Rotational Dynamics Formulas

Why This Matters

Rotational dynamics is where everything you learned about linear motion gets a powerful upgrade. The AP exam tests whether you can see the deep parallels between translational and rotational physics—force becomes torque, mass becomes moment of inertia, velocity becomes angular velocity—and apply these relationships to solve complex problems involving spinning objects, rolling motion, and conservation laws. Unit 5 builds directly on your understanding of forces from Unit 2, while Unit 6 extends energy and momentum concepts into rotating systems.

You're being tested on your ability to connect linear and rotational quantities, apply Newton's second law in rotational form, and use conservation of angular momentum and energy to analyze real physical systems. Don't just memorize these formulas—understand why each one mirrors its linear counterpart and when to apply each relationship. The FRQs love to combine rotation with translation (think rolling objects), so knowing what concept each formula represents will save you on exam day.

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The Rotational Analog of Newton's Second Law

Just as $F = ma$ governs linear motion, rotational motion has its own fundamental law connecting cause (torque) to effect (angular acceleration). The key insight is that moment of inertia plays the role of mass—it determines how hard it is to change an object's rotational state.

Torque

• $\tau = r \times F$ or $\tau = rF\sin\theta$—torque is the rotational equivalent of force, causing angular acceleration about an axis
• Lever arm $r_\perp = r\sin\theta$ determines effectiveness; force applied farther from the pivot or more perpendicular to the radius produces greater torque
• Right-hand rule gives direction: curl fingers from $\vec{r}$ to $\vec{F}$, thumb points along torque vector (positive or negative about axis)

Angular Acceleration

• $\alpha = \frac{\tau_{net}}{I}$—this is Newton's second law for rotation, the most important equation in Unit 5
• Directly proportional to net torque and inversely proportional to moment of inertia; same torque produces less acceleration for objects with mass distributed far from the axis
• Units are $\text{rad/s}^2$—always check that your torque is in N·m and moment of inertia in kg·m²

Moment of Inertia

• $I = \sum m_i r_i^2$ for discrete masses or $I = \int r^2 \, dm$ for continuous objects—quantifies rotational inertia
• Mass distribution matters: mass farther from the axis contributes more to $I$ (this is why a hoop has greater $I$ than a disk of equal mass and radius)
• Parallel-axis theorem $I = I_{cm} + Md^2$ lets you find $I$ about any axis parallel to one through the center of mass

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Rotational Kinematics: Describing the Motion

Before analyzing why things rotate, you need to describe how they rotate. These kinematic quantities have direct linear analogs and follow the same mathematical relationships for constant angular acceleration.

Angular Velocity

• $\omega = \frac{d\theta}{dt}$—the rate of change of angular position, measured in rad/s
• Sign convention: positive typically means counterclockwise when viewed from above; be consistent with your chosen axis direction
• Connects to linear velocity via $v = r\omega$ for any point at distance $r$ from the rotation axis

Angular Displacement

• $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$—for constant angular acceleration, directly analogous to $x = v_0 t + \frac{1}{2}at^2$
• Measured in radians—essential for using $s = r\theta$ to convert to arc length
• Full kinematic equations apply: $\omega^2 = \omega_0^2 + 2\alpha\theta$ and $\omega = \omega_0 + \alpha t$ complete the set

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Conservation Laws in Rotation

Conservation principles are your most powerful problem-solving tools. When external torques or non-conservative forces are absent, these quantities remain constant—giving you equations without needing to know the details of what happens during a collision or interaction.

Angular Momentum

• $L = I\omega$ for rigid bodies rotating about a fixed axis—the rotational analog of linear momentum $p = mv$
• Vector quantity: direction follows right-hand rule along the rotation axis; changes in $L$ require external torques
• For a point mass: $L = mvr\sin\theta$ or $\vec{L} = \vec{r} \times \vec{p}$—useful for objects in orbit or moving past a pivot

Conservation of Angular Momentum

• $L_i = L_f$ or $I_1\omega_1 = I_2\omega_2$ when net external torque is zero—the foundation of countless exam problems
• Explains the figure skater effect: decreasing $I$ (pulling arms in) increases $\omega$ to keep $L$ constant
• Applies to collisions involving rotation: if no external torques act during impact, total angular momentum is conserved

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Energy in Rotating Systems

Energy methods often provide the fastest path to a solution, especially when you don't care about time or acceleration. Rotational kinetic energy adds to your energy toolkit and is essential for problems involving rolling motion.

Rotational Kinetic Energy

• $K_{rot} = \frac{1}{2}I\omega^2$—energy stored in rotational motion, analogous to $K_{trans} = \frac{1}{2}mv^2$
• Rolling objects have both: total $K = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2$; use $v_{cm} = R\omega$ for rolling without slipping
• Different shapes roll differently: objects with larger $I$ (relative to $mR^2$) have more energy in rotation and less in translation at the same speed

Work-Energy Theorem for Rotation

• $W = \Delta K_{rot}$—work done by net torque equals change in rotational kinetic energy
• $W = \int \tau \, d\theta$ or $W = \tau \Delta\theta$ for constant torque—directly analogous to $W = F \cdot d$
• Connects to conservation of energy: include $K_{rot}$ in your energy bookkeeping for any rotating system

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Power and Rate of Energy Transfer

Power tells you how quickly energy is being transferred or work is being done. In rotating systems, this connects torque and angular velocity—critical for analyzing motors, engines, and any system where efficiency matters.

Power in Rotational Motion

• $P = \tau\omega$—instantaneous power delivered to a rotating object, analogous to $P = Fv$
• Units are watts (J/s)—same as linear power; this formula assumes torque and angular velocity are about the same axis
• Essential for efficiency problems: compare power input to useful power output to find energy losses

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Quick Reference Table

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Self-Check Questions

1. A disk and a hoop of equal mass and radius roll down the same incline. Which reaches the bottom first, and why in terms of how each distributes its kinetic energy?

2. An ice skater spinning with arms extended pulls her arms in. Using conservation of angular momentum, explain what happens to her angular velocity and rotational kinetic energy. Does energy conservation seem violated? Why or why not?

3. Two formulas both involve $I$ and $\omega$: $L = I\omega$ and $K = \frac{1}{2}I\omega^2$. Under what conditions is each quantity conserved, and what external influence would change each one?

4. A constant torque of 8 N·m acts on a wheel with moment of inertia 2 kg·m². Calculate the angular acceleration, and determine how much work is done after the wheel rotates through 10 radians from rest.

5. Compare the parallel-axis theorem to the concept of adding masses in a system. When would you need to use $I = I_{cm} + Md^2$, and how does this relate to physical pendulum problems?