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 × ${10}^{-5}{/}^{\circ }C$ )

• 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 tyre R = $\frac{994}{2}=497mm$

and change in diameter $△D=6mm$ so

$△R=\frac{6}{2}=3mm$

After increasing temperature by $△\theta$ tyre will fit onto wheel

Increment in the lenght ( circumference ) of the iron tyre

$△L=L\times \alpha \times △\theta =L\times \frac{\gamma }{3}\times △\theta$ [ As $\alpha =\frac{\gamma }{3}]$

$2\pi △R=2\pi R\left(\frac{\gamma }{3}\right)△\theta \Rightarrow △\theta =\frac{3}{\gamma }\frac{△R}{R}=\frac{3\times 3}{3.6\times {10}^{-5}\times 497}$

$\Rightarrow △\theta ={500}^{\circ }C$