AP Physics 1- 5.6 Newton’s Second Law in Rotational Form- Study Notes- New Syllabus

Newton’s Second Law in Rotational Form

Newton’s second law for rotation relates the net external torque acting on a rigid body to its angular acceleration. It is the rotational analog of Newton’s second law of linear motion.

\( \sum \tau = I \alpha \)

Where:

• \( \tau \) = Torque (Nm)
• \( I \) = Moment of inertia (kg·m²)
• \( \alpha \) = Angular acceleration (rad/s²)

Key Points:

• Just as \( F = ma \) governs linear motion, \( \tau = I \alpha \) governs rotational motion.
• The greater the rotational inertia \( I \), the harder it is to change the angular motion of the system.
• When torque is zero, angular acceleration is zero, meaning the object rotates with constant angular velocity.
• When torque is nonzero, the object experiences angular acceleration proportional to the net torque.

Relation with Linear Form:

Linear: \( F = m a \)    ↔    Rotational: \( \tau = I \alpha \)

Graphical Representation:

• The slope of an angular velocity vs. time graph gives angular acceleration \( \alpha \).
• The slope of an angular momentum vs. time graph gives the net torque \( \tau \).

Conditions for Change in Angular Velocity:

• A system’s angular velocity changes only when a net external torque acts on it.
• If \( \sum \tau \neq 0 \), then \( \alpha \neq 0 \), meaning angular velocity changes with time.
• If \( \sum \tau = 0 \), then \( \alpha = 0 \), and the system rotates with constant angular velocity (or remains at rest).