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Question

A large steel wheel is to be fitted onto 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 the coefficient of linear expansion of the steel to the constant over the required temperature range. αsteel=1.20×10-5K-1.


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Solution

Step-1: Given data

T1=27°C

Diameter of shaft l1=8.70cm

Diameter of central hole l2=8.69cm

The coefficient of linear expansion of the steel is αsteel=1.20×10-5K-1.

Step 2: Determine the change in diameter when the wheel slips on the shaft

If the shaft does wheel slip on the shaft, then change in diameter, l=l2-l1.

l=8.69-8.70cml=-0.01cm

Step 3: Determine the temperature at which the wheel slips on the shaft

Let the final temperature be T2.

Thus, the change in diameter can be given by l=l1αT2-T1.

-0.01=8.70×1.20×10-5T2-300-0.018.70×1.20×10-5=T2-300T2-300=-95.79T2=300-95.79T2=204.21

Thus, the final temperature is T2=204.21K.

T2=204.21-273°CT2=-68.79°C

Hence, the temperature of the shaft at which the wheel slip on the shaft is -68.79°C.


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