Rotational Motion, Equilibrium, and Dynamics Flashcards

Vocabulary flashcards covering key definitions, equations, and principles of rotational motion, rotational equilibrium, and dynamics.

Center of Gravity

The single point in a rigid body where the single equivalent force of all individual gravitational forces acting on its particles is located, calculated using coordinates $x_{cg} = \frac{\sum m_i x_i}{\sum m_i}$ and $y_{cg} = \frac{\sum m_i y_i}{\sum m_i}$.

Plumb Line Method

An experimental method used to determine the center of gravity of an object, where the center of gravity always coincides with a vertical line extended from a suspended string.

Torque ($\tau$)

A quantitative measure of the tendency of a force to cause or change a body's rotational motion ($SI\,unit: N \cdot m$), calculated as $\tau = r F_{\perp}$, where positive torque causes counterclockwise rotation and negative torque causes clockwise rotation.

Lever Arm ($r_{\perp}$)

The perpendicular distance from the rotational axis to the line of action of an applied force, used to calculate torque via $\tau = r_{\perp} F$.

Static Equilibrium of a Rigid Body

The state of a rigid body in which both the net external force is zero ($\sum F = 0$) and the net external torque about any axis is zero ($\sum \tau = 0$).

Moment of Inertia ($I$)

A measure of an object's resistance against changes in its rotational motion ($SI\,unit: kg \cdot m^2$), defined as $I = m r^2$ for a point mass or $I = \sum m_i r_i^2$ for a system of particles.

Newton's Second Law for Rotational Motion

The law stating that when a net external torque $\sum \tau$ acts on a rigid body with moment of inertia $I$, it produces an angular acceleration $\alpha$ given by $\sum \tau = I \alpha$.

Rotational Kinetic Energy ($KE_r$)

The kinetic energy possessed by an object rotating with angular speed $\omega$, expressed as $KE_r = \frac{1}{2} I \omega^2$.

Total Kinetic Energy of Rolling Motion

The total kinetic energy acquired by a rolling object without slipping, given by the sum of its rotational kinetic energy and translational kinetic energy: $KE = KE_r + KE_t = \frac{1}{2} I \omega^2 + \frac{1}{2} m v^2$.

Rotational Work ($W$)

The work done on a rigid body when it undergoes an angular displacement $\Delta \theta$ due to an applied torque $\tau$, given by $W = \tau \Delta \theta$.

Angular Momentum ($L$)

The rotational analog of linear momentum ($SI\,unit: kg \cdot m^2/s$), calculated as the product of an object's moment of inertia $I$ and its angular velocity $\omega$ ($L = I \omega$).

Law of Conservation of Angular Momentum

The principle stating that when a rigid body experiences no net external torque, its total angular momentum remains constant over time ($L_i = L_f$).

Radian ($rad$)

The SI unit of angular measurement defined as the angle subtended at the center of a circle by an arc whose length $s$ equals the radius $r$ ($s = r$), where $1\,rev = 2\pi\,rad = 360^\circ$.

Angular Displacement ($\Delta \theta$)

The angle through which a body rotates from an initial angular position $\theta_o$ to a final position $\theta$, defined as $\Delta \theta = \theta - \theta_o$ and related to arc length by $\Delta s = r \Delta \theta$.

Angular Velocity ($\omega$)

The rate of change of angular displacement with respect to time ($SI\,unit: rad/s$), with average angular velocity expressed as $\omega_{av} = \frac{\Delta \theta}{\Delta t}$.

Angular Acceleration ($\alpha$)

The rate of change of angular velocity with respect to time ($SI\,unit: rad/s^2$), with average angular acceleration expressed as $\alpha_{av} = \frac{\Delta \omega}{\Delta t}$.

Tangential Velocity ($v_t$)

The linear velocity of a point on a rotating body directed tangent to the circular path ($SI\,unit: m/s$), related to angular velocity by $v_t = r \omega$.

Tangential Acceleration ($a_t$)

The linear acceleration of a point on a rotating body directed tangent to its circular path ($SI\,unit: m/s^2$), caused by a change in tangential speed and calculated as $a_t = r \alpha$.

Centripetal Acceleration ($a_c$)

The acceleration directed toward the center of a circular path ($SI\,unit: m/s^2$) caused by a continuous change in the direction of velocity, given by $a_c = \frac{v^2}{r} = \omega^2 r$.

Uniform Circular Motion

A special case of circular motion in which an object moves in a circle at constant tangential speed, experiencing zero tangential acceleration ($a_t = 0$) and non-zero centripetal acceleration ($a_c \neq 0$).

Centrifugal Force

An apparent outward fictitious force felt by an object moving in a circular path, arising from Newton's first law of motion as the body tends to maintain straight-line motion.

Conical Pendulum

A pendulum consisting of a bob revolving in a horizontal circular path of radius $r$ at a string length $L$, with a period given by $Period = 2\pi \sqrt{\frac{L \cos(\theta)}{g}}$.

Banked Curve

A curve on a track or road tilted at an angle $\theta$ so that a component of the normal force provides the centripetal force required to turn safely without relying entirely on friction, with optimal speed $v = \sqrt{g r \tan(\theta)}$.