Conservation of Energy: Elastic Potential to Gravitational Potential

Overview

• Discussion about converting elastic potential energy to gravitational potential energy in a spring-mass system.
• A spring-mass system is involved where a mass attached to a compressed spring moves up an inclined plane when released.

Key Concepts

Conservation of Energy

• Total energy in a system remains constant if no external forces (e.g., friction) are acting.
• Energy transformation in this scenario:
  • Elastic Potential Energy (EPE) → Kinetic Energy (KE) → Gravitational Potential Energy (GPE)

Assumptions

• No friction between the block and the inclined plane.
• Energy transforms completely from one form to another without losses.

Energy Transformation

• At Compression (Initial State):
  • Energy stored as Elastic Potential Energy (EPE).
• At Maximum Height (Final State):
  • Energy is entirely Gravitational Potential Energy (GPE).
  • KE and EPE are zero at this point.

Formulae

• Elastic Potential Energy (EPE):
  EPE = 1/2 * k * x^2
  • k = spring constant
  • x = compression (displacement of spring)
• Gravitational Potential Energy (GPE):
  [ GPE = mgh ]
  • m = mass of the block
  • g = acceleration due to gravity
  • h = height

Calculation

• Solving for height (h):
  h = kx^2/2mg
• Given values:
  • Spring constant (k): 100 N/m
  • Compression (x): 20 cm (0.2 m in SI units)
  • Mass (m): 100 g (0.1 kg in SI units)
  • Gravitational acceleration (g): 9.8 m/s²
• Calculated height (h): 10.2 meters

Conclusion

• The block will move up 10.2 meters along the inclined plane, ignoring friction.
• The angle of inclination does not affect the height reached as long as friction is ignored.

Additional Notes

• If the incline was vertical, the scenario changes, which is not covered here.
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