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. 167⁰C
• B. 334⁰C
• C. 500⁰C
• D. 1000⁰C

Answer 1

Answer: (C)

500⁰C

Answer 2

Initial diameter of tyre = (1000 – 6) mm = 994 mm, so initial radius of tyre $R = \frac{994}{2} = 497mm$

and change in diameter ∆D = 6 mm so $\Delta R = \frac{6}{2} = 3mm$

After increasing temperature by ∆T tyre will fit onto wheel

Increment in the length (circumference) of the iron tyre

∆L = L × α × ∆T $= L \times \frac{\gamma}{3} \times \Delta T$ [As $\alpha = \frac{\gamma}{3}\rbrack$

⇒$2\pi\Delta R = 2\pi R\left( \frac{\gamma}{3} \right)\Delta T$

⇒ $\Delta T = \frac{3}{\gamma}\frac{\Delta R}{R} = \frac{3 \times 3}{3.6 \times 10^{- 5} \times 497}$

[As ∆R = 3 mm and R = 497 mm]

⇒ $\Delta T = 500^{o}C$