Question
Suppose a steel strut having a cross sectional area of 5.00×10⁴ m² and length 2.5 is bolted between two rigid bulkheads in the engine room of a submarine. Assume the density of the steel is the same as that of iron,
(a) calculate the change in temperature of the strut if it absorbs 3.00×10 to the power 5 j of energy.
(b) calculate the compressional stress of the strut
(a) calculate the change in temperature of the strut if it absorbs 3.00×10 to the power 5 j of energy.
(b) calculate the compressional stress of the strut
Answers
GPT-4o mini
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**)
### 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**)