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An aluminum plate fixed in a horizontal ...

An aluminum plate fixed in a horizontal position has a hole of diameter 2.000 cm. A steel sphere of diameter 2.005 cm rests on this hole. All the lengths refer to a temperature of `10^@C`.The temperature of the entire system is slowly increased. At what temperature will the ball fall down? coefficient of linear expansion of aluminum is `23 xx 10^(-6)C^(-1)` and that of steel is `11 xx 10^(-6)C^(-1)`.

Text Solution

Verified by Experts

As value of coefficient of linear expansion for aluminium is more than that for iron, it expands faster than iron. So at some higher temperature when diameter of hole will exactly become equal to that of iron sphere, the sphere will pass through the hole. Let it happen at some higher temperature T. Thus we have at this temperature T,
`("diameter of hole")_(Al)=("diameter of sphere")_("iron")`
`1[1+alpha_(Al)(T-25)]=1.004[1+alpha_("iron")(T-25)]`
`alpha_(Al)(T-25)=0.004+1.004alpha_("iron")(T-25)`
or `T=(0.004/(alpha_(Al)-1.004alpha_("iron"))+25)""^(@)C`
or `T=(0.004)/(1xx2.4xx10^(-5)-1.004xx1.2xx10^(-5))+25`
or `T=359.7^(@)C`
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