Unit 5.6: Newton’s Second Law in Rotational Form

Introduction

Newton’s Second Law in rotational form, Στ = Iα, describes how torque causes angular acceleration, much like F = ma relates force to linear acceleration. This law is crucial for understanding rotational dynamics—think of a spinning top slowing down due to friction or a wheel accelerating under a motor’s torque. Mastering this concept will help you analyze rotational motion problems effectively on the AP Physics 1 Exam.

Key Concepts

Newton’s Second Law for Rotation: The net torque (Στ) equals the product of rotational inertia (I) and angular acceleration (α): Στ = Iα.
Torque (τ): Rotational force, calculated as τ = r × F × sinθ. Sum all torques, accounting for direction (positive for counterclockwise, negative for clockwise).
Rotational Inertia (I): Resistance to angular acceleration, depending on mass and its distribution from the axis (e.g., I = ½MR² for a solid disk).
Angular Acceleration (α): Rate of change of angular velocity, measured in rad/s². Relates to linear acceleration via a = rα.

Mathematical Routines

To apply Στ = Iα:

Identify the axis of rotation.
Calculate rotational inertia (I) using the object’s shape and mass distribution.
Determine the net torque (Στ) by summing individual torques with correct signs.
Solve for angular acceleration: α = Στ / I.
Link to linear motion if needed using a = rα.

Tip: Double-check torque directions and ensure I matches the axis of rotation—mistakes here are common exam pitfalls.

Creating Representations

Visualizing rotational dynamics is key for the AP Exam. Practice:

Diagrams: Draw the object, label the axis, forces, lever arms, and torque directions (curved arrows for clockwise/counterclockwise).
Graphs: Sketch angular velocity vs. time to show constant α (a straight slope) under constant net torque.

Practical Reminder: Use diagrams to verify torque signs—clockwise torques should oppose counterclockwise ones in your sum.

Scientific Questioning & Argumentation

The AP Exam tests reasoning skills. Practice questions like:

“Why does a smaller rotational inertia lead to a larger angular acceleration for the same torque?” Answer: From Στ = Iα, if I decreases, α increases for constant Στ.
“How does mass distribution affect rotational motion?” Answer: More mass farther from the axis increases I, reducing α for a given torque.

Use Στ = Iα and diagrams to justify your reasoning.

Exam Tip: For free-response questions, explain how I and Στ determine α, tying calculations to physical principles like mass distribution.

Practice Activities

Activity 1: Angular Acceleration of a Disk

A solid disk with a mass of 4 kg and radius of 0.5 m has a net torque of 10 N·m applied about its center. Calculate its angular acceleration. Draw a diagram showing the setup.

Activity 2: Comparing Objects

A 2 kg hoop (radius 0.3 m) and a 2 kg solid disk (radius 0.3 m) each experience a net torque of 6 N·m. Calculate α for each and explain why they differ. Include diagrams.

Summary & Exam Preparation Tips

Newton’s Second Law for rotation, Στ = Iα, connects torque, rotational inertia, and angular acceleration. Key points:

Calculate I based on shape and mass distribution.
Sum torques (Στ) with correct signs and solve for α.
Use diagrams to visualize forces and torques.

For the AP Exam:

Practice problems linking torque to angular acceleration.
Check units (N·m for torque, kg·m² for I, rad/s² for α).
Justify answers with Στ = Iα and clear diagrams.

Regular practice will build confidence in solving rotational dynamics questions effectively.