Understanding Work Done in Stretched Wires
When two wires with different force constants are stretched through the same distance, the work done on each wire can be analyzed using the relationship between force, displacement, and the elastic properties of the wires.
Definitions
- Force Constant (K): This is a measure of a wire's stiffness. A higher K indicates a stiffer wire.
- Work Done (W): This is the energy expended in stretching the wire, given by the formula: W = (1/2) * K * x², where x is the displacement.
Work Done Calculation
- For Wire 1 with force constant K:
- W1 = (1/2) * K * x²
- For Wire 2 with force constant K':
- W2 = (1/2) * K' * x²
Key Points
- Same Displacement: Both wires are stretched through the same distance (x), which is crucial for comparing work done.
- Proportionality: The work done is directly proportional to the force constant of the wire. Hence, if K > K', then W1 > W2, meaning the stiffer wire requires more work to stretch.
- Total Work Done: The total work done when both wires are stretched can be expressed as:
- Total Work = W1 + W2 = (1/2) * K * x² + (1/2) * K' * x²
- This can be simplified to: Total Work = (1/2) * (K + K') * x²
Conclusion
The work done in stretching each wire depends on the respective force constants and the displacement. Understanding these relationships helps in analyzing mechanical properties of materials effectively.