Question

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

• A. ${167}^{\circ }C$
• B. ${334}^{\circ }C$
• C. ${500}^{\circ }C$
• D. ${1000}^{\circ }C$

Answer 1

The correct option is C ${500}^{\circ }C$

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