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

To solve the problem of fitting a large steel wheel onto a shaft, we need to determine the temperature at which the outer diameter of the shaft equals the inner diameter of the wheel. Here’s a step-by-step solution:

### Step 1: Understand the Given Data
- Outer diameter of the shaft at \(27^\circ C\) = \(8.70 \, \text{cm}\)
- Inner diameter of the wheel = \(8.69 \, \text{cm}\)
- Coefficient of linear expansion of steel, \(\alpha = 1.20 \times 10^{-5} \, \text{K}^{-1}\)
- Initial temperature, \(T_i = 27^\circ C = 300 \, \text{K}\)

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