Mastering Rotational Motion: Variables and Kinematics

Rigid Body Rotation

A rotation where objects do not stretch or deform and all points rotate with the same angular velocity.

Angular Position ($\theta$)

The angle through which a point, line, or body is rotated about a specified axis, measured in radians.

Standard Unit of Angular Position

Radians (rad) are the standard unit, though degrees and revolutions are often used.

Counter-Clockwise (CCW) Sign Convention

CCW rotation is considered positive (+).

Clockwise (CW) Sign Convention

CW rotation is considered negative (-).

Angular Velocity ($\omega$)

The rate of change of angular position, measured in radians per second (rad/s).

Angular Acceleration ($\alpha$)

The rate of change of angular velocity, measured in radians per second squared (rad/s²).

First Kinematic Equation for Angular Motion

\omegaf = \omegai + \alpha t is the first rotational kinematic equation.

Second Kinematic Equation for Angular Motion

\Delta \theta = \omega_i t + \frac{1}{2}\alpha t^2 is the second rotational kinematic equation.

Third Kinematic Equation for Angular Motion

\omegaf^2 = \omegai^2 + 2\alpha \Delta \theta is the third rotational kinematic equation.

Arc Length ($s$)

Distance traveled on the rim of a rotating object: s = r\Delta\theta.

Tangential Velocity ($v_t$)

Instantaneous linear speed of a point on the edge of a rotating object: v_t = r\omega.

Tangential Acceleration ($a_t$)

Acceleration along the circular path, linked to the angular acceleration: a_t = r\alpha.

Centripetal Acceleration ($a_c$)

Acceleration directed toward the center of the circular path, measured as a_c = \frac{v^2}{r} = \omega^2 r.

Total Linear Acceleration

The vector sum of tangential and centripetal acceleration, which are perpendicular to each other.

Rolling Without Slipping

Condition where linear distance traveled equals arc length, leading to $\Delta x_{cm} = R \Delta \theta$.

Linear Speed of Center of Mass ($v_{cm}$)

Linked to angular velocity: v_{cm} = R\omega.

Linear Acceleration of Center of Mass ($a_{cm}$)

Linked to angular acceleration: a_{cm} = R\alpha.

Angular Deceleration

When an object slows down its rotation, it has a negative angular acceleration ($\alpha < 0$).

Key Conversion Error

Remember that formulas involving angular motion necessitate angles in radians, not degrees or revolutions.

Centripetal vs Tangential Acceleration

Centripetal points towards the center; tangential affects the magnitude of velocity.

Sign Convention Consistency

Be consistent with rotation direction; CCW is often positive, CW is negative.

Frequency and Angular Velocity Relation

Frequency ($f$) in Hertz is related to angular velocity by $\omega = 2\pi f$.

Angular Position Change ($\Delta \theta$)

Measured in radians and represents the total angle rotated over a time interval.

Instantaneous Angular Velocity ($\omega$)

Represents the angular speed at a specific instant in time.

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

Defined as \omega_{avg} = \frac{\Delta \theta}{\Delta t} for the average over a time interval.