Rotational and Circular Motion

Vocabulary flashcards covering rotational and circular motion, including angular displacement, angular velocity, angular acceleration, kinematic relationships, tangential and radial components, and centripetal acceleration.

Angular Displacement ($\Delta\theta$)

The angle swept out by a reference line fixed to a rigid object as a particle travels from position A to position B in a time interval $t$, given by $\Delta\theta = \theta_f - \theta_i$.

Radian ($\text{rad}$)

The SI unit of angular displacement, defined as the angle subtended at the center of a circle by an arc with a length equal to the radius of the circle, where $1\,\text{rad} = \frac{360^\circ}{2\pi} \approx 57.3^\circ$.

Right Hand Rule (for Angular Displacement)

The rule specifying direction by holding the axis of rotation in the right hand with fingers curling in the direction of rotation; the thumb points in the direction of angular displacement.

Angular Velocity ($\omega$)

The time rate of change of angular displacement of a body, defined by $\omega = \frac{\Delta\theta}{\Delta t}$, with the SI unit of radian per second ($\text{rad s}^{-1}$).

Average Angular Velocity ($\omega_{av}$)

The total angular displacement $\theta$ of a body divided by the total time $t$, given by $\omega_{av} = \frac{\theta}{t}$.

Instantaneous Angular Velocity ($\omega_{inst}$)

The limiting value of the ratio between small angular displacement $\Delta\theta$ and small time interval $\Delta t$ as the time approaches zero: $\omega_{inst} = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}$.

Angular Acceleration ($\alpha$)

The time rate of change of angular velocity, given by $\alpha = \frac{\Delta\omega}{\Delta t}$, with the SI unit of $\text{rad/s}^2$.

Average Angular Acceleration ($\alpha_{av}$)

The total angular velocity $\omega$ of a body divided by the total time $t$, given by $\alpha_{av} = \frac{\omega}{t}$.

Instantaneous Angular Acceleration ($\alpha_{inst}$)

The limiting value of the ratio between small change in angular velocity $\Delta\omega$ and small time interval $\Delta t$ as time approaches zero: $\alpha_{inst} = \lim_{\Delta t \to 0} \frac{\Delta\omega}{\Delta t}$.

Positive Angular Acceleration

The condition where the angular velocity of a body increases, causing angular velocity and angular acceleration to point in the same direction.

Negative Angular Acceleration

The condition where the angular velocity of a body decreases, causing angular velocity and angular acceleration to be anti-parallel.

Linear and Angular Displacement Relation

The geometric relationship connecting arc length (linear displacement $S$) to radius $r$ and angular displacement $\theta$ (in radians): $S = r\theta$.

Tangential Velocity ($v$)

The linear velocity directed along the tangent to the circular path, related to angular velocity by $v = r\omega$, or in vector form by $\mathbf{v}_{\text{tangent}} = \boldsymbol{\omega} \times \mathbf{r}$.

Tangential Component of Acceleration ($a_T$)

The component of acceleration parallel to linear instantaneous velocity that occurs due to a change in the magnitude of linear velocity, given by $a_T = r\alpha$.

Radial Component of Acceleration ($a_R$)

The component of acceleration along the radius of the circular path that arises due to a change in the direction of linear instantaneous velocity; also referred to as centripetal acceleration.

Resultant Linear Acceleration Magnitude

The total magnitude of linear acceleration combining both tangential and radial components: $a = \sqrt{a_T^2 + a_R^2}$.

Centripetal Acceleration

The acceleration produced by the continuous change in direction of velocity of a particle moving in a circular path of radius $r$ at constant speed, directed towards the center of the circle.

Uniform Circular Motion

Circular motion in which a particle travels at constant speed, where velocities at initial and final times differ only in direction while their magnitudes are equal: $|v_i| = |v_f| = v$.