Question

A cube is made of a material of coefficient of linear expansion α and bulk modulus β . Pressure P is applied on the cube from all the direction. The rise in temperature which the cube should be subjected in order to maintain original volume is

(1) P/3αβ

(2) 3Pα/β

(3) 3αβ/P

(4) Pβ/3α

Answer 1

(1) P/3αβ

The decrease in volume due to volumetric strain of pressure P should be equal to increase in volume due to thermal expansion. We have

β = The Bulk modulus = p/volumetric strain = P/v/V = PV/v

∴ v = PV/β  ----(1)

v is decrease in volume

The volume V' when temperature is raised to t°C = V + γ Vt = V + 3α Vt  [∵ γ = 3α]

Increase in volume = v = V'–V = 3α Vt  .....(2)

From (1) and (2) we have