Rotational Motion, Torque, and Dynamics

Vocabulary practice flashcards covering rotational motion, torque, moments of inertia for various geometries, Newton's second law for rotation, angular momentum, and rotational kinetic energy.

Torque ($\tau$)

The tendency of a force to rotate an object about an axis of rotation, defined mathematically as $\tau = rF\sin(\theta) = Fd$, with SI units of $\text{N}\cdot\text{m}$.

Lever Arm

The perpendicular distance from the axis of rotation to the line along which the force acts (the line of action).

Factors Determining Torque Effectiveness

The three factors that determine how effectively an applied force opens a door or rotates an object: the magnitude of the force, the position of application of the force, and the angle at which the force is applied.

Torque Sign Convention

The directional standard where counterclockwise (anticlockwise) rotation corresponds to a positive torque ($+\tau$) and clockwise rotation corresponds to a negative torque ($-\tau$).

Moment of Inertia ($I$)

The tendency of a body to resist angular acceleration, describing how easily a body can be rotated about a given axis, defined as $I \equiv \sum m_i r_i^2$ for discrete particles and measured in $\text{kg}\cdot\text{m}^2$.

Factors Influencing Moment of Inertia

The three physical factors that govern a body's rotational inertia: total mass, the spatial distribution of mass, and the location/orientation of the axis of rotation.

Rotational Inertia of a Point Mass

The product of the particle's mass and the square of its distance from the axis of rotation: $I = mr^2$.

Moment of Inertia of a Hoop (Cylinder Axis)

The rotational inertia of a thin hoop or ring about its central cylindrical axis, given by the formula $I = MR^2$.

Moment of Inertia of a Solid Cylinder or Disk

The rotational inertia of a uniform solid cylinder or disk rotating about its central cylinder axis, given by $I = \frac{1}{2}MR^2$.

Moment of Inertia of a Solid Sphere

The rotational inertia of a uniform solid sphere rotating about any diameter passing through its center, given by $I = \frac{2}{5}MR^2$.

Moment of Inertia of a Thin Spherical Shell

The rotational inertia of a hollow spherical shell rotating about any diameter, given by $I = \frac{2}{3}MR^2$.

Moment of Inertia of a Thin Rod (Center Axis)

The rotational inertia of a thin rod rotated about an axis through its center perpendicular to its length, given by $I = \frac{1}{12}Ml^2$.

Moment of Inertia of a Thin Rod (End Axis)

The rotational inertia of a thin rod rotated about an axis through one of its ends perpendicular to its length, given by $I = \frac{1}{3}Ml^2$.

Newton's Second Law for Rotation

The dynamical equation relating net torque to the resulting angular acceleration of a body: $\tau_{\text{net}} = I\alpha$.

Angular Momentum ($L$)

The rotational analog of linear momentum for a rigid object, defined as the product of moment of inertia and angular velocity: $L = I\omega$ (or $L = mr^2\omega$ for a point mass), with SI units of $\text{kg}\cdot\text{m}^2/\text{s}$.

Torque and Rate of Change of Angular Momentum

The formulation expressing net torque as the time rate of change of angular momentum: $\tau = \frac{\Delta L}{\Delta t}$.

Law of Conservation of Angular Momentum

The principle stating that the total angular momentum of a rotating rigid body remains constant ($\frac{dL}{dt} = 0$ and $\Delta L = 0$) if the net external torque acting on it is zero.

Rotational Kinetic Energy ($KE_R$)

The kinetic energy possessed by an object due to its rotational motion, given by $KE_R = \frac{1}{2}I\omega^2$.

Total Kinetic Energy of a Rolling Object

The sum of translational and rotational kinetic energy for an object rolling without slipping: $KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$.

Work in Rotational Motion

The rotational equivalent of mechanical work, defined as the product of torque and angular displacement: $W = \tau\theta$.

Incline Rolling Race Speed Ranking

The comparative order of arrival at the bottom of an incline of height $h$ from fastest to slowest based on mass distribution: frictionless sliding box (fastest), solid sphere, solid cylinder, hollow cylinder / hoop (slowest).