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 coefficient of linear expansion of the steel to be constant over the required temperature range: α_steel = 1.20 × 10^–5 K^–1.

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, α_steel = 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 = D₁α_steel(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.