Question

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 (the coefficient of cubical expansion of iron is $3.6\times {10}^{-5}/{}^{o}C$)

• A. 167ºC
• B. 334ºC
• C. 500ºC
• D. 1000ºC

Answer 1

The correct option is C

500ºC

Initial diameter of tyre = (1000-6)mm =994 mm, so initial radius of typre $R=\frac{994}{2}=497mm$ and change in diameter $△D=6mmso△R=\frac{6}{2}=3mm$
After increasing temperature by $△\theta$ tyre will fit onto wheel
Increment in the length (circumference) of the iron tyre
$△L=L\times \alpha \times △\theta =L\times \frac{\gamma }{3}\times △\theta [As\alpha =\frac{\gamma }{3}]$
$2\pi △R=2\pi R(\frac{\gamma }{3})△\theta \Rightarrow △\theta =\frac{3}{\gamma }\frac{△R}{R}=\frac{3\times 3}{3.6\times {10}^{-5}\times 497}$
$\Rightarrow △\theta ={500}^{o}C$