Lecture Notes on Mechanical Energy

Introduction

• Mechanical Energy: Sum of kinetic and potential energies.
• Forces:
  • Conservative
  • Non-conservative
• Topics Covered:
  • Work-energy theorem
  • Work done by non-conservative forces
  • Conservation of mechanical energy

Kinetic and Potential Energies

• Kinetic Energy (KE):
  • Formula: ( KE = \frac{1}{2} mv^2 )
  • Represents energy of motion
  • Proportional to mass and square of velocity
• Potential Energy (PE):
  • Represents the potential to do work
  • Key example: Gravitational potential energy ( PE = mgh )
• Abbreviations: KE for kinetic energy, PE for potential energy (textbook preferences may vary)

Conservative vs Non-conservative Forces

• Conservative Forces:
  • Mechanical energy conserved if only these forces are present
  • Example: Gravity
  • Path-independent work
• Non-conservative Forces:
  • Example: Friction, air resistance
  • Opposes motion, converts mechanical energy to heat or sound

Work-Energy Theorem

• Concept: Network done on an object changes its kinetic energy
• Scenarios:
  • Positive work: Increase in kinetic energy and velocity
  • Negative work: Decrease in kinetic energy and velocity
  • Example discussed: Raising and lowering a marker at constant velocity

Conservation of Mechanical Energy

• Principle: Total mechanical energy (KE + PE) remains constant in a closed system with only conservative forces.
• Equation: ( KE_{initial} + PE_{initial} = KE_{final} + PE_{final} )
• Applications:
  • Calculating velocities and heights in various scenarios (e.g., falling objects, projectiles)

Example Problems

• Problem 1: Kinetic energy of a 1000kg car traveling at 20 m/s:
  • Solution: ( KE = 200,000 , J )
  • Network required to stop car: (-200,000 , J )
• Problem 2: Calculating final velocity of a falling stone from 7.35m:
  • Solution: ( v = 12.0 , m/s )
• Problem 3: Maximum height of a baseball thrown at 50 m/s:
  • Solution: ( h = 128 , m )
• Problem 4: Box sliding down a frictionless ramp:
  • Calculating final velocity using height from trigonometry
  • Example with friction shows decrease in mechanical energy

Conclusion

• Closing Remarks: Understanding mechanical energy, forces, and the work-energy theorem is crucial for solving physics problems related to motion and energy conservation.
• Further Learning: Practice problems available, explore more with Chad's prep courses.