Will the hole on a metal disc expand or contract upon heating? 
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Will a hole cut into a metal disk expand or shrink when the disc is heated? 

The metal disc of diameter D1 has a hole in it and the diameter of the hole is D2 (D2

Now, if I heat the metal disc, it expands. As far as I know, the metal expands along its free ends upon heating. In this case, the disc has got the scope to expand along both the inner and outer circumferences of the disc.
The question here is, in which way does the metal disc really expand and as a result, will the  hole on the disc expand or contract. To put in other words, assuming the new diameter of the hole on the disc to be D3, will D3 be greater than,  lesser than or equal to D2?
 A: It expands. Take a normal disc, and draw a circle on it. The circle expands. Take a new disc, draw a hole, and make a score on that circle. It still expands. Repeat with deeper scoring. It still expands. Keep making it deeper. It still expands. At one point, your razor(or whatever) will start poking through the metal. It still expands. Cut out the circle, but leave the cut out piece in the hole. Now, doesn't it seem logical that the circle+hole will expand? Remove the circle, it still expands.
So, to answer your question, a hole in a material behaves just like a circle of that same material. It expands on heating. 
What's actually happening is that if it tries to expand inwards (contract basically), it will have to compress itself, and increase its density. There will be resistance to it, so it tries to expand into a free region, namely the outside. And the inside expands just because it does when there is no hole.
A: Imagine that the hole is filled with another disk of exactly the same diameter as the hole (initially $D_2$) and made from the same material. Once both discs heat up, they will expand as if they were a single solid object. Since the small disk expands, we conclude that the hole in the large disc must expand too, $D_3>D_2$. 
