Properties Of Solids And Liquids Question 398
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}} , ^\circ\text{C}^{-1}$)
Options:
A) $ 167{}^\circ C $
B) $ 334{}^\circ C $
C) $ 500{}^\circ C $
D) $ 1000{}^\circ C $
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Answer:
Correct Answer: C
Solution:
[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} = 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 \times \Delta \theta =L\times \frac{\gamma }{3}\times \Delta \theta [ As\alpha =\frac{\gamma }{3} ] $
$ 2\pi \Delta R=2\pi R( \frac{\gamma }{3} )\Delta \theta \Rightarrow \Delta \theta = \frac{3\Delta R}{R\gamma} $
$ =\frac{3}{\gamma }\frac{\Delta R}{R}=\frac{3\times 3}{3.6\times {{10}^{-5}}\times 497}\Rightarrow \Delta \theta \simeq {{500}^{\circ}}C $
BETA
