Moment of Inertia

Introduction

• Moment of inertia (MOI) is a measure of an object's resistance to changes in its rotation.
• Important in Physics, especially in problems involving rotational motion.
• Similar role to mass in linear motion.
• SI unit: kg·m².

Definition

• Describes the amount of torque needed for a specific angular acceleration around a rotational axis.
• Also known as angular mass or rotational inertia.

Formula

• General formula: ( I = \sum m_i r_i^2 )
  • ( m_i ): mass of each particle
  • ( r_i ): distance from the axis of rotation
• Integral form: ( I = \int r^2 dm )
• Dimensional formula: ( M^1 L^2 T^0 )

Factors Affecting MOI

• Density of the material
• Shape and size of the body
• Axis of rotation (distribution of mass relative to the axis)

Systems

• Discrete systems (System of particles)
• Continuous systems (Rigid bodies)

Moment of Inertia for Systems

• For a system of particles: ( I = \sum m_i r_i^2 )
• For continuous mass distribution using integration.

Moment of Inertia for Various Objects

• Uniform Rod: ( I = \frac{ML^2}{12} )
• Circular Ring: ( I = MR^2 )
• Rectangular Plate: ( I = \frac{Ml^2}{12} )
• Uniform Circular Plate: ( I = \frac{MR^2}{2} )
• Thin Spherical Shell: ( I = \frac{2MR^2}{3} )
• Solid Sphere: ( I = \frac{2MR^2}{5} )

Theorems

Parallel Axis Theorem

• ( I = I_{cm} + Md^2 )
  • ( I_{cm} ): MOI about center of mass
  • ( d ): Distance between the axes

Perpendicular Axis Theorem

• Applicable for planar objects.

Radius of Gyration

• Represents the radial distance from the axis where entire mass can be assumed to be concentrated.

Solved Examples

• Example calculations for various systems and configurations.

FAQs

• MOI is scalar.
• MOI does not depend on rotational speed.
• A hollow cylinder has a larger MOI than a disc of the same radius.

Additional Resources

• Explore more about MOI for different shapes and configurations through links provided.