Physics 141 - Lecture 28: Center of Mass and Moment of Inertia

Vocabulary practice flashcards focusing on rotational motion kinematics, center of mass definitions for discrete and continuous systems, rotational kinetic energy, moment of inertia, and the parallel axis theorem.

Rigid Body

An extended object whose size and shape do not change as it moves, ensuring that every point on the rotating body shares the exact same angular velocity.

Angular Velocity ($\omega$)

The rate of change of angular position with respect to time, mathematically defined as $\omega = \frac{d\theta}{dt}$, where counterclockwise rotation is conventionally positive and clockwise rotation is negative.

Angular Acceleration ($\alpha$)

The rate of change of angular velocity with respect to time, given by the formula $\alpha = \frac{d\omega}{dt}$.

Translational Motion

A type of motion in which an object moves along a trajectory as a single whole without undergoing any rotation.

Rotational Motion

A type of motion in which an object turns about a fixed point or axis such that every individual point within the object traces out a circle.

Combination Motion

Motion in which an extended object simultaneously rotates about an internal axis while moving along an external trajectory.

Center of Mass (System of Particles)

The mass-weighted average position of a discrete collection of particles, calculated along the $x$-axis as $x_{cm} = \frac{1}{M}\sum_{i} m_i x_i$.

Center of Mass (Continuous Distribution)

The mass-weighted average position of an extended continuous solid, computed by integrating over infinitesimal mass elements as $x_{cm} = \frac{1}{M}\int x\,dm = \frac{\int x\mu(x)\,dx}{\int \mu(x)\,dx}$.

Linear Mass Density ($\mu$)

The mass per unit length of a one-dimensional object (such as a rod), having SI units of $\text{kg/m}$, where an infinitesimal mass segment is given by $dm = \mu(x)\,dx$.

Center of Mass of an Unconstrained Object

The specific internal point about which a free, unconstrained object (not mounted on an axle) rotates when no external net force acts upon it.

Rotational Kinetic Energy ($K_{\text{rot}}$)

The kinetic energy associated with the rotation of an extended object, calculated as $K_{\text{rot}} = \frac{1}{2}I\omega^2$.

Moment of Inertia ($I$)

The rotational analog of mass that quantifies an object's resistance to changes in rotational motion; defined as $I = \sum_{i} m_i r_i^2$ (or $\int r^2\,dm$) with SI units of $\text{kg}\cdot\text{m}^2$.

Moment of Inertia of a Point Mass

The rotational inertia of an individual particle of mass $m$ rotating at a perpendicular distance $r$ from the axis of rotation, defined as $I = mr^2$.

Parallel Axis Theorem

A theorem allowing the calculation of a body's moment of inertia about any axis parallel to an axis through its center of mass, expressed as $I = I_{cm} + Md^2$, where $M$ is the total mass and $d$ is the perpendicular distance between the two axes.