Question

The pressure applied from all directions on a cube is $P$. How much its temperature should be raised to maintain the original volume?
(The volume elasticity of the cube is $\beta$ and the coefficient of volume expansion is $\alpha$)

• A. $\frac{P}{\alpha \beta }$
• B. $\frac{P\alpha }{\beta }$
• C. $\frac{P\beta }{\alpha }$
• D. $\frac{\alpha \beta }{P}$

Answer 1

The correct option is A $\frac{P}{\alpha \beta }$

The volume explansion due to rise in temperature by $\Delta \theta$ is,
$\Delta V=V\alpha \Delta \theta$
$\Rightarrow \frac{\Delta V}{V}=\alpha \Delta \theta$
Volume elasticity of cube is equivalent to the bulk modules
$\Rightarrow$ Since, pressure $P$ has been applied from all directions, it will lead to reduction in volume. If the volume of cube has to be maintained constant, reduction in volume must be compensated by increase in volume due to temperature rise.
$\beta =-\frac{P}{\left(\frac{\Delta V}{V}\right)}$
$\Rightarrow \left|\beta \right|=\frac{P}{\left|\frac{\Delta V}{V}\right|}$
$\Rightarrow \beta =\frac{P}{\alpha \Delta \theta }$
$\therefore \Delta \theta =\frac{P}{\alpha \beta }$