Energy Stored in Railway Track Calculation

This solution details the calculation for energy stored per meter in a railway track, focusing on the effects of temperature variations.

Railway Track Thermal Expansion Data

• Area of cross-section, $A = 0.01 \text{ m}^2$
• Temperature variation, $\Delta T = 10^{\circ}C$
• Coefficient of linear expansion, $\alpha = 10^{-5} /^{\circ}C$
• Young's modulus, $E = 10^{11} \text{ Nm}^{-2}$
• Length considered, $L = 1 \text{ m}$ (for calculating energy per meter)

Calculating Thermal Strain

The change in length relative to the original length, known as strain ($\epsilon$), due to a temperature change ($\Delta T$) is:

$ \epsilon = \alpha \Delta T $

Substituting the given values:

$ \epsilon = (10^{-5} /^{\circ}C) \times (10^{\circ}C) = 10^{-4} $

Calculating Thermal Stress

Thermal stress ($\sigma$) arises from the resistance to this expansion/contraction. It is calculated using Young's modulus ($E$):

$ \sigma = E \epsilon $

Substituting the values:

$ \sigma = (10^{11} \text{ Nm}^{-2}) \times (10^{-4}) = 10^{7} \text{ Nm}^{-2} $

Calculating Strain Energy Density

The energy stored within the material per unit volume is called strain energy density ($u$). The formula is:

$ u = \frac{1}{2} \sigma \epsilon $

Or alternatively, using Young's modulus:

$ u = \frac{1}{2} E \epsilon^2 $

Using the calculated values:

$ u = \frac{1}{2} \times (10^{11} \text{ Nm}^{-2}) \times (10^{-4})^2 $

$ u = \frac{1}{2} \times 10^{11} \times 10^{-8} \text{ J/m}^3 $

$ u = 500 \text{ J/m}^3 $

Calculating Energy Stored Per Meter

To find the energy stored per meter ($U_{meter}$), we multiply the strain energy density ($u$) by the cross-sectional area ($A$) and the length ($L=1$ m):

$ U_{meter} = u \times A \times L $

Substituting the values:

$ U_{meter} = (500 \text{ J/m}^3) \times (0.01 \text{ m}^2) \times (1 \text{ m}) $

$ U_{meter} = 5 \text{ J/m} $

Energy Stored Result

The calculated energy stored per meter in the railway track is 5 J/m.