Rotational Forms of Work and Momentum

Rotational Work and Energy Transfer

• Energy Transfer: A torque can transfer energy into or out of a rigid system if it acts over a change in angular position or angular displacement ($\Delta \theta$).

• Rotational Work Equation: $W = \tau \times \Delta \theta$     * In AP Physics 1, the $\cos(\theta)$ term is omitted because torque and angular displacement are restricted to clockwise or counterclockwise directions.

• Sign Conventions:     * Positive Work: Torque and angular displacement are in the same direction.     * Negative Work: Torque and angular displacement are in opposite directions.

Graphical Representation of Work

• Force vs. Position Graph: Work is the signed area under the curve; area above the horizontal axis is positive, and area below is negative.

• Torque vs. negative position Graph: Work is the signed area under the curve.

Linear and Angular Momentum

• Linear Momentum ($p$): $p = m \times v$     * Values for $p$ and $v$ are vectors.

• Angular Momentum ($L$): There are two distinct equations depending on the system:     * Rigid Object with Shape: $L = I \times \omega$ (relative to an axis of rotation).     * Point Particle: $L = r \times m \times v \times \sin(\theta)$         * $r$: Distance from the axis of rotation or reference point to the particle.         * $m$: Mass of the particle.         * $v$: Magnitude of the particle's velocity.         * $\theta$: The angle between $r$ and $v$.

Questions & Discussion

• Bo's Translation of Work: Bo translated linear work ($W = F \times \Delta x \times \cos(\theta)$) to rotational work ($W = \tau \times \Delta \theta$). Bo noted that since rotational directions are limited to clockwise or counterclockwise in AP Physics 1, the $\cos(\theta)$ component is unnecessary.

• Mr. P's Correction on Sign: Mr. P emphasized that one must be careful with directions to determine if work done by torque is positive or negative.

• Bobby's Critique of Variables: Bobby noted that the letter choices for linear momentum ($p$) and angular momentum ($L$) do not intuitively match their names.

• Definition of Reference Points: Bobby and Mr. P clarified that the angular momentum of a point particle must always be calculated relative to an axis of rotation, which is also referred to as a reference point.