A large steel wheel is to be fitted on to a shaft of the same material. At 27 degree 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 coefficient of linear expansion of the steel to the constant over the required temperature range: αsteel = 1.20 x 10^-5 K^-1

Given

T = 270C

This can be written in Kelvin as,

27 + 273 = 300 K

Outer diameter of the steel shaft at T, d1 = 8.70 cm

Diameter of the central hole in the wheel at T, d2 = 8.69 cm

Coefficient of linear expansion of steel, αsteel = 1.20 x 10-5 K-1

After the shaft is cooled using dry ice its temperature becomes T1

The wheel will slip on that shaft, if the change in diameter,

Δd = 8.69 – 8.70 = -0.01 cm

Temperature T1 can be calculated from the relation

Δd = d1αsteel(T1 – T)

On substituting, we get,

-0.01 = 8.70 x 1.20 x 10-5 (T1 – 300)

(T1 – 300) = -95.78

Hence,

T1 = 204.21 K

= 204.21 – 273.16

We get,

= -68.95 degree C

Hence, the wheel will slip on the shaft when the temperature of the shaft is -69 degree C

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