Question

A wire of length 1 cm and a cube of side 1 cm of the same metal are heated through 1${}^o$C. If the change in length of the wire is x cm then the change in volume of the cube is :

• A. x${}^3$ cm${}^3$
• B. 3x cm${}^3$
• C. x cm${}^3$
• D. indeterminate as data are insufficient

Answer 1

The correct option is B 3x cm${}^3$

Given : Wire and cube are made of same metal.

length of wire $l=1cm,$

change in length of wire $\Delta l=xcm$

side of cube $l^{\prime }=1cm$ ,

increase in temperature $\Delta t=1^{o}C$

The coefficient of linear expansion is given by:

$\alpha =\frac{\Delta l}{l\times \Delta t}$

$\alpha =\frac{x}{1\times 1}=x$

Now, volume of cube is $V=1^3=1c{m}^3$

The coefficient of volume expansion is given by:

$\gamma =\frac{\Delta V}{V\times \Delta t}$

$\gamma =\frac{\Delta V}{1\times 1}=\Delta V$ (change in volume),

But we have,

$\gamma =3\alpha$

$\Delta V=3xc{m}^3$