A horizontal tube, open at both ends, contains a column of liquid. The length of this liquid column does not change with temperature. Let `gamma` : coefficient of volume expansion of the liquid and `alpha` : coefficient of linear expansion of the material of the tube

Options

A. `gamma= alpha`
B. `gamma = 2 alpha`
C. `gamma = 3 alpha`
D. `gamma= 0`

Answer

B

Explanation

Let `A_(0) and A_(T)` be the areas of cross-section of the tube at temperature `0^(@)C and T^(@)C` respectively.
L = length of the liquid column (constant)
Let `V_(0) and V_(T)` be the volumes of the liquid at temperature `0^(@)C and T^(@)C` respectively.
`:. V_(0) = LA_(0) and V_(T) = LA_(T)`
Also, `V_(T)= V_(0) (1 + gamm T) and A_(T) = A_(0) (1 + 2alpha T)`
`:. V_(T) = LA_(0) (1 + 2alpha T)` ...(i)
Also `V_(T) = V_(0) (1 + gamma T) = LA_(0) (1 + gammaT)` ...(ii)
Comparing (i) and (ii), we get
or `gamma = 2alpha`