Ncert Exercises & Solutions

11.7 A large steel wheel is to be fitted on to a shaft of the same material. At 27 °C, the outer diameter of the shaft is 8.70 cm and the diameter of the central hole in the wheel is 8.69 cm. The shaft is cooled using ‘dry ice’. At what temperature of the shaft does the wheel slip on the shaft? Assume the coefficient of linear expansion of the steel to be constant over the required temperature range :

$α_{steel} = 1.20 \times 10^{- 5} K^{- 1} .$

T = 27 + 273 = 300 K

The outer diameter of the steel shaft at T, d $_{1}$ = 8.70 cm

The diameter of the central hole in the wheel at T, d $_{2}$ = 8.69 cm

Coefficient of linear expansion of steel, $α_{steel}$ = 1.20 × $10^{- 5}$ 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, Δd = 8.69 – 8.70 = – 0.01 cm

Then,

Δd = $d_{1}$ $α_{steel}$ (T $_{1}$ – T)= 8.70 × 1.20 × $10^{- 5}$ (T $_{1}$ – 300)

(T $_{1}$ – 300) = 95.78

T $_{1}$ = 204.21 K = 204.21 – 273.16 = –68.95°C

Therefore, the wheel will slip on the shaft when the temperature of the shaft is –69 °C.

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