An iron tyre is to be fitted onto a wooden wh | vedclass

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(C) Initial diameter of the wheel $D_w = 1000 \, mm$.Initial diameter of the tyre $D_t = 1000 - 6 = 994 \, mm$.Initial radius of the tyre $R = 497 \,

Vedclass · Accepted Answer

$500$

Vedclass · Answer

(C) Initial diameter of the wheel $D_w = 1000 \, mm$.Initial diameter of the tyre $D_t = 1000 - 6 = 994 \, mm$.Initial radius of the tyre $R = 497 \, mm$.Change in radius required $\Delta R = 3 \, mm$.The coefficient of linear expansion $\alpha$ is related to the coefficient of cubical expansion $\gamma$ by $\alpha = \frac{\gamma}{3}$.Given $\gamma = 3.6 	imes 10^{-5} \, ^oC^{-1}$,so $\alpha = \frac{3.6 	imes 10^{-5}}{3} = 1.2 	imes 10^{-5} \, ^oC^{-1}$.The change in circumference $\Delta L$ is given by $\Delta L = 2\pi \Delta R = L \alpha \Delta 	heta = (2\pi R) \alpha \Delta 	heta$.Thus,$\Delta R = R \alpha \Delta 	heta$.$\Delta 	heta = \frac{\Delta R}{R \alpha} = \frac{3}{497 	imes 1.2 	imes 10^{-5}} \approx \frac{3}{0.005964} \approx 503 \, ^oC$.Rounding to the nearest provided option,the required temperature increase is $500 \, ^oC$.

$S$.No.	$l (m)$	$\Delta T (^{\circ}C)$	$\Delta l (m)$
$(1)$	$2$	$10$	$4 \times 10^{-4}$
$(2)$	$1$	$10$	$4 \times 10^{-4}$
$(3)$	$2$	$20$	$2 \times 10^{-4}$
$(4)$	$3$	$10$	$6 \times 10^{-4}$

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