4G Silver Academy Tamil - Parallel Axis Theorem Lecture Notes

Introduction

• Welcome students to the lecture on the Parallel Axis Theorem.
• Focus on rigid body dynamics and the concept of the center of gravity.

Center of Gravity

• For a rigid body with total mass (M):
  • Center of gravity (G) is critical in analyzing motion and gravitational forces.
• Distance is denoted as smaller distances between axes relevant to the theorem.

Parallel Axis Theorem

• Definition:
  • Relates the moment of inertia of a rigid body about one axis to that about a parallel axis.
• Equation:
  • Moment of Inertia at AB (I_AB) = Moment of Inertia at CD (I_CD) + M
    • (distance between the two axes)^2
  • Formula:
    • I_AB = I_G + M * d^2
    • Where I_G is the moment of inertia at the center of gravity and d is the distance between the axes.*

Derivation

• Step 1:
  • Calculate the gravitational force at the center of gravity.
• Step 2:
  • Use the formula for moment of inertia:
    • Moment of inertia (I) = Mass (m) × (distance from the axis)^2
  • Total moment of inertia (I_total):
    • I_total = Σm * r^2
  • For the rigid body at axis AB:
    • I_AB = Σm * (r_A^2 + 2 * r_A * d + d^2)
      • Where r_A is the distance of particles from axis A.
• Step 3:
  • Simplify and re-organize equations:
    • I_AB = Σm * r^2 + 2Σm * r * d + Σm * d^2
    • This leads to the conclusion:
      • I = I_G + M * d^2*

Conclusion

• Final Formula:
  • I = I_G + M * d^2
• The derivation illustrates the relationship between the moment of inertia about different axes, confirming the Parallel Axis Theorem.*