A large steel wheel is to be fitted on to a ...

Thermal Properties of Matter

Question : 7 of 22

Marks: +1, -0

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 :

\alpha_{\text{steel}} = 1.20 \times 10^{-5} \text{K}^{-1}.

Solution:

Here,

T_1 = 27^{\circ}\text{C} = 27 + 273 = 300 \text{K}

Let

L_1

and

L_2

be the linear dimensions of steel at temperatures

T_1

and

T_2

respectively.

Now

L_1 = 8.70 \text{ cm}, L_2 = 8.69 \text{ cm}

\alpha_{\text{steel}} = 1.20 \times 10^{-5} \text{K}^{-1}

Change in length

= L_2 - L_1 = L_1 \alpha (T_2 - T_1)

8.69 - 8.70 = 8.70 \times (1.20 \times 10^{-5}) \times (T_2 - 300)

(T_2 - 300) = \frac{0.01}{8.70 \times 1.2 \times 10^{-5}} = -95.78 \text{ K}

T_2 = 300 - 95.78 \text{ K} = 204.22 \text{ K} = -68.93^{\circ}\text{C}

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