Calculate the work (in kJ) when substance initially at 95 kPa, 19 m³ is expanded against a linear spring to 245 kPa, 67 m³.
Calculate the work (in kJ) when substance initially at 95 kPa, 19 m³ is expanded against a linear spring to 245 kPa, 67 m³.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Question
![**Calculate the work (in kJ) when a substance initially at 95 kPa, 19 m³ is expanded against a linear spring to 245 kPa, 67 m³.**
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### Explanation
This problem involves a calculation of work done by a substance expanding against a linear spring. The work done during this process can be visualized in the context of mechanical work, often represented graphically via pressure-volume (P-V) diagrams.
#### Key Points:
- **Initial Conditions:**
- Pressure: 95 kPa
- Volume: 19 m³
- **Final Conditions:**
- Pressure: 245 kPa
- Volume: 67 m³
In such cases, the linear relationship between pressure and volume during the process can be examined, where the work done is represented by the area under the curve in a P-V diagram.
### Methodology
To solve for the work done:
1. **Identify the type of process:** Linear spring expansion suggests a linear increase in pressure with volume.
2. **Calculate the work done:** The work is the integral of pressure with respect to volume, simplified to the area under the P-V curve for a linear spring.
#### Calculations of Work:
The work done \( W \) can be computed using the formula for the area under a straight line in a P-V graph:
\[
W = \frac{1}{2} (P_1 + P_2) \times (V_2 - V_1)
\]
Where:
- \( P_1 = 95 \text{ kPa} \)
- \( P_2 = 245 \text{ kPa} \)
- \( V_1 = 19 \text{ m}^3 \)
- \( V_2 = 67 \text{ m}^3 \)
This equation accounts for the triangular region under the linear curve from the initial to final state.
### Educational Perspective:
Understanding this concept requires comprehension of thermodynamic cycles and P-V diagrams, which are critical for visualizing work processes in mechanical systems. This scenario is a practical application of such principles, enhancing problem-solving skills in physics and engineering contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd30ea730-ba6b-4d49-90df-80ffc6e77ea4%2Ff5c2a6dd-1f19-4e93-9158-6c6524361de1%2Fo2ezvz9_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculate the work (in kJ) when a substance initially at 95 kPa, 19 m³ is expanded against a linear spring to 245 kPa, 67 m³.**
---
### Explanation
This problem involves a calculation of work done by a substance expanding against a linear spring. The work done during this process can be visualized in the context of mechanical work, often represented graphically via pressure-volume (P-V) diagrams.
#### Key Points:
- **Initial Conditions:**
- Pressure: 95 kPa
- Volume: 19 m³
- **Final Conditions:**
- Pressure: 245 kPa
- Volume: 67 m³
In such cases, the linear relationship between pressure and volume during the process can be examined, where the work done is represented by the area under the curve in a P-V diagram.
### Methodology
To solve for the work done:
1. **Identify the type of process:** Linear spring expansion suggests a linear increase in pressure with volume.
2. **Calculate the work done:** The work is the integral of pressure with respect to volume, simplified to the area under the P-V curve for a linear spring.
#### Calculations of Work:
The work done \( W \) can be computed using the formula for the area under a straight line in a P-V graph:
\[
W = \frac{1}{2} (P_1 + P_2) \times (V_2 - V_1)
\]
Where:
- \( P_1 = 95 \text{ kPa} \)
- \( P_2 = 245 \text{ kPa} \)
- \( V_1 = 19 \text{ m}^3 \)
- \( V_2 = 67 \text{ m}^3 \)
This equation accounts for the triangular region under the linear curve from the initial to final state.
### Educational Perspective:
Understanding this concept requires comprehension of thermodynamic cycles and P-V diagrams, which are critical for visualizing work processes in mechanical systems. This scenario is a practical application of such principles, enhancing problem-solving skills in physics and engineering contexts.
Expert Solution
Step 1: Given data
Given:
Initial pressure (P1) = 95 kPa
Initial volume (V1) = 19 m3
Final pressure (P2) = 245 kPa
Final volume (V2) = 67 m3
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