Rotational Inertia and Rolling Motion

Rotational Inertia and Extended Objects

Definition of Rotational Inertia (Moment of Inertia)

The moment of inertia (also called rotational inertia) quantifies an object's resistance to changes in its rotational motion about a specific axis. It depends on the mass distribution relative to the axis of rotation.

• For a system of discrete particles: The moment of inertia is given by the sum of each particle's mass times the square of its distance from the axis:

$I = \sum_{i} m_i r_i^2$

• For an extended object: The moment of inertia is calculated by integrating over the object's mass distribution:

$I = \int r^2 dm$

• Key Terms:

  • Axis of rotation: The line about which the object rotates.

  • dm: An infinitesimal mass element of the object.

  • r: The perpendicular distance from the axis of rotation to the mass element.

Parallel Axis Theorem

The parallel axis theorem relates the moment of inertia about any axis to the moment of inertia about a parallel axis through the center of mass:

$I = I_{cm} + md^2$

• $I_{cm}$: Moment of inertia about the center of mass axis.

• m: Total mass of the object.

• d: Perpendicular distance between the two axes.

Example: Calculating the moment of inertia of a disk about an axis parallel to its center but offset by a distance d.

Rotational Kinetic Energy

Rotational Kinetic Energy Formula

An object rotating with angular velocity $\omega$ has rotational kinetic energy:

$K_{rot} = \frac{1}{2}I\omega^2$

• Rotational kinetic energy is the energy due to the rotation of an object and depends on both its moment of inertia and angular velocity.

Work-Energy Principle for Rotation

For a constant torque $\tau_{ext}$ applied over an angular displacement $\Delta \theta$:

$\Delta K_{rot} = \left( \sum \tau_{ext} \right) \Delta \theta$

Total Kinetic Energy of Rolling Objects

For an object rolling without slipping, the total kinetic energy is the sum of translational and rotational kinetic energies:

$K = K_{cm} + K_{rot} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2$

• $v_{cm}$: Velocity of the center of mass.

• $\omega$: Angular velocity.

Example: A solid sphere rolling down an incline has both translational and rotational kinetic energy.

Rolling Motion and Rolling Without Slipping

Rolling Without Slipping

Rolling without slipping occurs when the point of contact between a rolling object and the surface has zero velocity relative to the surface. This means the object rolls in such a way that its rotation and translation are perfectly matched.

• Condition for rolling without slipping:

$v_{cm} = R\omega$

• R: Radius of the rolling object.

• At the point of contact: The velocity is zero relative to the ground.

Example: A wheel rolling on a road without skidding.

Velocity at Different Points on a Rolling Object

• Top of the object: $v_{top} = 2R\omega$

• Center of mass: $v_{cm} = R\omega$

• Bottom (point of contact): $v_{bottom} = 0$

Diagram Explanation: The velocity vectors at the top, center, and bottom of a rolling object illustrate how the combination of rotational and translational motion results in zero velocity at the contact point.

Sample Problems and Applications

Ranking Moments of Inertia (Three Balls)

A triangular shape is made from three identical balls connected by rigid rods of negligible mass. The moments of inertia about axes a, b, and c are compared.

• Ranking: $I_a < I_b = I_c$

• Reasoning: The distribution of mass relative to each axis determines the moment of inertia.

Calculating Rotational Inertia (Three Balls – Center of Mass)

Given a triangular arrangement of three balls (each of mass m), calculate the moment of inertia about an axis through the center of mass.

• Approach: Use the definition $I = \sum m_i r_i^2$ with appropriate distances from the center of mass.

Rod About End

Find the moment of inertia of a uniform rod of length $l$ and mass $m$ about an axis through one end perpendicular to its length.

• Formula:

$I = \frac{1}{3}ml^2$

Example: This formula is commonly used for pendulums and rotating beams.

Summary Table: Key Formulas and Concepts

Concept	Formula	Description
Moment of inertia (particles)	$I = \sum m_i r_i^2$	Sum over all particles, $r_i$ is distance to axis
Moment of inertia (extended object)	$I = \int r^2 dm$	Integral over mass distribution
Parallel axis theorem	$I = I_{cm} + md^2$	Relates inertia about any axis to that through center of mass
Rotational kinetic energy	$K_{rot} = \frac{1}{2}I\omega^2$	Energy due to rotation
Total kinetic energy (rolling)	$K = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2$	Sum of translational and rotational energies
Rolling without slipping	$v_{cm} = R\omega$	Condition for no skidding
Rod about end	$I = \frac{1}{3}ml^2$	Uniform rod, axis at one end

Additional Info

• Rolling without slipping is a key concept in dynamics, especially for analyzing wheels, cylinders, and spheres in contact with surfaces.

• Friction is necessary for rolling without slipping; it provides the torque needed for rotation.

• In problems involving composite objects, use the parallel axis theorem and sum moments of inertia for each part.