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alpha:beeta:gamma=1:2:3
Asked by neha khan | 12 Oct, 2013, 08:57: PM Expert Answer
The linear expansion at temperature t is given as Lt = L0(1 + ?t)
Here, ? is the co-efficient of linear expansion.

We know that area is length multiplied by length.
Thus, expansion increase in area, that is, surface expansion is given as length × length.

Thus, area A will expand to ? A (1 + ?t)(1 + ?t) ? A (1 + 2?t)                  ...... (1)

Note: Expansion of (1 + ?t)2 = 1 + 2?t + (?t)2.
Since, (?t)2 is very small, it can be ignored.

Now, we know that expansion in area is given as At = A0 (1 + ?t)                ...... (2)
Where, ? is the co-efficient of superficial expansion.

Thus, on comparing (1) and (2), we get   ? = 2?

Similarly, we know that volume is (length)3.

Thus, volume increases as  V ? V (1 + ?t)(1 + ?t)(1 + ?t) ? A (1 + 3?t)                  ...... (3)

Expansion of (1 + ?t)3 = 1 + 3?t, ignoring higher powers of ?t.  In higher mathematics there is a formula for binomial expansion where in we can say that (1 + ?t)3 = 1 + 3?t. At this level, you can directly assume it to be true.

We know that Vt = V0 (1 + ?t)                                    ...... (4)

Where ? is the co-efficient of cubical expansion.

Thus, on comparing (3) and (4), we get   ? = 3?

Thus, we have ? : ? : ? = 1 : 2 : 3

Note: You need to consider area and volume expansion only. It is then a simple mathematical expansion.
Answered by Romal Bhansali | 13 Oct, 2013, 11:36: AM