Dynamics of Rigid Bodies Lecture Notes

Overview

• Topic: Dynamics of Rigid Bodies
• Previous Subjects: Engineering Mechanics (Solid Mechanics & Fluid Mechanics)
• Current Focus: Rigid bodies after Statics (bodies are moving but remain rigid)

Key Concepts

• Definitions:
  • Rigid Bodies: Do not deform under applied forces.
  • Dynamics: Study of bodies in motion (unlike Statics, where bodies are at rest).
  • Deformable Bodies: Bodies that change shape under force.

1. Dynamics Breakdown

• Components of Dynamics:
  • Kinematics (motion without forces): consistent motion paths.
  • Kinetics (motion with forces): actions that influence motion.

2. Kinematics

• Definition: Motion in a straight line without considering forces.
• Key Terms:
  • Displacement: Change in position.
    • Differs from Distance (total length traveled).
  • Velocity: Change in position over time.
    Formula:
    [ V = \frac{Distance}{Time} ]
  • Instantaneous Velocity: Derivative of position over time.
    Formula:
    [ V = \frac{dS}{dt} ]
  • Acceleration: Change in velocity over time.
    Formula:
    [ A = \frac{dV}{dt} ]

3. Formulas to Remember

• Velocity Relationship:
  [ V = \frac{dS}{dt} ]
• Acceleration Relationship:
  [ A = \frac{dV}{dt} ]
• Instantaneous Relationship:
  • Equate time intervals:
    [ dt = \frac{dS}{V} = \frac{dV}{A} ]
• Main Kinematic Equations (for constant acceleration):
  1. [ V = V_0 + A \cdot t ]
  2. [ S = S_0 + V_0 t + \frac{1}{2} A t^2 ]
  3. [ V^2 = V_0^2 + 2 A(S - S_0) ]

4. Sample Problems

• Example with a car's velocity equation:
  • Given: [ V = 3t^2 + 2t ]
  • To find position [ S ] and acceleration at [ t = 3s ]
• Solution Steps:
  1. Write equations and given parameters.
  2. Differentiate velocity function to find acceleration.
  3. Integrate velocity to derive position.

5. Gravity Considerations

• Important when analyzing vertical motion:
  • Gravity acts downwards (9.81 m/s²).

Sample Acceleration Problem Analysis

• Analyze rocket motion under gravity:

Summary of Key Examples:

1. Determine maximum height and speed before hitting the ground from given velocity data.
2. Always check for zero velocity to determine motion changes (max height).
3. Average velocity & speed generation from total distance traveled.

Conclusion

• Essential takeaways include understanding displacement vs. distance, applying kinematic formulas correctly, and differential calculus applications in physics.
• Activity 1: Work further sample problems for real-world application.

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