alpha:beta:gamma=1:2:3

Asked by neha khan | 10th Oct, 2013, 10:47: PM

Expert Answer:

The linear expansion at temperature t is given as
 
Lt = L0(1 + ?t)
 
Here, ? is the co-efficient of linear expansion.
 
Now, for an isotropic expansion, expansion increase in area, that is surface expansion is given as length × length
 
Thus, A ? A (1 + ?t)(1 + ?t) ? A (1 + 2?t)                  ...... (1)
Expansion of (1 + ?t)2 = 1 + 2?t + (?t)2. Since, (?t)2 is very small, it can be ignored.
 
We know that At = A0 (1 + ?t)                                 ...... (2)
Where ? is the co-efficient of superficial expansion.
 
Thus, on comparing (1) and (2), we get
 
? = 2?
 
Similarly, 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.
 
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

Answered by Romal Bhansali | 11th Oct, 2013, 09:42: AM

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