AP Physics C: Rotational Kinematics Review Notes

Angular velocity is defined in two ways: average and instantaneous.
Average Angular Velocity: $ω_{average} = \frac{\Delta \theta}{\Delta t}$
Instantaneous Angular Velocity: $ω_{instantaneous} = \frac{d\theta}{dt}$
Common units for angular velocity include radians per second ( $\text{rad/s}$ ) and revolutions per minute ( $\text{rev/min}$ ).

Angular acceleration describes the rate of change of angular velocity.
Average Angular Acceleration: $\alpha_{average} = \frac{\Delta \omega}{\Delta t}$
Instantaneous Angular Acceleration: $\alpha_{instantaneous} = \frac{d\omega}{dt}$
The unit for angular acceleration is radians per second squared ( $\text{rad/s}^2$ ).

Uniformly Angularly Accelerated Motion occurs when the angular acceleration is constant ( $\alpha = \text{constant}$ ).
This motion involves five variables: $\Delta \theta$ (angular displacement), $\omega_i$ (initial angular velocity), $\omega_f$ (final angular velocity), $\alpha$ (angular acceleration), and $\Delta t$ (time interval).
There are four primary kinematic equations for UαM. If three variables are known, the other two can be calculated using these equations: - $\omega_f = \omega_i + \alpha \Delta t$ - $\Delta \theta = \omega_i \Delta t + \frac{1}{2} \alpha \Delta t^2$ - $\omega_f^2 = \omega_i^2 + 2 \alpha \Delta \theta$ - $\Delta \theta = \frac{1}{2}(\omega_i + \omega_f) \Delta t$

Arc length (represented by the symbol $s$ ) is the linear distance traveled when an object moves along a circle or a portion of a circle. It is defined as the linear length of the traveled arc.
Key Equation: $s = r\Delta \theta$ , where arc length equals the radius ( $r$ ) of the object multiplied by the angular displacement ( $\Delta \theta$ ).
Important Constraints and Units for Arc Length: - Radians must be used for $\Delta \theta$ when applying this equation. - Arc length is a linear dimension measuring physical distance, with units such as meters ( $\text{m}$ ). - This equation is not provided on the standard AP equation sheet. - The lecturer uses a lowercase cursive $ℯ$ for arc length because a standard printed $s$ often resembles the number $5$ .
Example of Arc Length: The formula for circumference ( $C$ ) is a specific instance of the arc length formula where the angular displacement is one full revolution, or $2\pi$ radians: $C = r(2\pi)$ .
Angular Conversion: $1 \text{ revolution} = 360^\circ = 2\pi \text{ radians}$ .

Tangential Velocity ( $v_t$ ): This is the linear velocity of an object moving in a circular path. - Equation: $v_t = r\omega$ - Derivation: Taking the derivative of the arc length equation with respect to time: $\frac{d}{dt}(s = r\Delta \theta) \rightarrow \frac{ds}{dt} = r \frac{d\theta}{dt} \rightarrow v_t = r\omega$ . - This equation is included on the AP equation sheet. - Units are meters per second ( $\text{m/s}$ ).
Tangential Acceleration ( $a_t$ ): This is the linear acceleration of an object moving in a circular path. - Equation: $a_t = r\alpha$ - Derivation: Taking the derivative of the tangential velocity equation with respect to time: $\frac{d}{dt}(v_t = r\omega) \rightarrow \frac{dv_t}{dt} = r \frac{d\omega}{dt} \rightarrow a_t = r\alpha$ . - This equation is not included on the AP equation sheet. - Units are meters per second squared ( $\text{m/s}^2$ ).
Constraints for Tangential Equations: - Both formulas assume the radius ( $r$ ) of the circular path remains constant. - Radians must be used for angular velocity ( $\omega$ ) and angular acceleration ( $\alpha$ ). - Tangential quantities are geometrically tangent to the circle, meaning they are always perpendicular to the radius of the circle at that point.

Uniform Circular Motion defines a scenario where objects move in a circle with an angular acceleration of zero ( $\alpha = 0$ ).
Velocity and Acceleration: - The magnitude of the velocity (speed) does not change. - However, the direction of the velocity is constantly changing, meaning the velocity vector is not constant. - Because the velocity is changing, an acceleration must exist. This specific acceleration is called centripetal acceleration ( $a_c$ ).
Centripetal Acceleration ( $a_c$ ): - Formula: $a_c = \frac{v_t^2}{r} = r\omega^2$ - Units are meters per second squared ( $\text{m/s}^2$ ). - Direction: Centripetal translates to "center seeking," and the acceleration vector is always directed toward the center of the circle described by the object's path.

According to Newton’s Second Law ( $\sum F = ma$ ), an acceleration requires a net force. In circular motion, the centripetal acceleration necessitates a centripetal force ( $F_{in}$ ).
Formula: $F_{in} = ma_c$
Characteristics of Centripetal Force: - Centripetal force is defined as the net force acting in the "in" direction (toward the center). - It is not a distinct or "new" physical force (like gravity or friction); rather, it is a category for the net center-seeking force. - It should never be drawn as a separate force on a Free Body Diagram (FBD). - Sign Convention: The "in" direction toward the center is defined as positive, while the "out" direction away from the center is negative. - Example: See the "conical pendulum" example from AP Physics 1 Kinematics Review for practical application.

Non-Uniform Circular Motion occurs when the angular acceleration is non-zero ( $\alpha \neq 0$ ).
Acceleration Components: - There is a tangential acceleration ( $a_t$ ) which is parallel to the tangential velocity. - There is a centripetal acceleration ( $a_c$ ) which is normal (perpendicular) to the tangential acceleration.
Net Acceleration: The total or net acceleration vector for an object in non-uniform circular motion is the vector sum of these components: $\mathbf{a}_{net} = \mathbf{a}_c + \mathbf{a}_t$

The period ( $T$ ) of an object moving in a circle is defined as the time required to complete one full revolution.
Derivation: Based on $\omega = \frac{\Delta \theta}{\Delta t}$ , for one revolution $\Delta \theta = 2\pi$ and $\Delta t = T$ .
Formula: $\omega = \frac{2\pi}{T} \rightarrow T = \frac{2\pi}{\omega}$
The period is measured in units of seconds ( $\text{s}$ ).