Question

If $\alpha$ represent linear expansion, $\beta$ represent superficial expansion and $\gamma$ represent cubical expansion, then choose the correct statement :

• A. $\alpha :\beta :\gamma ::1:3:2$
• B. $\alpha :\beta :\gamma ::2:3:1$
• C. $\alpha :\beta :\gamma ::3:1:1$
• D. $\alpha :\beta :\gamma ::1:2:3$

Answer 1

Answer: (D)

$\alpha :\beta :\gamma ::1:2:3$

Whenever there is an increase in the dimensions of a body due to heating, then the body is said to be expanded and the phenomenon is known as expansion. Solids undergo three types of expansions:

a) Linear expansion $\alpha$

b) Superficial expansion $\beta$

c) Cubical expansion $\gamma$

Relation between $\alpha$, $\beta$ and $\gamma$- From definitions of $\alpha$, $\beta$ and $\gamma$, it is clear that increase in length, area and volume with rise in temperature $△T$ will be respectively.

$△L=L\alpha △T△A=A\beta △T△V=V\gamma △T$

and so final length $L^{\prime }$, area $A^{\prime }$ and volume $V^{\prime }$ will be

$L^{\prime }=L(1+\alpha △T)A^{\prime }=A(1+\beta △T)V^{\prime }=V(1+\gamma △T)...........\left(1\right)$

i.e. change of final value length, area or volume depends on its initial value.

Change in temperature and nature of material

As the area is 2D while volume is 3D so

$\frac{A^{\prime }}{A}={\left(\frac{L^{\prime }}{L}\right)}^2$

and $\frac{V^{\prime }}{V}={\left(\frac{L^{\prime }}{L}\right)}^3$

but $L^{\prime }=L(1+\alpha △T)$

i.e. $\frac{L^{\prime }}{L}=1+\alpha △T$

Similarly, $\frac{A^{\prime }}{A}={(1+\alpha △T)}^2$

and $\frac{V^{\prime }}{V}={(1+\alpha △T)}^3$

or $A^{\prime }=A(1+2\alpha △T)V^{\prime }=V(1+3\alpha △T)$

Comparing these equations for $A^{\prime }$ and $V^{\prime }$ with that of equation $\left(1\right)$

We find $\beta =2\alpha$ and $\gamma =3\alpha$

i.e. $\frac{\alpha }{1}=\frac{\beta }{2}=\frac{\gamma }{3}\alpha :\beta :\gamma =1:2:3$