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 `8.70cm` and the diameter of the central hole in the wheel is `8.69cm` . 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: `alpha_(steel) = 1.20xx10^(-5)K^(-1)`.

Explanation

Diameter of shaft should be equal to the diameter of the hole of wheel to fit into it.
`d_1 = 8.70 cm , T_1 = 27^@C = 300K`
` d_2 = 8.69 cm , T_2 = ? `
As ` Deltad = alpha d_1 Delta T`
`d_2 - d_1 = alpha d_1 (T_2 - T_1)`
` 8.69 - 8.70 = (1.20 xx 10^(-5) ) xx 8.70 (T_2 - 300)`
`T_2 - 300 = - 95.78 K`
` rArr T_2 = 204.2 K " or " - 68.8^@C`