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 that
At temperature T 1 = 27 o C diameter of outer shaft D 1 = 8.70 cm
At temperature T 2 , the diameter of the shaft changes to D 2 = 8.69 cm
Coefficient of linear expansion, α = 1.2 × 10 − 5 K -1
Change in diameter is given by relation,
Δ D = D 1 α Δ T D 2 − D 1 = D 1 α Δ T
Substitute the values.
8.69 − 8.70 = 8.70 × 1.2 × 10 − 5 × ( T 2 − 27 ) ( T 2 − 27 ) = − 95.78 T 2 = − 68 .8 o C ≃ − 69 o C
Thus, the wheel can slip on the shaft when the shaft is cooled to a temperature of − 69 ° C .
Given that
At temperature
At temperature
Coefficient of linear expansion,
Change in diameter is given by relation,
Substitute the values.
Thus, the wheel can slip on the shaft when the shaft is cooled to a temperature of
