A large steel wheel is to befitted 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 αsteel= 1-20 x 10-5K-1.

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Now, let's tackle the problem. We have a steel wheel to be fitted onto a steel shaft. The key here is to understand that the wheel and the shaft will expand or contract with changes in temperature, and we need to find the temperature at which the wheel slips on the shaft.

We know that the outer diameter of the shaft at 27°C is 8.70 cm, and the diameter of the central hole in the wheel is 8.69 cm. This implies that at 27°C, the shaft and the wheel are in a snug fit.

Given that we're cooling the shaft using dry ice, which is at a temperature of approximately -78.5°C, we need to find out at what temperature the diameter of the shaft contracts enough for the wheel to slip.

To do this, we'll use the formula for linear expansion:

ΔL=αLΔTΔL=αLΔT

Where:

• ΔL is the change in length (or diameter in this case)
• α is the coefficient of linear expansion
• L is the original length (or diameter)
• ΔT is the change in temperature

We'll start by finding the change in temperature required for the shaft to contract enough to let the wheel slip.

ΔT=ΔLαLΔT=αLΔL

Given:

• Initial diameter of the shaft (at 27°C), L0=8.70 cmL0=8.70cm
• Final diameter (when the wheel slips), Lf=8.69 cmLf=8.69cm
• Coefficient of linear expansion for steel, αsteel=20×10−6 K−1αsteel=20×10−6K−1

ΔL=Lf−L0=8.69 cm−8.70 cm=−0.01 cmΔL=Lf−L0=8.69cm−8.70cm=−0.01cm

ΔT=−0.01 cm20×10−6 K−1×8.70 cmΔT=20×10−6K−1×8.70cm−0.01cm

ΔT≈−5.75 KΔT≈−5.75K

So, the temperature at which the wheel slips on the shaft would be approximately 27∘C−5.75∘C27∘C−5.75∘C, which is approximately 21.25∘C21.25∘C.

Therefore, at around 21.25∘C21.25∘C, the wheel will start slipping on the shaft due to contraction caused by cooling.