Answer :

Given,

Initial temperature, T1 = 27°C = 27 + 273.15 = 300.15 K


Outer diameter of the steel shaft at T1, D1 = 8.70 cm


Diameter of the central hole in the wheel at T1, d1 = 8.69 cm


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


Let the temperature be T2 after the shaft is cooled using ‘dry ice’.


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


Δd = 8.69 – 8.70 = – 0.01 cm


Temperature T2 can be calculated as follows.


Δd = D1αsteel(T2 – T1)


-0.01 cm = 8.70 cm × 1.20 × 10-5 K-1 × (T2 – 300.15 K)


(T2 – 300.15 K) = -95.78 K


T2 = 204.37 K = 204.37 – 273.15 = -68.78°C


Therefore, the wheel will slip on the shaft when the temperature of the shaft is –68.78°C.


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