The Physics
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Opus in profectus

Rotational Statics

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Discussion

Equilibrium of Extended Bodies

Discuss.

∑τ⟲ = ∑τ⟳

  1. counter clockwise is positive
  2. clockwise is negative
Translational and rotational quantities compared
concept translation connection rotation
equilibrium
∑F = 0 ⇒  ∑F+x = ∑F−x
∑F+y = ∑F−y
∑F+z = ∑F−z
∑τ = 0 ⇒  ∑τ+x = ∑τ−x
∑τ+y = ∑τ−y
∑τ+z = ∑τ−z

Center of mass

The center of mass is computed from the mass distribution. The center of gravity is computed from the weight distribution. The distinction between the two is suaully unimportant.

Discrete.

rcm =  ∑miri
∑mi

Continuous.

rcm =  1 ⌠⌠⌠
⌡⌡⌡		 r ρ dV
m

Continuous and uniform.

rcm =  1 ⌠⌠⌠
⌡⌡⌡		 r dV
V

stability

Stability of Equilibrium

Stability of Equilibrium
stable
equilibrium		 unstable
equilibrium		 neutral
equilibrium
∑F(x + ∆x) ∝ −∆x
restoring force		 ∑F(x + ∆x) ∝ +∆x
repelling force		 ∑F(x + ∆x) = 0
no force
d2U/dx2 > 0
concave up		 d2U/dx2 < 0
concave down		 d2U/dx2 = 0
flat
center of gravity is below pivot center of gravity is above pivot center of gravity is at the pivot
small displacement raises center of gravity small displacement lowers center of gravity small displacement does not raise or lower center of gravity