A large steel wheel is to be fitted on to a shaft of the same material. At $27\,^{\circ} C ,$ the outer diameter of the shaft is $8.70\; cm$ and the diameter of the centrall hole in the wheel is $8.69 \;cm$. The shaft is cooled using 'dry ice'. At what temperature (in $^oC$) of the shaft does the wheel slip on the shaft? Assume coefficient of linear expansion of the steel to be constant over the required temperature range: $\alpha_{steel} =1.20 \times 10^{-3} \;K ^{-1}$

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The given temperature, $T=27^{\circ} C$ can be written in Kelvin as

$27+273=300 K$

Outer diameter of the steel shaft at $T, d_{1}=8.70 cm$

Diameter of the central hole in the wheel at $T, d_{2}=8.69 cm$

Coefficient of linear expansion of steel, $\alpha$ steel $=1.20 \times 10^{-5} K ^{-1}$

After the shaft is cooled using "dry ice', its temperature becomes $T_{1}$.

The wheel will slip on the shaft, if the change in diameter, $\Delta d=8.69-8.70$

$=-0.01 cm$

Temperature $T_{1},$ can be calculated from the relation:

$\Delta d=d_{1} \alpha_{\text {steel }}\left(T_{1}-T\right)$

$=8.70 \times 1.20 \times 10^{-5}\left(T_{1}-300\right)$

$\left(T_{1}-300\right)=95.78$

$\therefore T_{1}=204.21 K$

$=204.21-273.16$

$=-68.95^{\circ} C$

Therefore, the wheel will slip on the shaft when the temperature of the shaft is $-69\,^{\circ} C$

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