Rotational Equilibrium

Summary

Statics is the branch of mechanics dealing with forces acting on an object at rest.
Static Equilibrium (Extended Bodies)
- An extended body is in static equilibrium when the net force and net torque on it are zero.


∑ F = 0 ⇒		⎧ ⎪ ⎨ ⎪ ⎩		∑ F_+x = ∑ F_−x		∑ τ = 0 ⇒		⎧ ⎪ ⎨ ⎪ ⎩		∑ τ_+x = ∑ τ_−x
				∑ F_+y = ∑ F_−y						∑ τ_+y = ∑ τ_−y
				∑ F_+z = ∑ F_−z						∑ τ_+z = ∑ τ_−z
No Net Force in Any Direction						No Net Torque About Any Axis

The translational and rotational acceleration of an object in equilibrium equal zero.
- It will remain at rest or continue moving with a constant translational velocity.
- It will remain at rest or continue rotating with a constant angular velocity.

Center of Mass (Center of Gravity)
- is the effective location of the force of gravity acting on an extended body
- is the pivot about which an extended body will always be in equilibrium in a gravitational field
- is the point about which an extended body will always rotate when free of external torques
- may lie outside of an extended body

The center of mass is computed from the mass distribution.			The center of gravity is computed from the weight distribution.

r_cm =	∑ m_ir_i	= (x, y, z)	r_cg =	∑ W_ir_i	= (x, y, z)
	∑ m_i			∑ W_i
Discrete Collection of Objects

r_cm =

1

⌠⌠⌠
⌡⌡⌡

r dm =

1

⌠⌠⌠
⌡⌡⌡

r dV

r_cg =

1

⌠⌠⌠
⌡⌡⌡

r dW =

1

⌠⌠⌠
⌡⌡⌡

r dV

m

V

W

V

Continuous Distribution of Matter

Stability of Equilibrium
- The equilibrium of an object in a gravitational field is said to be …
  - stable if any small disturbance would raise its center of mass.
  - unstable if any small disturbance would lower its center of mass.
  - neutral if any small disturbance would neither raise nor lower its center of mass.

The Physics Hypertextbook

No condition is permanent.

© 1998–2011 Glenn Elert