Problems on Linear, Area, and Cubic Expansivity

Note: ( β = 2α , γ = 3α)

Remember

Area Expansivity formula: A₂ = A₁ +  β A₁( θ₂ – θ₁)

Cubic Expansivity formula: \( \scriptsize V_2 = V_1 \: + \: \gamma V_1(\theta_2 \: – \: \theta_1)\)

Therefore, Area and cubical expansivity can also be written as 

A₂ = A₁ + A₁2α ( θ₂ –  θ₁)

or

A₂ = A₁(1 + 2α( θ₂ –  θ₁))

V₂ = V₁ + V₁3α ( θ₂ –  θ₁)

or

V₂ = V₁(1 + 3α( θ₂ –  θ₁))

Example 10.6.1:

If the cubic expansivity of brass between 27°C and 1000°C is 5.7 × 10^-5 K^-1, what is its linear expansivity?

Solution

Cubic expansivity, γ = 5.7 × 10^-5 K^-1

linear expansivity, α = ?

Using the formula, γ = 3α

Therefore, 5.7 × 10^-5 = 3α

\( \scriptsize \alpha = \normalsize \frac{5.7 \: \times \: 10^{-5}}{3} \)

\( \scriptsize \alpha = 1.95 \: \times \: 10^{-5}\)

Example 10.6.2:

A blacksmith heated a metal whose cubic expansivity is 6.3 × 10^-6 K^-1. What is the area expansivity?

In the question we are given the value for cubic expansivity. We can use this value to get the linear expansivity and then use the value of linear expansivity to calculate the area expansivity.

\( \scriptsize \gamma = 6.3 \: \times \: 10^{-6}\:K^{-1}\)

\( \scriptsize \gamma = 3 \alpha\)

\( \scriptsize 6.3 \: \times \: 10^{-6}\:K^{-1} = 3 \: \times \: \alpha \)

\( \scriptsize \therefore \alpha = \normalsize \frac{6.3 \: \times \: 10^{-6}\:K^{-1}}{3}\)

\( \scriptsize \alpha = 2.1 \: \times \: 10^{-6}\:K^{-1}\)

Area expansivity \( \scriptsize \beta = 2 \alpha \)

\(\scriptsize \beta = 2 \: \times \: 2.1 \: \times \: 10^{-6}\: K^{-1}\)

\(\scriptsize \beta = 4.2 \: \times \: 10^{-6}\: K^{-1}\)

Example 10.6.3:

The linear expansivity of a metal cube is 1.8 × 10^-6 K^-1. If the length of each side of the cube is 15 cm, find the area and volume of the cube when its temperature is increased by 60°.

Solution

L = 15 cm, θ = 60°

α = 1.8 × 10^-6 K^-1

Initial area = L × b = 15 × 15 = 225 cm²

Initial volume = L × b × h  = 15 × 15 × 15  = 3375 cm³

Area expansivity  β = 2α =   2 × 1.8 × 10^-6 K^-1  = 3.6 × 10^-6 K^-1  

Cubic expansivity γ = 3α = 3 × 1.8 × 10^-6 K^-1   = 5.4 × 10^-6 K^-1  

New Area, A₂ = A₁(1 + 2 αθ)

= 225(1 + 3.6 × 10^-6 × 60)

= 225(1 + 0.000216)

= 225(1.000216)

= 225 × 1.000216 = 225.0486 cm²

New volume, V₂ = V₁(1 + 3αθ)

      = 3375 (1 + 5.4 × 10^-6 × 60)

      = 3375 (1 + 0.000324)

      = 3375 (1.000324)

      = 3375 × 1.000324

      = 3376.0935 cm³