The Physics
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Opus in profectus

# 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.
• No net force in any direction.  ∑F = 0 ⇒ ⎧⎨⎩ ∑F+x = ∑F−x ∑F+y = ∑F−y ∑F+z = ∑F−z
• No net torque about any axis.  ∑ τ = 0 ⇒ ⎧⎨⎩ ∑ τ+x = ∑ τ−x ∑ τ+y = ∑ τ−y ∑ τ+z = ∑ τ−z
• The translational and angular 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.  Discrete Collection of Objects rcm = ∑ miri = (x̅, y̅, z̅) ∑ mi Continuous Distribution of Matter rcm = 1 ⌠⌠⌠⌡⌡⌡ r dm m rcm = 1 ⌠⌠⌠⌡⌡⌡ r dV V
• The center of gravity is computed from the weight distribution.  Discrete Collection of Objects rcg = ∑ Wiri = (x̅, y̅, z̅) ∑ Wi Continuous Distribution of Matter rcg = 1 ⌠⌠⌠⌡⌡⌡ r dW W rcg = 1 ⌠⌠⌠⌡⌡⌡ r dV V
• 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.