The Physics
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Opus in profectus

# Rotational Statics

## Summary

• Statics is the branch of mechanics dealing with forces acting on objects that are not accelerating.
• Equilibrium is a general concept that can refer to states of…
• static equilibrium, which includes…
• translational equilibrium (the equilibrium of forces)
• rotational equilibrium (the equilibrium of torques)
• dynamic equilibrium, which includes but is not limited to…
• thermodynamic equilibrium (the equilibrium of internal energy transfer)
• An adjective before the noun is not needed when the type of equilibrium is known from context.
• This section of this book is about static equilibrium only.
• Dynamic equilibrium will be dealt with in other sections of this book.
• Static equilibrium (a.k.a. the equilibrium of extended bodies)
• The translational and angular acceleration of an extended body in static equilibrium are both zero.
• An object in static equilibrium will…
• remain at rest or continue moving with a constant translational velocity and
• remain at rest or continue rotating with a constant angular velocity.
• An extended body is in static equilibrium when the net force and net torque on it are zero.
 ∑F−x = ∑F+x ⎫⎪⎬⎪⎭ ⇒ ∑F = 0 ∑F−y = ∑F+y ∑F−z = ∑F+z
 ∑τ−x = ∑τ+x ⎫⎪⎬⎪⎭ ⇒ ∑τ = 0 ∑τ−y = ∑τ+y ∑τ−z = ∑τ+z
• An extended body is in static equilibrium when the forces and torques on it are balanced.
 ∑F−x = ∑F+x ⎫⎪⎬⎪⎭ ⇒ ∑F = 0 ∑F−y = ∑F+y ∑F−z = ∑F+z
 ∑τ−x = ∑τ+x ⎫⎪⎬⎪⎭ ⇒ ∑τ = 0 ∑τ−y = ∑τ+y ∑τ−z = ∑τ+z
• Center of mass (center of gravity)
• The center of mass of an object…
• is the effective location of an extended body for for the purposes of some calculations
• is the point about which an extended body will always rotate when free of external torques
• is computed from the mass distribution
• may lie outside of an extended body
• is represented by the symbol rcm or rcom
• The center of gravity of an object…
• is the effective location for the force of gravity acting on an extended body
• is the pivot about which an extended body will be in stable rotational equilibrium in a gravitational field
• is computed from the weight distribution
• is in the same location as the center of mass when the gravitational field is uniform
• is represented by the symbol rcg or rcog
• Equations
• For a discrete collection of objects  rcm = ∑miri = 1 ⎛⎜⎝ ∑mixi î + ∑miyi ĵ + ∑mizi k̂ ⎞⎟⎠ ∑mi m
Where…  rcm = the location of the center of mass for the set of objects mi = the mass of the ith object in the set ri = the position vector of the ith object in the set m = the total mass of the set xi, yi, zi = the coordinates of the ith object in the set î, ĵ, k̂ = the unit vectors of the coordinate system used
• For a single object with…
 rcm = 1 ⌠⌠⌠⌡⌡⌡ r dV V
 rcm = 1 ⌠⌠⌠⌡⌡⌡ ρr dV m
Where…  rcm = the location of the center of mass of the object m = the total mass of the object V = the total volume of the object r = the position vector of any infinitesimal piece of the object dV = an infinitesimal volume element in the object ρ = a density function
• Stability of equilibrium for extended bodies
• The equilibrium of an extened body in a gravitational field is said to be…
• stable if…
• any small disturbance results in a net torque that tries to restore its equilibrium orientation
• any small disturbance would raise its center of mass
• its center of gravity is below the pivot point
• unstable if…
• any small disturbance results in a net torque that drives it away from its equilibrium orientation
• any small disturbance would lower its center of mass
• its center of gravity is above the pivot point
• neutral if…
• any small disturbance does not result in a net force
• any small disturbance would neither raise nor lower its center of mass
• its center of gravity is at the pivot point