Physics Lecture: External and Internal Torque, and Conservation of Angular Momentum

External Torque and Pottery Wheel Example

• Scenario: Torque applied by hands on a pottery wheel
  • Side view of the wheel with hands applying frictional force
  • Initial angular momentum transitions to final angular momentum due to external torque
• Key Concept: This is not a conservation of angular momentum problem because the focus is solely on the wheel, not the hands
• Equation: Use torque
  • ( \tau = \frac{dL}{dt} )
• Diagram: View from above
  • Two frictional forces oppose the wheel's motion
  • Determine the direction of forces and calculate resulting torques
• Example: Calculate torque with 90-degree forces
  • Left hand: negative torque due to downward force
  • Right hand: negative torque due to downward force
  • Sum of torques: ( - F_{friction} \cdot r )
• Relation to linear momentum: Similar calculation to ( F = \frac{dP}{dt} )

No External Torque and Angular Momentum Conservation

• Scenario: Squirrel jumps onto a bird feeder
  • Bird feeder: 3 kg, 1.5 m string
  • Squirrel: 2 kg, jumps horizontally at 1 m/s
• Key Concept: Conservation of Angular Momentum Problem
  • Identify internal forces, ignore external torque
• Before and After Collision:
  • Before: Bird feeder is not rotating (angular momentum is 0)
  • After: Combined system of feeder and squirrel
• Equation: Total angular momentum before equals total angular momentum after
  • ( L_{before} = L_{after} )
  • ( L_{squirrel, initial} + L_{feeder, initial} = (I_{feeder} + I_{squirrel}) \cdot \omega )
• Find Moments of Inertia:
  • Feeder: ( I = m_{feeder} \cdot r^2 )
  • Squirrel: ( I = m_{squirrel} \cdot r^2 )
• Example Calculation: Insert values and solve for final angular velocity (( \omega ))

Applications of Conservation of Angular Momentum

• Varied Examples:
  • Ice skater pulling in arms: Decreased moment of inertia increases angular velocity
  • Dog running to center of merry-go-round: Changing moment of inertia affects rotational speed
  • Divers tucking in for spins and extending to stop
• Key Principle: ( I_{initial} , \omega_{initial} = I_{final} , \omega_{final} )

Demonstration and Explanation

• Example 1: Spinning chair with arm extension and retraction
  • Demonstrates the change in angular velocity
• Example 2: Spinning disk with direction change
  • Demonstrates conservation of angular momentum with torque and rotational direction changes

Conclusion

• Angular Momentum Conservation: Fundamental concept for rotating systems without external torques
• Practical Examples: Diverse scenarios help in understanding the broad application
• Hands-on Demonstrations: Visualize and reinforce understanding of the physical principles