A steel rod of length 1.0000 m and cross-sectional area 3.63·10-4m2 is placed snugly against two immobile end points. The rod is initially placed when the temperature is 0°C. Find the stress in the rod when the temperature rises to 38.9°C.
Thermal expansion equals stress compression. Solve for the stress, sigma.
L*(alpha)*(delta T) = (sigma/E)*L
sigma = E*(alpha)*(deltaT)
E is Young's modulus for steel
delta T = 38.9 C
alpha is the linear coefficient of thermal expansionm of styel
L cancels out and is not needed.
To find the stress in the rod when the temperature rises to 38.9°C, we need to consider the thermal expansion of the rod.
The formula for thermal stress is given by:
Stress = Young's Modulus * Coefficient of Thermal Expansion * Change in Temperature
1. Determine the change in temperature:
The change in temperature is given by the difference between the final temperature (38.9°C) and the initial temperature (0°C):
Change in Temperature = Final Temperature - Initial Temperature
= 38.9°C - 0°C
= 38.9°C
2. Determine the coefficient of thermal expansion:
The coefficient of thermal expansion is a material-specific constant that describes how much a material expands or contracts when the temperature changes. For steel, the coefficient of thermal expansion is typically around 12 x 10^-6 per °C.
Coefficient of Thermal Expansion = 12 x 10^-6 per °C
3. Determine the Young's modulus for steel:
Young's modulus is a measure of the stiffness and elasticity of a certain material. For steel, the Young's modulus is typically around 200 GPa (Gigapascals), which is equal to 200 x 10^9 in Pascal.
Young's Modulus = 200 x 10^9 Pascal
4. Calculate the stress:
Using the formula for thermal stress and the values we have determined:
Stress = Young's Modulus * Coefficient of Thermal Expansion * Change in Temperature
Stress = (200 x 10^9 Pascal) * (12 x 10^-6 per °C) * (38.9°C)
Now, let's calculate:
Stress = (200 x 10^9) * (12 x 10^-6) * (38.9)
Stress = 92,760 Pascal (or N/m²)
Therefore, the stress in the rod when the temperature rises to 38.9°C is 92,760 Pascal (N/m²).