To solve the problem, we will begin by calculating the change in temperature of the strut (part a) and then proceed to calculate the compressional stress (part b).
Given Data:
- Cross-sectional area of the steel strut (A) = \( 5.00 \times 10^{-4} , \text{m}^2 \)
- Length of the strut (L) = 2.5 m
- Energy absorbed (Q) = \( 3.00 \times 10^{5} , \text{J} \)
- Density of steel (ρ), assumed to be equal to iron = \( 7874 , \text{kg/m}^3 \) (approximate value for the density of steel)
(a) Change in Temperature
To find the change in temperature of the strut when it absorbs energy, we can use the formula relating the energy absorbed to the change in temperature:
\[ Q = mc\Delta T \]
where:
- \( Q \) is the heat absorbed by the material,
- \( m \) is the mass of the strut,
- \( c \) is the specific heat capacity of steel (assumed to be similar to iron, \( c \approx 450 , \text{J/(kg} \cdot \text{°C)} \)),
- \( \Delta T \) is the change in temperature.
Step 1: Calculate the mass of the strut
\[ m = \rho \times V \]
where \( V \) (volume) is calculated as:
\[ V = A \times L \]
Substituting the values:
\[ V = (5.00 \times 10^{-4} , \text{m}^2) \times (2.5 , \text{m}) = 1.25 \times 10^{-3} , \text{m}^3 \]
Next, calculate the mass:
\[ m = 7874 , \text{kg/m}^3 \times 1.25 \times 10^{-3} , \text{m}^3 = 9.843 , \text{kg} \]
Step 2: Rearranging for \(\Delta T\) and substituting the values
Now rearranging the heat equation for \(\Delta T\):
\[ \Delta T = \frac{Q}{mc} \]
Substituting the values:
\[ \Delta T = \frac{3.00 \times 10^5 , \text{J}}{(9.843 , \text{kg})(450 , \text{J/(kg} \cdot \text{°C)})} \]
Calculating:
\[ \Delta T = \frac{3.00 \times 10^5}{4428.35} \approx 67.7 , \text{°C} \]
So, the change in temperature \(\Delta T\) is approximately 67.7 °C.
(b) Compressional Stress
The compressional stress \(\sigma\) can be calculated using the following formula:
\[ \sigma = \frac{F}{A} \]
where:
- \( F \) (force) is calculated using the weight (acting force) on the strut:
- \( A \) is the cross-sectional area of the strut.
Assuming the strut is under its own weight alone, the force due to the weight can be calculated as follows:
\[ F = m \cdot g \]
where \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity).
Step 1: Calculate \( F \)
\[ F = 9.843 , \text{kg} \cdot 9.81 , \text{m/s}^2 \approx 96.7 , \text{N} \]
Step 2: Calculate the compressional stress
Now substituting the values into the stress formula:
\[ \sigma = \frac{96.7 , \text{N}}{5.00 \times 10^{-4} , \text{m}^2} \approx 193400 , \text{Pa} \]
So, the compressional stress in the strut is approximately 193,400 Pa (or 193.4 kPa).
Summary:
(a) Change in Temperature: 67.7 °C
(b) Compressional Stress: 193,400 Pa (or 193.4 kPa)