Q7 of 26 Page 294

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: αsteel = 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, 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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A brass wire 1.8 m long at 27 °C is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of –39 °C, what is the tension developed in the wire, if its diameter is 2.0 mm? Co-efficient of linear expansion of brass = 2.0 × 10–5 K–1; Young’s modulus of brass = 0.91 × 1011 Pa.