Rotation of Rigid Bodies

Comprehensive vocabulary flashcards covering rotational kinematics, linear-angular relations, moment of inertia formulas, rotational kinetic energy, gravitational potential energy, and the parallel-axis theorem.

Angular Velocity ($\omega$)

The rate of change of angular position with respect to time, defined instantaneously as $\omega = \frac{d\theta}{dt}$ and measured in units of $\text{rad/s}$.

Angular Acceleration ($\alpha$)

The rate of change of angular velocity with respect to time, defined instantaneously as $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ and measured in units of $\text{rad/s}^2$.

Right-Hand Rule for Angular Velocity

The convention stating that if the fingers of the right hand are curled in the direction of rotation, the extended thumb points along the rotation axis in the direction of the angular velocity vector $\vec{\omega}$.

Sign Convention for Fixed-Axis Angular Velocity ($\omega_z$)

When viewed from above along the $z$-axis, counterclockwise rotation corresponds to $\omega_z > 0$ (pointing in the positive $z$-direction), whereas clockwise rotation corresponds to $\omega_z < 0$ (pointing in the negative $z$-direction).

Rotational Speeding Up vs. Slowing Down Condition

Rotation speeds up when the angular acceleration vector $\vec{\alpha}$ and angular velocity vector $\vec{\omega}$ point in the same direction; rotation slows down when $\vec{\alpha}$ and $\vec{\omega}$ point in opposite directions.

Rotational Kinematic Equations for Constant $\alpha$

The set of equations describing rotational motion with constant angular acceleration: $\omega_z = \omega_{0z} + \alpha_z t$, $\theta = \theta_0 + \omega_{0z}t + \frac{1}{2}\alpha_z t^2$, $\omega_z^2 = \omega_{0z}^2 + 2\alpha_z(\theta - \theta_0)$, and $\theta - \theta_0 = \frac{1}{2}(\omega_{0z} + \omega_z)t$.

Radian Measure Requirement

The rule establishing that in any kinematic or dynamic equation relating linear quantities to angular quantities (such as $s = r\theta$, $v = r\omega$, or $a_{\text{tan}} = r\alpha$), the angle $\theta$ must always be measured in radians.

Tangential Speed ($v$)

The linear speed of a point at a distance $r$ from the rotation axis of a rigid body, related to angular speed by $v = r\omega$.

Tangential Acceleration ($a_{\text{tan}}$) of a Rotating Rigid Body

The linear acceleration component parallel to the circular path of a rotating point, representing the rate of change of linear speed: $a_{\text{tan}} = \frac{dv}{dt} = r\alpha$.

Radial (Centripetal) Acceleration ($a_{\text{rad}}$) of a Rotating Rigid Body

The linear acceleration component directed perpendicular to the velocity and toward the rotation axis, given by $a_{\text{rad}} = \frac{v^2}{r} = \omega^2 r$.

Moment of Inertia ($I$)

The rotational analog of mass that measures a rigid body's resistance to rotational acceleration about a given axis, defined discretely as $I = \sum \Delta m_i r_i^2$ and with SI units of $\text{kg}\,\text{m}^2$.

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

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

Gravitational Potential Energy of an Extended Body

The total gravitational potential energy of an extended body in a uniform gravitational field, which equals $U = Mgy_{\text{cm}}$, where $M$ is total mass and $y_{\text{cm}}$ is the vertical position of the center of mass.

Parallel-Axis Theorem (Huygens-Steiner Theorem)

The theorem stating that the moment of inertia $I$ of a body about any axis parallel to an axis passing through its center of mass is given by $I = I_{\text{cm}} + Md^2$, where $d$ is the perpendicular distance between the parallel axes.

Continuous Moment-of-Inertia Integral

The definite integral used to calculate the moment of inertia for a continuous body with distributed mass: $I = \int r^2\,dm$, where $r$ is the perpendicular distance from the infinitesimal mass element $dm$ to the rotation axis.

Moment of Inertia: Solid Cylinder or Disk

The moment of inertia of a uniform solid cylinder or disk of mass $M$ and radius $R$ about its central cylinder axis: $I = \frac{1}{2}MR^2$.

Moment of Inertia: Thin Hoop (Ring)

The moment of inertia of a thin-walled hollow hoop or ring of mass $M$ and radius $R$ about its central axis: $I = MR^2$.

Moment of Inertia: Thick-Walled Hollow Cylinder

The moment of inertia of a hollow cylinder of mass $M$, inner radius $R_1$, and outer radius $R_2$ about its central axis: $I = \frac{1}{2}M(R_1^2 + R_2^2)$.

Moment of Inertia: Solid Sphere

The moment of inertia of a uniform solid sphere of mass $M$ and radius $R$ about any diameter: $I = \frac{2}{5}MR^2$.

Moment of Inertia: Thin-Walled Hollow Sphere

The moment of inertia of a thin-walled hollow spherical shell of mass $M$ and radius $R$ about any diameter: $I = \frac{2}{3}MR^2$.

Moment of Inertia: Slender Rod (Axis Through Center)

The moment of inertia of a uniform slender rod of mass $M$ and length $L$ about an axis perpendicular to the rod passing through its center of mass: $I = \frac{1}{12}ML^2$.

Moment of Inertia: Slender Rod (Axis Through One End)

The moment of inertia of a uniform slender rod of mass $M$ and length $L$ about an axis perpendicular to the rod passing through one end: $I = \frac{1}{3}ML^2$.

Moment of Inertia: Uniform Rectangular Plate (Axis Through Center)

The moment of inertia of a thin rectangular plate of mass $M$ with side lengths $a$ and $b$ about an axis perpendicular to its plane passing through its center: $I = \frac{1}{12}M(a^2 + b^2)$.

Moment of Inertia: Thin Rectangular Plate (Axis Along Edge)

The moment of inertia of a thin rectangular plate of mass $M$ and side lengths $a$ and $b$ about an axis lying along the edge of length $b$: $I = \frac{1}{3}Ma^2$.