} \times \Delta \theta $ [As $\alpha = \frac{\gamma }{3}]$
$2\pi \,\Delta R = 2\pi \,R\,\left( {\frac{\gamma }{3}} \right)\,\Delta \theta $$⇒$$\Delta \theta = \frac{3}{\gamma }\frac{{\Delta R}}{R} = \frac{{3\, \times 3}}{{3.6 \times {{10}^{ – 5}} \times 497}}$
$⇒$$\Delta \theta = {500^o}C$
