Rotational Equilibrium

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Rotational Equilibrium

Equilibrium of Extended Bodies

Discuss.

∑ τ = ∑ τ

  1. counter clockwise is positive
  2. clockwise is negative
Translational and rotational quantities compared
concepttranslationconnectionrotation
equilibrium
∑F = 0 ⇒ ∑ F+x = ∑ Fx
∑ F+y = ∑ Fy
∑ F+z = ∑ Fz
∑ τ = 0 ⇒ ∑ τ+x = ∑ τx
∑ τ+y = ∑ τy
∑ τ+z = ∑ τz

Center of Mass

The center of mass is computed from the mass distribution.

Discrete.

rcm = ∑ miri = ()
∑ mi

Continuous.

rcm = 1⌠⌠⌠
⌡⌡⌡
dm = ()
m

Continuous and uniform.

rcm = 1⌠⌠⌠
⌡⌡⌡
dV = ()
V

The center of gravity is computed from the weight distribution.

Discrete.

rcg = ∑ Wiri = ()
∑ Wi

Continuous.

rcg = 1⌠⌠⌠
⌡⌡⌡
dW = ()
W

Continuous and uniform.

rcg = 1⌠⌠⌠
⌡⌡⌡
dV = ()
V

stability

Stability of Equilibrium

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

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