Rotational Dynamics Formulas to Know for AP Physics C: Mechanics (2025)

Rotational dynamics focuses on how objects rotate and the forces that cause this motion. Key concepts include torque, angular acceleration, and moment of inertia, which help explain how rotational motion behaves and how energy is conserved in these systems.

1. Torque: τ = r × F

   • Torque is the rotational equivalent of linear force, causing an object to rotate about an axis.
   • The direction of torque is determined by the right-hand rule.
   • Torque depends on both the magnitude of the force and the distance from the pivot point (lever arm).

2. Angular acceleration: α = τ / I

   • Angular acceleration measures how quickly an object's rotational speed changes.
   • It is directly proportional to the applied torque and inversely proportional to the moment of inertia.
   • A larger moment of inertia means more torque is needed to achieve the same angular acceleration.

3. Moment of inertia: I = Σ mr²

   • Moment of inertia quantifies an object's resistance to changes in its rotational motion.
   • It depends on the mass distribution relative to the axis of rotation; mass farther from the axis increases I.
   • Different shapes have specific formulas for calculating their moment of inertia.

4. Angular momentum: L = I ω

   • Angular momentum is a measure of the rotational motion of an object and is conserved in a closed system.
   • It is the product of the moment of inertia and angular velocity.
   • Changes in angular momentum occur due to external torques acting on the system.

5. Conservation of angular momentum: L₁ = L₂

   • In the absence of external torques, the total angular momentum of a system remains constant.
   • This principle explains phenomena such as a figure skater spinning faster by pulling in their arms.
   • It applies to both isolated systems and systems with internal interactions.

6. Rotational kinetic energy: KE = ½ I ω²

   • Rotational kinetic energy is the energy due to an object's rotation.
   • It is analogous to translational kinetic energy but incorporates the moment of inertia and angular velocity.
   • Understanding this concept is crucial for solving problems involving energy conservation in rotational systems.

7. Work-energy theorem for rotation: W = ΔKE

   • The work done on a rotating object results in a change in its rotational kinetic energy.
   • This theorem connects the concepts of work and energy in rotational dynamics.
   • It can be used to analyze systems where forces cause rotational motion.

8. Power in rotational motion: P = τ ω

   • Power in rotational motion measures the rate at which work is done or energy is transferred.
   • It is the product of torque and angular velocity, indicating how quickly energy is being converted.
   • Understanding power is essential for analyzing the efficiency of rotating systems.

9. Angular velocity: ω = dθ/dt

   • Angular velocity describes how fast an object rotates and is measured in radians per second.
   • It is the rate of change of angular displacement over time.
   • Angular velocity can be positive or negative, indicating the direction of rotation.

10. Angular displacement: θ = ½ α t²

• Angular displacement measures the angle through which an object has rotated in a given time.
• It is influenced by angular acceleration and time, particularly in uniformly accelerated rotational motion.
• This formula is useful for solving problems involving rotational motion under constant angular acceleration.