Rotational Statics

Summary

• Statics is the branch of mechanics dealing with forces acting on objects that are not accelerating.
• Equilibrium is a general concept that can refer to states of…
  • static equilibrium, which includes…
    • translational equilibrium (the equilibrium of forces)
    • rotational equilibrium (the equilibrium of torques)
  • dynamic equilibrium, which includes but is not limited to…
    • thermodynamic equilibrium (the equilibrium of internal energy transfer)
• An adjective before the noun is not needed when the type of equilibrium is known from context.
  • This section of this book is about static equilibrium only.
  • Dynamic equilibrium will be dealt with in other sections of this book.
• Static equilibrium (a.k.a. the equilibrium of extended bodies)
  • The translational and angular acceleration of an extended body in static equilibrium are both zero.
    • An object in static equilibrium will…
      • remain at rest or continue moving with a constant translational velocity and
      • remain at rest or continue rotating with a constant angular velocity.
  • An extended body is in static equilibrium when the net force and net torque on it are zero.

    no net force

    ∑F−x = ∑F+x		⎫ ⎪ ⎬ ⎪ ⎭	⇒	∑F = 0
    ∑F−y = ∑F+y
    ∑F−z = ∑F+z

    no net torque

    ∑τ−x = ∑τ+x		⎫ ⎪ ⎬ ⎪ ⎭	⇒	∑τ = 0
    ∑τ−y = ∑τ+y
    ∑τ−z = ∑τ+z

  • An extended body is in static equilibrium when the forces and torques on it are balanced.

    balanced forces

    ∑F−x = ∑F+x		⎫ ⎪ ⎬ ⎪ ⎭	⇒	∑F = 0
    ∑F−y = ∑F+y
    ∑F−z = ∑F+z

    balanced torques

    ∑τ−x = ∑τ+x		⎫ ⎪ ⎬ ⎪ ⎭	⇒	∑τ = 0
    ∑τ−y = ∑τ+y
    ∑τ−z = ∑τ+z

• Center of mass (center of gravity)
  • The center of mass of an object…
    • is the effective location of an extended body for for the purposes of some calculations
    • is the point about which an extended body will always rotate when free of external torques
    • is computed from the mass distribution
    • may lie outside of an extended body
    • is represented by the symbol r_cm or r_com
  • The center of gravity of an object…
    • is the effective location for the force of gravity acting on an extended body
    • is the pivot about which an extended body will be in stable rotational equilibrium in a gravitational field
    • is computed from the weight distribution
    • is in the same location as the center of mass when the gravitational field is uniform
    • is represented by the symbol r_cg or r_cog
  • Equations
    • For a discrete collection of objects

      rcm =	∑miri	=	1	⎛ ⎜ ⎝	∑mixi î + ∑miyi ĵ + ∑mizi k̂	⎞ ⎟ ⎠
      	∑mi		m

      Where…

      rcm =	the location of the center of mass for the set of objects
      mi =	the mass of the ith object in the set
      ri =	the position vector of the ith object in the set
      m =	the total mass of the set
      xi, yi, zi =	the coordinates of the ith object in the set
      î, ĵ, k̂ =	the unit vectors of the coordinate system used

    • For a single object with…

      uniform density

      rcm =	1	⌠⌠⌠ ⌡⌡⌡	r dV
      	V

      nonuniform density

      rcm =	1	⌠⌠⌠ ⌡⌡⌡	ρr dV
      	m

      Where…

      rcm =	the location of the center of mass of the object
      m =	the total mass of the object
      V =	the total volume of the object
      r =	the position vector of any infinitesimal piece of the object
      dV =	an infinitesimal volume element in the object
      ρ =	a density function

• Stability of equilibrium for extended bodies
  • The equilibrium of an extened body in a gravitational field is said to be…
    • stable if…
      • any small disturbance results in a net torque that tries to restore its equilibrium orientation
      • any small disturbance would raise its center of mass
      • its center of gravity is below the pivot point
    • unstable if…
      • any small disturbance results in a net torque that drives it away from its equilibrium orientation
      • any small disturbance would lower its center of mass
      • its center of gravity is above the pivot point
    • neutral if…
      • any small disturbance does not result in a net force
      • any small disturbance would neither raise nor lower its center of mass
      • its center of gravity is at the pivot point