Rotational Equilibrium

Discussion

Equilibrium of Extended Bodies

Discuss.

∑ τ = ∑ τ

  1. counter clockwise is positive
  2. clockwise is negative
concept translation connection rotation
equilibrium
∑ F = 0 ⇒ 

∑ F+x = ∑ Fx
∑ F+y = ∑ Fy
∑ F+z = ∑ Fz
   
∑ τ = 0 ⇒ 

∑ τ+x = ∑ τx
∑ τ+y = ∑ τy
∑ τ+z = ∑ τz
Translational and rotational quantities compared

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

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
Stability of Equilibrium