Question

An anisotropic material has coefficient of linear thermal expansion α₁, α₂ and α₃ along x, y and z-axis respectively. Coefficient of cubical expansion of its material will be equal to ______.

विकल्प

• α₁ + α₂ + α₃

• α₁ + 2α₂ + 3α₃

• 3α₁ + 2α₂ + α₃

• `(alpha_1 + alpha_2 + alpha_3)/3`

Answer 1

An anisotropic material has coefficient of linear thermal expansion α₁, α₂ and α₃ along x, y and z-axis respectively. Coefficient of cubical expansion of its material will be equal to α₁ + α₂ + α₃.

Explanation:

V = xyz

V' = (x + Δx) (y + Δy) (z + Δz)

Neglecting terms

V' = xyz + xy Δz + (Δx) yz + xΔyz

`V_2^' = V + (VDeltaz)/z + (VDeltax)/x + (VDeltay)/y`

`V_2^'` = V[1 + (α₁ + α₂ + α₃)ΔT]

`("We know that"  alpha xx DeltaT = (Deltaℓ)/ℓ)`

= V(1 + α_eqΔT)

⇒ α_eq = (α₁ + α₂ + α₃)