PHYS 1500 Torque and Equilibrium Notes

Torque and Related Concepts

Key Vocabulary

  • Torque: A measure of the rotational force acting on an object, defined mathematically by the equation ⃗τ = rF sin ϕ.
  • Moment of Inertia: A scalar quantity that determines the torque needed for a desired angular acceleration about a rotational axis; represented as I in the equation ⃗τA = IA⃗αA.
  • Rotation Axis: The line about which an object rotates; critical to calculate torque.
  • Lever Arm: The perpendicular distance from the axis of rotation to the line of action of the force; denoted as ⃗r⊥.
  • Center of Gravity: The point where the weight of an object is equally distributed in all directions; affects stability and torque calculations.

Types of Equilibrium

  • Stable Equilibrium: An object returns to its original position after being displaced slightly; e.g., a ball in a bowl.
  • Unstable Equilibrium: An object moves away from its original position when slightly displaced; e.g., a pencil balanced on its tip.
  • Neutral Equilibrium: An object remains in its new position after being displaced; e.g., a ball on a flat surface.

Important Equations

  • Torque Equation:
    ⃗τ = rF sin ϕ

  • r: Distance from the rotation axis to the point of force application

  • F: Magnitude of the force

  • ϕ: Angle between the lever arm and the force direction

  • Torque and Angular Acceleration:
    ⃗τA = IA⃗αA

  • IA: Moment of inertia of object A

  • αA: Angular acceleration of object A

Questions & Concept Explanations

  1. Forces and Torques:
  • Similar: Both cause motion/change in state; both rely on distance (lever arm for torque).
  • Different: Forces produce linear motion; torques produce rotational motion.
  1. Cheater Bar Mechanics:
  • A longer handle increases the lever arm, reducing the force needed to apply the same torque, making it easier to loosen bolts.
  1. Force-Torque Examples:
  • Large torque from small force: Using a long wrench on a bolt.
  • Large force generating small torque: A heavy person pushing downward at the center of a merry-go-round.
  • Large force generating no torque: A force applied directly at the axis of rotation (e.g., a person lifting a barbell straight up).
  1. Torque Calculation Guidelines:
  • Use position vector ⃗r for non-perpendicular forces.
  • Use lever arm ⃗r⊥ for perpendicular forces to simplify calculations.
  • Use force F for total force in torque calculation; use perpendicular component F⃗⊥ for specific components.
  1. Effect of Rotation Axis on Torque:
  • The choice of axis does not change the result of the sum of torques in static equilibrium, but the individual values may vary depending on the axis chosen.
  1. Equilibrium Differences:
  • Stable example: A pendulum (returns to the middle)
  • Unstable example: A pyramid with the apex at the top (falls over easily)
  • Neutral example: A marble on a horizontal surface (stays wherever it moves).

Introductory Problems

  1. Leg Movement and Torque Calculation:
  • Sketch your leg in a forward position to visualize forces:
  • Include position vector ⃗rG for gravitational force.
  • Draw gravitational force F⃗G.
  • Label angle ϕG used in the torque calculation.