A well-known technique for achieving a very tight fit between two components is to "expand by heating and then cool to shrink fit." For example,an aluminum ring of inner radius 5.98 cm needs to be firmly bonded to a cylindrical shaft of radius 6.00 cm (Measurements are at 20 $\circ$ C).Calculate the minimum temperature to which the aluminum ring needs to be heated before it can be slipped over the shaft for fitting. (Coefficient of linear expansion for Al = 24 $\times$ 10^-6/K.) A)140 $\circ$ C B)850 $\circ$ C C)120 $\circ$ C D)160 $\circ$ C E)180 $\circ$ C

Question

Quizplus · Accepted Answer

The minimum temperature can be calculated using the formula for linear expansion, $\Delta L = L_0 \alpha \Delta T$, where $\Delta L$ is the change in length, $L_0$ is the original length, $\alpha$ is the coefficient of linear expansion, and $\Delta T$ is the change in temperature. Here, the change in radius needed is $6.00\, \text{cm} - 5.98\, \text{cm} = 0.02\, \text{cm}$. The original radius is $5.98\, \text{cm}$, and the coefficient of linear expansion for aluminum is $24 \times 10^{-6}/K$. Solving for $\Delta T$, we find that the minimum temperature increase needed is approximately $160^\circ C$.

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