Conservation of Energy with Friction

Introduction

• Discussed a conservation of energy problem with friction.
• Previous problems assumed only conservative forces.
• Introduction to non-conservative forces like friction.

Problem Setup

• Source: University of Oregon's zebu.uoregon.edu.
• Problem: 90 kg bike and rider.
• Start at rest from the top of a 500 meter long hill with a 5-degree incline.

Key Elements

• Mass: 90 kg (bike and rider combined).
• Hill Description: 500 meters long with a 5-degree incline.
• Friction Force: Average friction force of 60 newtons.
• Objective: Find the speed of the bike at the bottom of the hill.

Energy Calculations

Initial Energy

• Only potential energy as the system starts from rest.
• Potential Energy (PE):
  • Formula: ( PE = mgh )
  • Mass (m) = 90 kg
  • Gravity (g) = 9.8 m/s²
  • Height (h) derived using trigonometry:
    • 500 m hill with a 5-degree incline.
    • Height = 500 * sin(5°) = 43.6 meters.
  • Calculate PE: ( 90 \times 9.8 \times 43.6 = 38,455 ) Joules.*

Energy Conversion

• Energy Loss Due to Friction:

  • Work done by friction is negative: ( -60 \times 500 ) meters = (-30,000 ) Joules.

• Final Energy:

  • Initial energy minus energy lost to friction:
  • ( 38,455 - 30,000 = 8,455 ) Joules.

Final Speed Calculation

• At the bottom, energy is all kinetic since the rider is at 'ground level'.
• Kinetic Energy (KE):
  • Formula: ( KE = \frac{1}{2} mv^2 )
  • Set KE = 8,455 Joules and solve for velocity (v):
  • ( \frac{1}{2} \times 90 \times v^2 = 8,455 )
  • ( v^2 = \frac{8,455}{45} = 187.9 )
  • ( v = \sqrt{187.9} \approx 13.7 ) m/s.

Conclusion

• Final velocity at the bottom of the hill: 13.7 m/s.
• Energy not fully conserved due to friction.
• Energy converted to heat, demonstrating non-conservative forces.
• Application of conservation of energy principles with friction.