Explanation of Moment of Inertia

Introduction

• Moment of Inertia (I) plays the same role in rotational motion as mass does in linear motion.
• It is also known as the mass moment of inertia.

Definition

• The Moment of Inertia of a mass M, located at a distance R from an axis, is defined as MR².

Equivalent of Linear and Rotational Motion

• In linear motion, displacement S, velocity V, and acceleration A have equivalents in rotational motion as θ, ω (omega), and α (alpha).
• Instead of force there is torque, and instead of mass there is moment of inertia I.

Relation between Torque and Force

• Torque (τ) = I × α (angular acceleration)
• By multiplying F = ma by r, the definition of torque τ = Iα is obtained.

Components of Moment of Inertia

• Moment of Inertia does not depend solely on mass but also on the distance from the axis and the orientation of the axis.
• It is a scalar quantity and can be calculated by adding equivalent moments of inertia.

Unit and Calculation

• The unit of Moment of Inertia is kg m².

Questions and Examples

Question 1

• Four masses m are located at the four corners:
  • Moment of inertia = ma²

Question 2

• M mass on the sides of an equilateral triangle:
  • Moment of inertia = 5/4 ML²

Question 3

• Moment of inertia for masses on x-y plane with axis through one mass:
  • Moment of inertia = 2 ML²

Question 4

• Various masses and their distances are given:
  • Calculation: Σ(mr²)

Conclusion

• Understanding the proper axis and correct perpendicular distance is essential for calculating Moment of Inertia.
• The next video will cover the moment of inertia for continuous bodies.