Static Equilibrium Notes

Lesson Outcome

  • Describe the conditions of static equilibrium.
  • Solve simple problems regarding static equilibrium.

Static Equilibrium

  • Definition: An object at rest experiencing no net force or net torque.
  • This means the object maintains a constant position without acceleration.
  • All forces are balanced, with no translational or rotational motion.

Conditions of Static Equilibrium

  • Condition 1: The vector sum of all forces acting on the object must be zero.
    • \sum F = 0
  • Condition 2: The sum of all torques about any axis acting on the object must be zero.
    • \sum \tau = 0

Translational Motion

  • \sum F = 0 implies no acceleration (a = 0) and constant velocity.

Rotational Motion

  • \sum \tau_{tot} = 0 implies no angular acceleration (\alpha = 0).
  • Torque is defined as t = r \times F.
  • Angular momentum is defined as L = r \times p.

Center of Mass (CM)

  • An object can be divided into many small particles, each with specific mass and coordinates.
  • The x-coordinate of the center of mass is given by: (Similar expressions exist for y-coordinates).

Center of Gravity (CG)

  • All gravitational forces acting on mass elements are equivalent to a single gravitational force through the center of gravity (CG).

Axis of Rotation

  • The choice of axis for calculating torques is arbitrary.
  • If an object is in translational equilibrium and the net torque is zero about one axis, it is zero about any other axis.
  • Choose a rotation axis to simplify problems.

Problem-Solving Strategy

  • Choose a convenient axis for calculating net torque.
  • Choose an origin that simplifies calculations i.e a force that acts along a line passing through the origin produces a zero torque.
  • Be careful of sign with respect to rotational axis:
    • Positive if force tends to rotate object in counter-clockwise (CCW) direction.
    • Negative if force tends to rotate object in clockwise (CW) direction.
    • Zero if force is on the rotational axis.
  • Apply the second condition for equilibrium.
  • Solve the equations simultaneously.