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Unit 5: Rotational Motion

5.5: Rotational Equilibrium and Newton’s First Law in Rotational Form

Introduction

Rotational equilibrium occurs when an object is not rotating or is rotating at a constant angular velocity—similar to how an object in linear equilibrium has no net force acting on it. Newton’s First Law in rotational form states that an object at rest or in constant angular motion stays that way unless acted upon by a net torque. This concept applies to everyday scenarios like a balanced seesaw or a stationary Ferris wheel, and it’s critical for solving equilibrium problems on the AP Physics 1 Exam.

Key Concepts

  • Newton’s First Law for Rotation: If the net torque on an object is zero (Στ = 0), its angular velocity remains constant (including zero, meaning no rotation).
  • Rotational Equilibrium:
    • Condition: The sum of all torques about any axis must be zero (Στ = 0).
    • Implies no angular acceleration (α = 0).
  • Translational Equilibrium: For full equilibrium, the net force must also be zero (ΣF = 0), alongside zero net torque.
  • Sign Convention: Positive torques (counterclockwise) and negative torques (clockwise) must balance out.

Mathematical Routines

To analyze rotational equilibrium:

  • Choose a pivot point (axis of rotation)—often where forces act or where unknowns can be eliminated.
  • Calculate torques: τ = r × F × sinθ for each force, noting direction (positive or negative).
  • Set the sum of torques equal to zero: Στ = 0.
  • Check translational equilibrium (ΣF = 0) if forces are involved.
Tip: Pick the pivot wisely—choosing a point where an unknown force acts can simplify equations by making its lever arm zero, thus eliminating its torque.

Creating Representations

Diagrams are essential for equilibrium problems on the AP Exam. Practice:

  • Free-Body Diagrams: Draw the object, label the pivot, forces (with distances from the pivot), and torque directions (curved arrows for clockwise/counterclockwise).
  • Force and Torque Tables: List forces, their lever arms, and resulting torques to ensure Στ = 0.
Practical Reminder: Clearly label distances and force angles in your diagram to avoid mistakes when calculating torques.

Scientific Questioning & Argumentation

AP Physics 1 emphasizes reasoning. Practice questions like:

  • “Why must the net torque be zero for an object to remain at rest?” Answer: If Στ ≠ 0, there’s an angular acceleration, causing rotation per τ = Iα.
  • “How does the choice of pivot affect equilibrium analysis?” Answer: The condition Στ = 0 holds for any pivot; different pivots just change the equations but not the physical outcome.

Support answers with Στ = 0, diagrams, and torque calculations.

Exam Tip: In free-response questions, justify your pivot choice and show how torques balance, using both math and physical reasoning.

Practice Activities

Activity 1: Balancing a Seesaw

A 20 kg child sits 2 m from the pivot of a seesaw. Where must a 30 kg child sit on the opposite side to balance it? Assume the seesaw is massless. Draw a diagram and calculate.

Activity 2: Equilibrium with Multiple Forces

A 5 m long, 100 N beam is pivoted at one end. A 200 N force acts downward 2 m from the pivot, and an unknown upward force acts at the other end. Find the unknown force for equilibrium. Draw a diagram showing forces and torques.

Summary & Exam Preparation Tips

Rotational equilibrium requires Στ = 0 (Newton’s First Law for rotation), often alongside ΣF = 0 for full stability. Key points:

  • Calculate torques with τ = r × F × sinθ and ensure they sum to zero.
  • Use diagrams to track forces and lever arms.
  • Practice choosing pivots to simplify problem-solving.

For the AP Exam, master equilibrium problems, show all steps, and explain your reasoning with diagrams and equations.