Rotational Equilibrium

Discussion

Equilibrium of Extended Bodies

Translational and Rotational Quantities Compared
  translational connections rotational
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.

rcm =  ∑ miri  = (x, y, z)   rcg =  1 ⌠⌠⌠
⌡⌡⌡		 r dW =  1 ⌠⌠⌠
⌡⌡⌡		 r dV = (x, y, z)
∑ mi W V

The center of gravity is computed from the weight distribution.

rg =  ∑ Wiri  = (x, y, z)   rcg =  1 ⌠⌠⌠
⌡⌡⌡		 r dW =  1 ⌠⌠⌠
⌡⌡⌡		 r dV = (x, y, z)
∑ Wi W V

Stability of Equilibrium

Stability of Equilibrium
equilibrium
condition		 stable
equilibrium		 unstable
equilibrium		 neutral
equilibrium
∑ F = 0
no net force		 ∑ F(x + Δx) ∝ −Δx
restoring force		 ∑ F(x + Δx) ∝ +Δx
repelling force		 ∑ F(x + Δx) = 0
no force
dU/dx = 0
local extrema		 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

The Physics Hypertextbook

  • No condition is permanent.